`What is a Thing?': Topos Theory in the Foundations of Physics
Andreas Doering, Chris Isham
Introduction
Many people who work in quantum gravity would agree that a deep change in our understanding of foundational issues will occur at some point along the path. However, opinions differ greatly on whether a radical revision is necessary at the very beginning of the process, or if it will emerge ‘along the way’ from an existing, or future, research programme that is formulated using the current paradigms. For example, many (albeit not all) of the current generation of string theorists seem inclined to this view, as do a, perhaps smaller, fraction of those who work in loop quantum gravity.
In this article we take the iconoclastic view that a radical step is needed at the very outset. However, for anyone in this camp the problem is always knowing where to start. It is easy to talk about a ‘radical revision of current paradigms’—the phrase slips lightly off the tongue—but converting this pious hope into a concrete theoretical structure is a problem of the highest order.
For us, the starting point is quantum theory itself. More precisely, we believe that this theory needs to be radically revised, or even completely replaced, before a satisfactory theory of quantum gravity can be obtained.
In this context, a striking feature of the various current programmes for quantising gravity—including superstring theory and loop quantum gravity—is that, notwithstanding their disparate views on the nature of space and time, they almost all use more-or-less standard quantum theory. Although understandable from a pragmatic viewpoint (since all we have is more-or-less standard quantum theory) this situation is nevertheless questionable when viewed from a wider perspective.
For us, one of the most important issues is the use in the standard quantum formalism of critical mathematical ingredients that are taken for granted and yet which, we claim, implicitly assume certain properties of space and/or time. Such an a priori imposition of spatio-temporal concepts would be a major category The philosophy of Kant runs strongly in our veins. error if they turn out to be fundamentally incompatible with what is needed for a theory of quantum gravity.
A prime example is the use of the continuum When used in this rather colloquial way, the word ‘continuum’ suggests primarily the cardinality of the sets concerned, and, secondly, the topology that is conventionally placed on these sets. by which, in this context, is meant the real and/or complex numbers. These are a central ingredient in all the various mathematical frameworks in which quantum theory is commonly discussed. For example, this is clearly so with the use of (i) Hilbert spaces or -algebras; (ii) geometric quantisation; (iii) probability functions on a non-distributive quantum logic; (iv) deformation quantisation; and (v) formal (i.e., mathematically ill-defined) path integrals and the like. The a priori imposition of such continuum concepts could be radically incompatible with a quantum-gravity formalism in which, say, space-time is fundamentally discrete: as, for example, in the causal-set programme.
As we shall argue later, this issue is closely connected with the question of what is meant by the ‘value’ of a physical quantity. In so far as the concept is meaningful at all at the Planck scale, why should the value be a real number defined mathematically in the usual way?
Another significant reason for aspiring to change the quantum formalism is the peristalithic problem of deciding how a ‘quantum theory of cosmology’ could be interpreted if one was lucky enough to find one. Most people who worry about foundational issues in quantum gravity would probably place the quantum-cosmology/closed-system problem at, or near, the top of their list of reasons for re-envisioning quantum theory. However, although we are deeply interested in such conceptual issues, the primary motivation for our research programme is not to find a new interpretation of quantum theory. Rather, our main goal is to find a novel structural framework within which new types of theories of physics can be constructed.
However, having said that, in the context of quantum cosmology it is certainly true that the lack of any external ‘observer’ of the universe ‘as a whole’ renders inappropriate the standard Copenhagen interpretation with its instrumentalist use of counterfactual statements about what would happen if a certain measurement is performed. Indeed, the Copenhagen interpretation is inapplicable for any The existence of the long-range, and all penetrating, gravitational force means that, at a fundamental level, there is only one truly closed system, and that is the universe itself. system that is truly ‘closed’ (or ‘self-contained’) and for which, therefore, there is no ‘external’ domain in which an observer can lurk. This problem has motivated much research over the years and continues to be of wide interest.
The philosophical questions that arise are profound, and look back to the birth of Western philosophy in ancient Greece, almost three thousand years ago. Of course, arguably, the longevity of these issues suggests that these questions are ill-posed in the first place, in which case the whole enterprise is a complete waste of time! This is probably the view of most, if not all, of our colleagues at Imperial College; but we beg to differ Of course, it is also possible that our colleagues are right..
When considering a closed system, the inadequacy of the conventional instrumentalist interpretation of quantum theory encourages the search for an interpretation that is more ‘realist’ in some way. For over eighty years, this has been a recurring challenge for those concerned with the conceptual foundations of modern physics. In rising to this challenge we join our Greek ancestors in confronting once more the fundamental question: “What is a thing?” is the title of one of the more comprehensible of Heidegger’s works . By this, we mean comprehensible to the authors of the present article. We cannot speak for our colleagues across the channel: from some of them we may need to distance ourselves.
Of course, as written, the question is itself questionable. For many philosophers, including Kant, would assert that the correct question is not “What is a thing?” but rather “What is a thing as it appears to us?” However, notwithstanding Kant’s strictures, we seek the thing-in-itself, and, therefore, we persevere with Heidegger’s form of the question.
Nevertheless, having said that, we can hardly ignore the last three thousand years of philosophy. In particular, we must defend ourselves against the charge of being ‘naive realists’. If we were professional philosophers this would be a terrible insult. :-) At this point it become clear that theoretical physicists have a big advantage over professional philosophers. For we are permitted/required to study such issues in the context of specific mathematical frameworks for addressing the physical world; and one of the great fascinations of this process is the way in which various philosophical positions are implicit in the ensuing structures. For example, the exact meaning of ‘realist’ is infinitely debatable but, when used by a classical physicist, it invariably means the following:
The idea of ‘a property of the system’ (for example, ‘the value of a physical quantity at a certain time’) is meaningful, and mathematically representable in the theory.
Propositions about the system (typically asserting that the system has this or that property) are handled using Boolean logic. This requirement is compelling in so far as we humans are inclined to think in a Boolean way.
There is a space of ‘microstates’ such that specifying a microstate In simple non-relativistic systems, the state is specified at any given moment of time. Relativistic systems (particularly quantum gravity!) require a more sophisticated understanding of ‘state’, but the general idea is the same. leads to unequivocal truth values for all propositions about the system: i.e., a state We are a little slack in our use of language here and in what follows by frequently referring to a microstate as just a ‘state’. The distinction only becomes important if one wants to introduce things like mixed states (in quantum theory), or macrostates (in classical physics) all of which are often just known as ‘states’. Then one must talk about microstates (pure states) to distinguish them from the other type of state. encodes “the way things are”. This is a natural way of ensuring that the first two conditions above are satisfied.
The standard interpretation of classical physics satisfies these requirements and provides the paradigmatic example of a realist philosophy in science. Heidegger’s answer to his own question adopts a similar position :
“A thing is always something that has such and such properties, always something that is constituted in such and such a way. This something is the bearer of the properties; the something, as it were, that underlies the qualities.”
In quantum theory, the situation is very different. There, the existence of any such realist interpretation is foiled by the famous Kochen-Specker theorem . This asserts that it is impossible to assign values to all physical quantities at once if this assignment is to satisfy the consistency condition that the value of a function of a physical quantity is that function of the value. For example, the value of ‘energy squared’ is the square of the value of energy.
Thus, from a conceptual perspective, the challenge is to find a quantum formalism that is ‘realist enough’ to provide an acceptable alternative to the Copenhagen interpretation, with its instrumentally-construed intrinsic probabilities, whilst taking on board the implications of the Kochen-Specker theorem.
So, in toto what we seek is a formalism that is (i) free of prima facie prejudices about the nature of the values of physical quantities—in particular, there should be no fundamental use of the real or complex numbers; and (ii) ‘realist’, in at least the minimal sense that propositions are meaningful, and are assigned ‘truth values’, not just instrumentalist probabilities of what would happen if appropriate measurements are made.
However, finding such a formalism is not easy: it is notoriously difficult to modify the mathematical framework of quantum theory without destroying the entire edifice. In particular, the Hilbert space structure is very rigid and cannot easily be changed; and the formal path-integral techniques do not fare much better.
In the spirit of general abstraction, one might one wonder if this formalism can be generalised to a structure in which is represented by an arrow where and are objects in some category, , other than the category of sets, ? In such a theory, one would seek to represent propositions about the ‘values’ (whatever that might mean) of physical quantities with sub-objects of , just as in classical physics propositions are represented by subsets of the state space (see Section 2.2 for more detail of this).
Our central conceptual idea is that such a categorial structure constitutes a generalisation of the concept of ‘realism’ in which the ‘values’ of a physical quantity are coded in the arrow .
Clearly the propositions will play a key role in any such theory, and, presumably, the minimum required is that the associated sub-objects of form some sort of ‘logic’, just as the subsets of form a Boolean algebra.
This rules out most categories since, generically, the sub-objects of an object do not have any logical structure. However, if the category is a ‘topos’ then the sub-objects of any object do have this property, and hence the current research programme.
Our suggestion, therefore, is to try to construct physical theories that are formulated in a topos other than . This topos will depend on both the theory-type and the system. More precisely, if a theory-type (such as classical physics, or quantum physics) is applicable to a certain class of systems, then, for each system in this class, there is a topos in which the theory is to be formulated. For some theory-types the topos is system-independent: for example, classical physics always uses the topos of sets. For other theory-types, the topos varies from system to system: as we shall see, this is the case in quantum theory.
In somewhat more detail, any particular example of our suggested scheme will have the following ingredients:
Propositions about a system are represented by sub-objects of the state-object . These sub-objects form a Heyting algebra (as indeed do the sub-objects of any object in a topos): a distributive lattice that differs from a Boolean algebra only in that the law of excluded middle need not hold, i.e., . A Boolean algebra is a Heyting algebra with strict equality: .
Generally speaking (and unlike in set theory), an object in a topos may not be determined by its ‘points’. In particular, this may be so for the state-object, in which case the concept of a microstate is not so useful. In quantum theory, the state-object has no points/microstates at all. As we shall see, this statement is equivalent to the Kochen-Specker theorem. Nevertheless, truth values can be assigned to propositions with the aid of a ‘truth object’ (or ‘pseudo-state’). These truth values lie in another Heyting algebra.
Of course, it is not instantly obvious that quantum theory can be written in this way. However, as we shall see, there is a topos reformulation of quantum theory, and this has two immediate implications. The first is that we acquire a new type of ‘realist’ interpretation of standard quantum theory. The second is that this new approach suggests ways of generalising quantum theory that make no fundamental reference to Hilbert spaces, path integrals, etc. In particular, there is no prima facie reason for introducing standard continuum quantities. As emphasised above, this is one of our main motivations for developing the topos approach. We shall say more about this later.
From a conceptual perspective, a central feature of our scheme is the ‘neo-realist’ structure reflected mathematically in the three statements above. This neo-realism is the conceptual fruit of the fact that, from a categorial perspective, a physical theory expressed in a topos ‘looks’ like classical physics expressed in the topos of sets.
Evidently the suggested mathematical structures could be used in two different ways. The first is that of the ‘conventional’ theoretical physicist with little interest in conceptual matters. For him/her, what we and our colleagues are developing is a new tool-kit with which to construct novel types of theoretical model. Whether or not Nature has chosen such models remains to be seen, but, at the very least, the use of topoi certainly suggests new techniques.
For those physicists who are interested in conceptual issues, the topos framework gives a radically new way of thinking about the world. The neo-realism inherent in the formalism is described mathematically using the internal language that is associated with any topos. This describes how things look from ‘within’ the topos: something that should be particularly useful in the context of quantum cosmology In this context see the work of Markopoulou who considers a topos description of the universe as seen by different observers who live inside it ..
On the other hand, the pragmatic theoretician with no interest in conceptual matters can use the ‘external’ description of the topos in which the category of sets provides a metalanguage with which to formulate the theory. From a mathematical perspective, the interplay between the internal and external languages of a topos is one of the fascinations of the subject. However, much remains to be said about the significance of this interaction for real theories of physics.
This present article is partly an amalgam of a series of four papers that we placed on the ArXiv server These are due to published in Journal of Mathematical Physics in the Spring of 2008. in March, 2007 . However, we have added a fair amount of new material, and also made a few minor corrections (mainly typos). Some of the more technical theorems have been placed in the Appendix with the hope that this makes the article a little easier to read. We have also added some remarks about developments made by researchers other than ourselves since the ArXiv preprints were written. Of particular importance to our general programme is the work of Heunen and Spitters which adds some powerful ingredients to the topoi-in-physics toolkit. Finally, we have included some background material from the earlier papers that formed the starting point for the current research programme .
We must emphasise that this is not a review article about the general application of topos theory to physics; this would have made the article far too long. For example, there has been a fair amount of study of the use of synthetic differential geometry in physics. The reader can find references to much of this on the, so-called, ‘Siberian toposes’ web site This is http://users.univer.omsk.su/˜topoi/. See also Cecilia Flori’s website that deals more generally with topos theory and physics: http://topos-physics.org/. There is also the work by Mallios and collaborators on ‘Abstract Differential Geometry’ . Of course, as always these days, Google will speedily reveal all that we have omitted.
But even less is this paper a review of the use of category theory in general in physics. For there any many important topics that we do not mention at all. For example, Baez’s advocation of -categories ; ‘categorial quantum theory’ ; Takeuti’s theory Takeuti’s work is not exactly about category theory applied to quantum theory: it is more about the use of formal logic, but the spirit is similar. For a recent paper in this genre see . of ‘quantum sets’ ; and Crane’s work on categorial models of space-time .
Finally, a word about the style in which this article is written. We spent much time pondering on this, as we did before writing the four ArXiv preprints. The intended audience is our colleagues who work in theoretical physics, especially those whose interests included foundational issues in quantum gravity and quantum theory. However, topos theory is not an easy branch of mathematics, and this poses the dilemma of how much background mathematics should be assumed of the reader, and how much should be explained as we go along. The references that we have found most helpful in our research are . We have approached this problem by including a short mathematical appendix on topos theory. However, reasons of space precluded a thorough treatment, and we hope that, fairly soon, someone will write an introductory review of topos theory in a style that is accessible to a typical theoretical-physicist reader.
This article is structured in the following way. We begin with a discussion of some of the conceptual background, in particular the role of the real numbers in conventional theoretical physics. Then in Section 3 we introduce the idea of attaching a propositional language, , to each physical system . The intent is that each theory of corresponds to a particular representation of . In particular, we show how classical physics satisfies this requirement in a very natural way.
Propositional languages have limited scope (they lack the quantifiers ‘’ and ‘’), and in Section 4 we propose the use of a higher-order language . Languages of this type are a central feature of topos theory and it is natural to consider the idea of representing in different topoi. Classical physics always takes place in the topos, , of sets but our expectation is that other areas of physics will use a different topos.
This expectation is confirmed in Section 5 where we discuss in detail the representation of for a quantum system (the representation of is discussed in Section 8). The central idea is to represent propositions as sub-objects of the ‘spectral presheaf’ which belongs to the topos, , of presheaves (set-valued, contravariant functors) on the category, , of abelian sub-algebras of the algebra of all bounded operators on . This representation employs the idea of ‘daseinisation’ in which any given projection operator is represented at each context/stage-of-truth in by the ‘closest’ projector to it in . There are two variants of this: (i) ‘outer’ daseinisation, in which is approached from above (in the lattice of projectors in ); and (ii) ‘lower’ daseinisation, in which is approached from below.
The next key move is to discuss the ‘truth values’ of propositions in a quantum theory. This requires the introduction of some analogue of the microstates of classical physics. We say ‘analogue’ because the spectral presheaf —which is the quantum topos equivalent of a classical state space—has no global elements, and hence there are no microstates at all: this is equivalent to the Kochen-Specker theorem. The critical idea is that of a ‘truth object’, or ‘pseudo-state’ which, as we show in Section 6, is the closest one can get in quantum theory to a microstate.
In Section 10 we discuss the way in which unitary operators act on the quantum topos objects. Then, in Sections 11, 12 and 13 we discuss the problem of handling ‘all’ possible systems in a single coherent scheme. This involves introducing a category of systems which, it transpires, has a natural monoidal structure. We show in detail how this scheme works in the case of classical and quantum theory.
Finally, in Section 14 we discuss/speculate on some properties of the state object, quantity-value object, and truth objects that might be present in any topos representation of a physical system.
To facilitate reading this long article, some of the more technical material has been put in Appendix 1. In Appendix 2 there is a short introduction to some of the relevant parts of topos theory.
The Conceptual Background of our Scheme
As mentioned in the Introduction, one of the main goals of our work is to find new tools with which to develop theories that are significant extensions of, or developments from, quantum theory but without being tied a priori to the use of the standard real or complex numbers.
In this context we note that real numbers arise in theories of physics in three different (but related) ways: (i) as the values of physical quantities; (ii) as the values of probabilities; and (iii) as a fundamental ingredient in models of space and time (especially in those based on differential geometry). All three are of direct concern vis-a-vis our worries about making unjustified, a priori assumptions in quantum theory. We shall now examine them in detail.
One reason for assuming physical quantities are real-valued is undoubtedly grounded in the remark that, traditionally (i.e., in the pre-digital age), they are measured with rulers and pointers, or they are defined operationally in terms of such measurements. However, rulers and pointers are taken to be classical objects that exist in the physical space of classical physics, and this space is modelled using the reals. In this sense there is a direct link between the space in which physical quantities take their values (what we call the ‘quantity-value space’) and the nature of physical space or space-time .
If conceded, this claim means the assumption that physical quantities are real-valued is problematic in any theory in which space, or space-time, is not modelled by a smooth manifold. Admittedly, if the theory employs a background space, or space-time—and if this background is a manifold—then the use of real-valued physical quantities is justified in so far as their value-space can be related to this background. Such a stance is particularly appropriate in situations where the background plays a central role in giving meaning to concepts like ‘observers’ and ‘measuring devices’, and thereby provides a basis for an instrumentalist interpretation of the theory.
But even here caution is needed since many theoretical physicists have claimed that the notion of a ‘space-time point in a manifold’ is intrinsically flawed. One argument (due to Penrose) is based on the observation that any attempt to localise a ‘thing’ is bound to fail beyond a certain point because of the quantum production of pairs of particles from the energy/momentum uncertainty caused by the spatial localisation. Another argument concerns the artificiality The integers, and associated rationals, have a ‘natural’ interpretation from a physical perspective since we can all count. On the other hand, the Cauchy-sequence and/or the Dedekind-cut definitions of the reals are distinctly un-intuitive from a physical perspective. of the use of real numbers as coordinates with which to identify a space-time point. There is also Einstein’s famous ‘hole argument’ in general relativity which asserts that the notion of a space-time point (in a manifold) has no physical meaning in a theory that is invariant under the group of space-time diffeomorphisms.
Another cautionary caveat concerning the invocation of a background is that this background structure may arise only in some ‘sector’ of the theory; or it may exist only in some limiting, or approximate, sense. The associated instrumentalist interpretation would then be similarly limited in scope. For this reason, if no other, a ‘realist’ interpretation is more attractive than an instrumentalist one.
In fact, in such circumstances, the phrase ‘realist interpretation’ does not really do justice to the situation since it tends to imply that there are other interpretations of the theory, particularly instrumentalism, with which the realist one can contend on a more-or-less equal footing. But, as we just argued, the instrumentalist interpretation may be severely limited as compared to the realist one. To flag this point, we will sometimes refer to a ‘realist formalism’, rather than a ‘realist interpretation’. Of course, such discussions are unnecessary in classical physics since, there, if knowledge of the value of a physical quantity is gained by making a (ideal) measurement, the reason why we obtain the result that we do, is because the quantity possessed that value immediately before the measurement was made. In other words, “epistemology models ontology”.
1.2 Why Are Probabilities Required to Lie in the Interval [0,1][0,1]?
The motivation for using the subset $NN_{i}/NN_{i}N_{i}/N01N\rightarrow\infty$ are obtained.
The relative-frequency interpretation of probability is natural in instrumentalist theories of physics, but it is not meaningful if there is no classical spatio-temporal background in which the necessary measurements could be made; or, if there is a background, it is one to which the relative-frequency interpretation cannot be adapted.
In the absence of a relativity-frequency interpretation, the concept of ‘probability’ must be understood in a different way. In the physical sciences, one of the most discussed approaches involves the concept of ‘potentiality’, or ‘latency’, as favoured by Heisenberg , Margenau , and Popper (and, for good measure, Aristotle). In this case there is no compelling reason why the probability-value space should necessarily be a subset of the real numbers. The minimal requirement is that this value-space is an ordered set, so that one proposition can be said to be more or less probable than another. However, there is no prima facie reason why this set should be totally ordered: i.e., there may be pairs of propositions whose potentialities cannot be compared—something that seems eminently plausible in the context of non-commensurable quantities in quantum theory.
By invoking the idea of ‘potentiality’, it becomes feasible to imagine a quantum-gravity theory with no spatio-temporal background but where probability is still a fundamental concept. However, it could also be that the concept of probability plays no fundamental role in such circumstances, and can be given a meaning only in the context of a sector, or limit, of the theory where a background does exist. This background could then support a limited instrumentalist interpretation which would include a (limited) relative-frequency understanding of probability.
In fact, most modern approaches to quantum gravity aspire to a formalism that is background independent . So, if a background space does arise, it will be in one of the restricted senses mentioned above. Indeed, it is often asserted that a proper theory of quantum gravity will not involve any direct spatio-temporal concepts, and that what we commonly call ‘space’ and ‘time’ will ‘emerge’ from the formalism only in some appropriate limit . In this case, any instrumentalist interpretation could only ‘emerge’ in the same limit, as would the associated relative-frequency interpretation of probability.
In a theory of this type, there will be no prima facie link between the values of physical quantities and the nature of space or space-time, although, of course, this cannot be totally ruled out. In any event, part of the fundamental specification of the theory will involve deciding what the ‘quantity-value space’ should be.
These considerations suggest that quantum theory must be radically changed if one wishes to accommodate situations where there is no background space/space-time, manifold within which an instrumentalist interpretation can be formulated. In such a situation, some sort of ‘realist’ formalism is essential.
These reflections also suggest that the quantity-value space employed in an instrumentalist realisation of a theory—or a ‘sector’, or ‘limit’, of the theory—need not be the same as the quantity-value space in a neo-realist formulation. At first sight this may seem strange but, as is shown in Section 8, this is precisely what happens in the topos reformulation of standard quantum theory.
2 The Genesis of Topos Ideas in Physics
Even setting aside the more exotic considerations of quantum gravity, one can still query the use of real numbers to model space and/or time. One might argue that (i) the use of (triples of) real numbers to model space is based on empirically-based reflections about the nature of ‘distances’ between objects; and (ii) the use of real numbers to model time reflects our experience that ‘instants of time’ appear to be totally ordered, and that intervals of time are always divisible These remarks are expressed in the context of the Newtonian view of space and time, but it is easy enough to generalise them to special relativity..
However, what does it really mean to say that two particles are separated by a distance of, for example, cms? From an empirical perspective, it would be impossible to make a measurement that could unequivocally reveal precisely that value from among the continuum of real numbers that lie around it. There will always be experimental errors of some sort: if nothing else, there are thermodynamical fluctuations in the measuring device; and, ultimately, uncertainties arising from quantum ‘fluctuations’. Similar remarks apply to attempts to measure time.
Thus, from an operational perspective, the use of real numbers to label ‘points’ in space and/or time is a theoretical abstraction that can never be realised in practice. But if the notion of a space/time/space-time ‘point’ in a continuum, is an abstraction, why do we use it? Of course it works well in theories used in normal physics, but at a fundamental level it must be seen as questionable.
These operational remarks say nothing about the structure of space (or time) ‘in itself’, but, even assuming that this concept makes sense, which is debatable, the use of real numbers is still a metaphysical assumption with no fundamental justification.
Traditionally, we teach our students that measurements of physical quantities that are represented theoretically by real numbers, give results that fall into ‘bins’, construed as being subsets of the real line. This suggests that, from an operational perspective, it would be more appropriate to base mathematical models of space or time on a theory of ‘regions’, rather than the real numbers themselves.
From a physical perspective, the use of open subsets as models of regions is attractive as it leaves a certain, arguably desirable, ‘fuzziness’ at the edges, which is absent for closed sets. Thus, following this path, we would axiomatise that a mathematical model of space or time (or space-time) involves an algebra of entities called ‘regions’, and with operations that are the analogue of unions and intersections for subsets of a set. This algebra would allow arbitrary ‘unions’ and finite ‘intersections’, and would distribute If the distributive law is dropped we could move towards the quantum-set ideas of ; or, perhaps, the ideas of non-commutative geometry instigated by Alain Connes . over these operations. In effect, we are axiomatising that an appropriate mathematical model of space-time is an object in the category of locales.
However, a locale is the same thing as a complete Heyting algebra (for the definition see below), and, as we shall, Heyting algebras are inexorably linked with topos theory.
2.2 Another Possible Role for Heyting Algebras
The use of a Heyting algebra to model space/time/space-time is an attractive possibility, and was the origin of the interest in topos theory of one of us (CJI) some years ago. However, there is another motivation which is based more on logic, and the desire to construct a ‘neo-realist’ interpretation of quantum theory.
It is easy to see how the logical calculus of propositions arises in this picture. For let and be propositions, represented by the subsets and respectively, and consider the proposition “ and ”. This is true if, and only if, both and are true, and hence the subset of states that represents this logical conjunction consists of those states that lie in both and —i.e., the set-theoretic intersection . Thus “ and ” is represented by . Similarly, the proposition “ or ” is true if either or (or both) are true, and hence this logical disjunction is represented by those states that lie in plus those states that lie in —i.e., the set-theoretic union . Finally, the logical negation “not ” is represented by all those points in that do not lie in —i.e., the set-theoretic complement .
In this way, a fundamental relation is established between the logical calculus of propositions about a physical system, and the Boolean algebra of subsets of the state space. Thus the mathematical structure of classical physics is such that, of necessity, it reflects a ‘realist’ philosophy, in the sense in which we are using the word.
One way to escape from the tyranny of Boolean algebras and classical realism is via topos theory. Broadly speaking, a topos is a category that behaves very much like the category of sets; in particular, the collection of sub-objects of an object forms a Heyting algebra, just as the collection of subsets of a set form a Boolean algebra. Our intention, therefore, is to explore the possibility of associating physical propositions with sub-objects of some object (the analogue of a classical state space) in some topos.
A Heyting algebra, , is a distributive lattice with a zero element, , and a unit element, , and with the property that to each pair there is an implication , characterized by
The negation is defined as and has the property that the law of excluded middle need not hold, i.e., there may exist , such that or, equivalently, there may exist such that . This is the characteristic property of an intuitionistic logic. Here, is nothing but the category-theoretical exponential and is the product . The definition uses the adjunction between the exponential and the product, . A slightly easier, albeit ‘less categorical’ definition is: a Heyting algebra, , is a distributive lattice such that for any two elements , the set has a maximal element, denoted by . A Boolean algebra is the special case of a Heyting algebra in which there is the strict equality: i.e., for all . It is known from Stone’s theorem that each Boolean algebra is isomorphic to an algebra of (clopen, i.e., closed and open) subsets of a suitable (topological) space.
The elements of a Heyting algebra can be manipulated in a very similar way to those in a Boolean algebra. One of our claims is that, as far as theories of physics are concerned, Heyting logic is a viable The main difference between theorems proved using Heyting logic and those using Boolean logic is that proofs by contradiction cannot be used in the former. In particular, this means that one cannot prove that something exists by arguing that the assumption that it does not leads to contradiction; instead it is necessary to provide a constructive proof of the existence of the entity concerned. Arguably, this does not place any major restriction on building theories of physics. Indeed, over the years, various physicists (for example, Bryce DeWitt) have argued that constructive proofs should always be used in physics. alternative to Boolean logic.
To give some idea of the difference between a Boolean algebra and a Heyting algebra, we note that the paradigmatic example of the former is the collection of all measurable subsets of a measure space . Here, if represents a proposition, the logical negation, , is just the set-theoretic complement .
On the other hand, the paradigmatic example of a Heyting algebra is the collection of all open sets in a topological space . Here, if is open, the logical negation is defined to be the interior of the set-theoretical complement . Therefore, the difference between in the topological space , and in the measurable space generated by the topology of , is just the ‘thin’ boundary of the closed set .
2.3 Our Main Contention about Topos Theory and Physics
The conceptual interpretation of this formalism is ‘neo-realist’ in the following sense:
In what follows, and are called the ‘state object’, and the ‘quantity-value object’, respectively.
Propositions about the system are represented by sub-objects of . These sub-objects form a Heyting algebra.
Once the topos analogue of a state (a ‘truth object’) has been specified, these propositions are assigned truth values in the Heyting logic associated with the global elements of the sub-object classifier, , in the topos .
Thus a theory expressed in this way looks very much like classical physics except that whereas classical physics always employs the topos of sets, other theories—including quantum theory and, we conjecture, quantum gravity—use a different topos.
One deep result in topos theory is that there is an internal language associated with each topos. In fact, not only does each topos generate an internal language, but, conversely, a language satisfying appropriate conditions generates a topos. Topoi constructed in this way are called ‘linguistic topoi’, and every topos can be regarded as a linguistic topos. In many respects, this is one of the profoundest ways of understanding what a topos really ‘is’. This aspect of topos theory is discussed at length in the books by Bell , and Lambek and Scott .
These results are exploited in Section 4 where we introduce the idea that, for any applicable theory-type, each physical system is associated with a ‘local’ language, . The application of the theory-type to is then involves finding a representation of in an appropriate topos; this is equivalent to finding a ‘translation’ of into the internal language of that topos.
Closely related to the existence of this linguistic structure is the striking fact that a topos can be used as a foundation for mathematics itself, just as set theory is used in the foundations of ‘normal’ (or ‘classical’) mathematics. In this context, the key remark is that the internal language of a topos has a form that is similar in many ways to the formal language on which normal set theory is based. It is this internal, topos language that is used to interpret the theory in a ‘neo-realist’ way.
The main difference with classical logic is that the logic of the topos language does not satisfy the principle of excluded middle, and hence proofs by contradiction are not permitted. This has many intriguing consequences. For example, there are topoi in which there exist genuine infinitesimals that can be used to construct a rival to normal calculus. The possibility of such quantities stems from the fact that the normal proof that they do not exist is a proof by contradiction.
Thus each topos carries its own world of mathematics: a world which, generally speaking, is not the same as that of classical mathematics.
Consequently, by postulating that, for a given theory-type, each physical system carries its own topos, we are also saying that to each physical system plus theory-type there is associated a framework for mathematics itself! Thus classical physics uses classical mathematics; and quantum theory uses ‘quantum mathematics’—the mathematics formulated in the topoi of quantum theory. To this we might add the conjecture: “Quantum gravity uses ‘quantum gravity’ mathematics”!
Propositional Languages and Theories of Physics
Attempts to construct a naïve realist interpretation of quantum theory founder on the Kochen-Specker theorem. However, if, despite this theorem, some degree of realism is still sought, there are not that many options.
One approach is to focus on a particular, maximal commuting subset of physical quantities and declare by fiat that these are the ones that ‘have’ values; essentially, this is what is done in ‘modal’ interpretations of quantum theory. However, this leaves open the question of why Nature should select this particular set, and the reasons proposed vary greatly from one scheme to another.
In our work, we take a completely different approach and try to formulate a scheme which takes into account all these different choices for commuting sets of physical quantities; in particular, equal ontological status is ascribed to all of them. This scheme is grounded in the topos-theoretic approach that was first proposed in . This uses a technique whose first step is to construct a category, , the objects of which can be viewed as contexts in which the quantum theory can be displayed: in fact, they are just the commuting sub-algebras of operators in the theory. All this will be explained in more detail in Section 5.
In this earlier work, it was postulated that the logic for handling quantum propositions from this perspective is that associated with the topos of presheaves In quantum theory, the category is just a partially-ordered set, which simplifies many manipulations. (contravariant functors from to ), . The idea is that a single presheaf will encode quantum propositions from the perspective of all contexts at once. However, in the original papers, the crucial ‘daseinisation’ operation (see Section 5) was not known and, consequently, the discussion became rather convoluted in places. In addition, the generality and power of the underlying procedure was not fully appreciated by the authors.
For this reason, in the present article we return to the basic questions and reconsider them in the light of the overall topos structure that has now become clear.
From a conceptual perspective, the proposition can be read in two, very different, ways:
The (naïve) realist interpretation: “The physical quantity has a value, and that value lies in .”
The instrumentalist interpretation: “If a measurement is made of , the result will be found to lie in .”
The former is the familiar, ‘commonsense’ understanding of propositions in both classical physics and daily life. The latter underpins the Copenhagen interpretation of quantum theory. Of course, the instrumentalist interpretation can also be applied to classical physics, but it does not lead to anything new. For, in classical physics, what is measured is what is the case: “Epistemology models ontology”.
We will now study the role of propositions in physics more carefully, particularly in the context of ‘realist’ interpretations.
2 The Propositional Language 𝒫ℒ(S){\cal PL}(S)
We are going to construct a formal language, , with which to express propositions about a physical system, , and to make deductions concerning them. Our intention is to interpret these propositions in a ‘realist’ way: an endeavour whose mathematical underpinning lies in constructing a representation of in a Heyting algebra, , that is part of the mathematical framework involved in the application of a particular theory-type to .
We denote the set of all such strings by . Note that what has been here called a ‘physical quantity’ could better (but more clumsily) be termed the ‘name’ of the physical quantity. For example, when we talk about the ‘energy’ of a system, the word ‘energy’ is the same, and functions in the same way in the formal language, irrespective of the details of the actual Hamiltonian of the system.
The primitive propositions are used to define ‘sentences’. More precisely, a new set of symbols is added to the language, and then a sentence is defined inductively by the following rules (see Ch. 6 in ):
Each primitive proposition in is a sentence.
If is a sentence, then so is .
If and are sentences, then so are , , and .
The collection of all sentences, , is an elementary formal language that can be used to express and manipulate propositions about the system . Note that, at this stage, the symbols , , , and have no explicit meaning, although of course the implicit intention is that they should stand for ‘not’, ‘and’, ‘or’ and ‘implies’, respectively. This implicit meaning becomes explicit when a representation of is constructed as part of the application of a theory-type to (see below). Note also that is a propositional language only: it does not contain the quantifiers ‘’ or ‘’. To include them requires a higher-order language. We shall return to this in our discussion of the language .
The next step arises because is not only a vehicle for expressing propositions about the system : we also want to reason with it about the system. To achieve this, a series of axioms for a deductive logic must be added to . This could be either classical logic or intuitionistic logic, but we select the latter since it allows a larger class of representations/models, including representations in topoi in which the law of excluded middle fails.
The axioms for intuitionistic logic consist of a finite collection of sentences in (for example, ), plus a single rule of inference, modus ponens (the ‘rule of detachment’) which says that from and the sentence may be derived.
Thus, along with the axioms of intuitionistic logic and detachment, we might be tempted to add the following axioms:
These axioms are consistent with the intuitionistic logical structure of .
We shall see later the extent to which the axioms (3.2–3.3) are compatible with the topos representations of classical and quantum physics. However, the other obvious proposition to consider in this way—“It is not the case that belongs to ”—is clearly problematical.
However, applying ‘’ to both sides of (3.4) gives
2.2 Representations of 𝒫ℒ(S){\cal PL}(S).
To use a language ‘for real’ for some specific physical system one must first decide on the set of physical quantities that are to be used in describing . This language must then be represented in the concrete mathematical structure that arises when a theory-type (for example: classical physics, quantum physics, DI-physics,…) is applied to . Such a representation, , maps each primitive proposition, , in to an element, of some Heyting algebra (which could be Boolean), , whose specification is part of the theory of . For example, in classical mechanics, the propositions are represented in the Boolean algebra of all (Borel) subsets of the classical state space.
The representation of the primitive propositions can be extended recursively to all of with the aid of the following rules :
Note that, on the left hand side of (3.6–3.9), the symbols are elements of the language , whereas on the right hand side they denote the logical connectives in the Heyting algebra, , in which the representation takes place.
This extension of from to is consistent with the axioms for the intuitionistic, propositional logic of the language . More precisely, these axioms become tautologies: i.e., they are all represented by the maximum element, , in the Heyting algebra. By construction, the map is then a representation of in the Heyting algebra . A logician would say that is an -valuation, or -model, of the language .
Clearly, a major consideration in using the language is choosing the Heyting algebra in which the representation is to take place. A fundamental result in topos theory is that the set of all sub-objects of any object in a topos is a Heyting algebra, and these are the Heyting algebras with which we will be concerned.
Of course, beyond the language, , and its representation , lies the question of whether or not a proposition is ‘true’. This requires the concept of a ‘state’ which, when specified, yields ‘truth values’ for the primitive propositions in . These can then be extended recursively to the rest of . In classical physics, the possible truth values are just ‘true’ or ‘false’. However, as we shall see, the situation in topos theory is more complex.
2.3 Using Geometric Logic
The inductive definition of given above means that sentences can involve only a finite number of primitive propositions, and therefore only a finite number of disjunctions (‘’) or conjunctions (‘’). An interesting variant of this structure is the, so-called, ‘propositional geometric logic’. This is characterised by modifying the language and logical axioms so that:
There are arbitrary disjunctions, including the empty disjunction (‘0’).
There are finite conjunctions, including the empty conjunction (‘1’)
Conjunction distributes over arbitrary disjunctions; disjunction distributes over finite conjunctions.
This structure does not include negation, implication, or infinite conjunctions.
From a conceptual viewpoint, this set of rules is obtained by considering what it means to actually ‘affirm’ the propositions in . A careful analysis of this concept is given by Vickers ; the idea itself goes back to work by Abramsky . The conclusion is that the set of ‘affirmable’ propositions should satisfy the rules above.
Clearly such a logic is tailor-made for seeking representations in the open sets of a topological space—the paradigmatic example of a Heyting algebra. The phrase ‘geometric logic’ is normally applied to a first-order logic with the properties above, and we will return to this in our discussion of the typed language . What we have here is just the propositional part of this logic.
The restriction to geometric logic would be easy to incorporate into our languages : for example, the axiom (3.3) (if added) could be extended to read Note that the bi-implication used in, for example, (3.2–3.3), is not available if there is no implication symbol. Thus we have assumed that we are now working with a logical structure in which ‘equality’ is a meaningful concept; hence the introduction of ‘’ in (3.10).
The move to geometric logic is motivated by a conception of truth that is grounded in the actions of making real measurements. This resonates strongly with the logical positivism that seems still to lurk in the collective unconscious of the physics profession, and which, of course, was strongly affirmed by Bohr in his analysis of quantum theory. However, our drive towards ‘neo-realism’ involves replacing the idea of observation/measurement with that of ‘the way things are’, albeit in a more sophisticated interpretation than that of the ubiquitous cobbler-in-the-market. Consequently, the conceptual reasons for using ‘affirmative’ logic are less compelling. This issue deserves further thought: at the moment we are open-minded about it.
The use of geometric logic becomes more interesting in the context of the typed language , and we shall return to this in Section 4.2.2
2.4 Introducing Time Dependence
In addition to describing ‘the way things are’ there is also the question of how the-way-things-are changes in time. In the form presented above, the language may seem geared towards a ‘canonical’ perspective in so far as the propositions concerned are implicitly taken to be asserted at a particular moment of time. As such, deals with the values of physical quantities at that time. In other words, the underlying spatio-temporal perspective seems thoroughly ‘Newtonian’.
However, this is only partly true since the phrase ‘physical quantity’ can have meanings other than the canonical one. For example, one could talk about the ‘time average of momentum’, and call that a physical quantity. In this case, the propositions would be about histories of the system, not just ‘the way things are’ at a particular moment in time.
In the former case, would naturally include history propositions of the form
and other obvious variants of this. Here we assume that .
The sequential proposition in (3.11) is to be interpreted (in a realist reading) as asserting that “ ‘The physical quantity has a value that lies in at time ’ and ‘the physical quantity has a value that lies in at time ’ and and ‘the physical quantity has a value that lies in at time ’ ”. Clearly what we have here is a type of temporal logic. Thus this would be an appropriate structure with which to discuss the ‘consistent histories’ interpretation of quantum theory, particularly in the, so-called, HPO (history projection formalism) . In that context, (3.11) represents a, so-called, ‘homogeneous’ history.
In the second approach, where there is only one time label, the representation will map “” to a time-dependent element, , of the Heyting algebra, ; one could say that this is a type of ‘Heisenberg picture’.
This suggests another option, which is to keep the language free of any time labels, but allow the representation to be time-dependent. In this case, is a time-dependent member of . Perhaps we should also consider the possibility that the Heyting algebra is time dependent, in which case is a member of .
A different approach is to ascribe time dependence to the ‘truth objects’ in the theory: this corresponds to a type of Schrödinger picture. The concept of a truth object is discussed in detail in Section 6.
2.5 The Representation of 𝒫ℒ(S){\cal PL}(S) in Classical Physics
Let us now look at the representation of that corresponds to classical physics. In this case, the topos involved is just the category, , of sets and functions between sets.
Then the representation maps the primitive proposition to the subset of given by
and hence, in classical physics, all three conditions (3.2–3.4) that we discussed earlier can be added consistently to the language .
Consider now the assignment of truth values to the propositions in this theory. This involves the idea of a ‘microstate’ which, in classical physics, is simply an element of the state space . Each microstate assigns to each primitive proposition , a truth value, , which lies in the set (which we identify with ) and is defined as
2.6 The Failure to Represent 𝒫ℒ(S){\cal PL}(S) in Standard Quantum Theory.
The procedure above that works so easily for classical physics fails completely if one tries to apply it to standard quantum theory.
In quantum physics, a physical quantity is represented by a self-adjoint operator on a Hilbert space , and the proposition is represented by the projection operator which projects onto the subset of the spectrum of ; i.e.,
Of course, the set of all projection operators, , in has a ‘logic’ of its own---the ‘quantum logic’ For an excellent survey of quantum logic see . This includes a discussion of a first-order axiomatisation of quantum logic, and with an associated sequent calculus. It is interesting to compare our work with what the authors of this paper have done. We hope to return to this at some time in the future. of the Hilbert space —but this is incompatible with the intuitionistic logic of the language and the representation (3.17).
while, on the other hand, the logical bi-implication
can be deduced from the axioms of the language .
This failure of distributivity bars any naïve realist interpretation of quantum logic. If an instrumentalist interpretation is used instead, the spectral projectors now represent propositions about what would happen if a measurement is made, not propositions about what is ‘actually the case’. And, of course, when a state is specified, this does not yield actual truth values but only the Born-rule probabilities of getting certain results.
A Higher-Order, Typed Language for Physics
We want now to consider the possibility of representing the physical quantities of a system by arrows in a topos other than .
The physical meaning of such an arrow is not clear, a priori. Nor is it even clear what it is that is being represented in this way. However, what is clear is that in such a situation it is not correct to assume that the quantity-value object is necessarily the real-number object in the topos (assuming that there is one). Rather, the target-object, , has to be determined for each topos, and is therefore an important part of the ‘representation’.
A powerful technique for allowing the quantity-value object to be system-dependent is to add a symbol ‘’ to the system language. Developing this line of thinking suggests that ‘’, too, should be added to the language, as should a set of symbols of the form ‘’, to be construed as ‘what it is’ (hopefully a physical quantity) that is represented by arrows in a topos. Similarly, there should be a symbol ‘’, to act as the linguistic precursor to the sub-object classifier in the topos; in the topos , this is just the set .
The clean way of doing all this is to construct a ‘local language’ . Our basic assumption is that such a language, , can be associated with each system . A physical theory of then corresponds to a representation of in an appropriate topos.
We first consider the minimal set of symbols needed to handle elementary physics. For more sophisticated theories in physics it will be necessary to change, or enlarge, this set of ‘ground-type’ symbols.
The symbols for the local language, , are defined recursively as follows:
The basic type symbols are . The last two, and , are known as ground-type symbols. They are the linguistic precursors of the state object, and quantity-value object, respectively.
If , , are type symbols, then so is By definition, if then . .
If is a type symbol, then so is .
For each type symbol, , there is associated a countable set of variables of type .
To each pair of type symbols there is associated a set, , of function symbols. Such a symbol, , is said to have signature ; this is indicated by writing .
Some of these sets of function symbols may be empty. However, in our case, particular importance is attached to the set, , of function symbols , and we assume this set is non-empty.
The function symbols represent the ‘physical quantities’ of the system, and hence will depend on the system . In fact, the only parts of the language that are system-dependent are these function symbols. The set is the analogue of the set, , of physical quantities associated with the propositional language .
For example, if is a point particle moving in one dimension, the set of physical quantities could be chosen to be which represent the position, momentum, and energy of the system. On the other hand, if is a particle moving in three dimensions, we could have to allow for three-dimensional position and momentum (with respect to some given Euclidean coordinate system). Or, we could decide to add angular momentum too, to give the set . A still further extension would be to add the quantities and for all unit vectors and ; and so on.
Note that, as with the propositional language , the fact that a given system has a specific Hamiltonian It must be emphasised once more that the use of a local language is not restricted to standard, canonical systems in which the concept of a ‘Hamiltonian’ is meaningful. The scope of the linguistic ideas is much wider than that and the canonical systems are only an example. Indeed, our long-term interest is in the application of these ideas to quantum gravity where the local language is likely to be very different from that used here. However, we anticipate that the basic ideas will be the same.—expressed as a particular function of position and momentum coordinates—is not something that is to be coded into the language: instead, such system dependence arises in the choice of representation of the language. This means that many different systems can have the same local language.
The next step is to enumerate the ‘terms’ in the language, together with their associated types :
For each type symbol , the variables of type are terms of type .
A term of type is called a formula; a formula with no free variables is called a sentence.
If is function symbol with signature , and is a term of type , then is term of type .
In particular, if is a physical quantity, and is a term of type , then is a term of type .
If are terms of type , then is a term of type .
If is a term of type , and if , then is a term of type .
If are terms of the same type, then ‘’ is a term of type .
If are terms of type respectively, then is a term of type .
Note that the logical operations are not included in the set of symbols. Instead, they can all be defined using what is already given. For example, (i) ; and (ii) if and are terms of type , then The parentheses are not symbols in the language, they are just a way of grouping letters and sentences. The same remark applies to the inverted commas ‘’. Thus, in terms of the original set of symbols, we have
To make the language into a deductive system we need to add a set of appropriate axioms and rules of inference. The former are expressed using sequents: defined as expressions of the form where is a formula (a term of type ) and is a set of such formula. The intention is that ‘’ is to be read intuitively as “the collection of formula in ‘imply’ ”. If is empty we just write .
For applications in physics we could, and presumably should, add extra axioms (in the form of sequents). For example, perhaps the quantity-value object should always be an abelian-group object, or at least a semi-group One could go even further and add the axioms for real numbers. However, the example of quantum theory suggests that this is inappropriate: in general, the quantity-value object will not be the real-number object .? This can be coded into the language by adding the axioms for an abelian group structure for . This involves the following steps:
A ‘unit’ function symbol ; this will be the linguistic analogue of the unit element in an abelian group.
An ‘addition’ function symbol .
An ‘inverse’ function symbol
For another example, consider a point particle moving in three dimensions, with the function symbols . As stands, there is no way to specify, for example, that ‘’. Such relations can only be implemented in a representation of the language. However, if this relation is felt to be ‘universal’ (i.e., if it is expected to hold in all physically-relevant representations) then it could be added to the language with the use of extra axioms.
One of the delicate decisions that has to be made about is what extra axioms to add to the base language. Too few, and the language lacks content; too many, and representations of potential physical significance are excluded. This is one of the places in the formalism where a degree of physical insight is necessary!
2 Representing ℒ(S)\mathcal{L}({S}) in a Topos
The construction of a theory of the system involves choosing a representation The word ‘interpretation’ is often used in the mathematical literature, but we want to reserve that for use in discussions of interpretations of quantum theory, and the like./model, , of the language in a topos A more comprehensive notation is , which draws attention to the system under discussion; similarly, the state object could be written as , and so on. This extended notation is used in Section 11 where we are concerned with the relations between different systems, and then it is essential to indicate which system is meant. However, in the present article, only one system at a time is being considered, and so the truncated notation is fine. . The choice of both topos and representation depend on the theory-type being used.
We now list the -representation of the most significant symbols and terms in our language, (we have picked out only the parts that are immediately relevant to our programme: for full details see ).
The ground type symbols and are represented by objects and in . These are identified physically as the state object and quantity-value object, respectively.
The symbol , is represented by , the sub-object classifier of the topos .
The symbol , is represented by , the terminal object in .
For each type symbol , we have , the power object of the object in .
In particular, and .
Each function symbol in (i.e., each physical quantity) is represented by an arrow in .
We will generally require the representation to be faithful: i.e., the map is one-to-one.
We see that the analogue of the ‘’ used in the -proposition is played by sub-objects of (i.e., global elements of ) in the domain of the arrow in (4.22). These objects are, of course, representation-dependent (i.e., they depend on ).
A term, , of type with no free variables is represented by a global element . These will typically act as ‘truth values’ for propositions about the system.
Any axioms that have been added to the language are required to be represented by the arrow .
We should emphasise that the decision to focus on the particular type of language that we have, is not an arbitrary one. Indeed, there is a deep connection between such languages and topos theory.
Furthermore, each local language, , gives rise to an associated topos, , whose objects are equivalence classes of -sets, where is defined to mean that the equation (i.e., a term of type with no free variables) can be proved using the sequent calculus of the language with its axioms. From this perspective, a representation of the system-language in a topos is equivalent to a functor from the topos to .
2.2 Theory Construction as a Translation of Languages
Conversely, for each topos there is a local language, , whose ground-type symbols are the objects of , and whose function symbols are the arrows in . It then follows that a representation of a local language, , in is equivalent to a ‘translation’ of in .
Thus constructing a theory of physics is equivalent to finding a suitable translation of the system language, , to the language, , of an appropriate topos .
As we will see later, the idea of translating one local language into another plays a central role in the discussion of composite systems and sub-systems.
In the case of spoken languages, one can translate from, say, (i) English to German; or (ii) from English to Greek, and then from Greek to German. However, no matter how good the translators, these two ways of going from English to German will generally not agree. This is partly because the translation process is not unique, but also because each language possesses certain intrinsic features that simply do not admit of translation.
There is an interesting analogous question for the representation of the local languages . Namely, suppose and are two different topos theories of the same system (these could be, say, classical physics and quantum physics). The question is if/when will there be a translation such that
In terms of the representation functors from the topos to the topoi and , the question is if there exists an interpolating functor from to .
In Section 12.2, we will introduce a certain category, , whose objects are topoi and whose arrows are geometric morphisms between topoi. It would be natural to require the arrow from to (if it exists) to be an arrow in this category.
It is at this point that ‘geometric logic’ enters the scene (cf. Section 3.2.3). A formula in is said to be positive if it does not contain the symbols Here, the formula is defined as ; is defined as ; where or . These conditions imply that is also absent. In fact, a positive formula uses only and . A disjunction can have an arbitrary index set, but a conjunction can have only a finite index set. A sentence of the form is said to be a geometric implication if both and are positive. Then a geometric logic is one in which only geometric implications are present in the language.
The advantage of using just the geometric part of logic is that geometric implications are preserved under geometric morphisms. This makes it appropriate to ask for the existence of ‘geometric translations’ , as in (4.24), since these will preserve the logical structure of the language .
The notion of ‘toinvariance’ introduced recently by Landsmann can be interpreted within our structures as asserting that the translations should always exist; or, at least, they should under appropriate conditions. Of course, the significance of this depends on how much information about the system is reflected in the language and how much in the individual representations.
For example, in the case of classical and quantum physics, one might go so far as to include information about the dynamics of the system within the local language . If the topoi and are those for the classical and quantum physics of respectively (so that is and is ), then an interpolating translation would be a nice realisation of Landsmann’s long-term goal of regarding quantisation as some type of functorial operation.
Of course, introducing dynamics raises interesting questions about the status of the concept of ‘time’ (cf the discussion in Section 3.2.4). In particular, is time to be identified as an object in representing topos, or is it an external parameter, like the ‘’ quantities in the propositional languages ?
3 Classical Physics in the Local Language ℒ(S)\mathcal{L}({S})
The quantum theory representation of is studied in Section 5. Here we will look at the concrete form of the expressions above for the example of classical physics. In this case, for all systems , and all classical representations, , the topos is . This representation of has the following ingredients:
The ground-type symbol is represented by a symplectic manifold, , that is the state-space for the system .
The type symbol is represented by the set, , of all To be super precise, we really need to use the collection of all Borel subsets of . subsets of the state space .
The type symbol , is represented by : the sub-object classifier in .
The type symbol , is represented by the singleton set: i.e., , the terminal object in .
4 Adapting the Language ℒ(S)\mathcal{L}({S}) to Other Types of Physical System
Our central contention in this work is that (i) each physical system, , can be equipped with a local language, ; and (ii) constructing an explicit theory of in a particular theory-type is equivalent to finding a representation of in a topos which may well be other than the topos of sets.
However, there are other situations where the relationship between the language and its representations is more complicated than this. In particular, there is the critical question about what features of the theory should go into the language, and what into the representation. The first step in adding new features is to augment the set of ground-type symbols. This is because these represent the entities that are going to be of generic interest (such as a state object or quantity-value object). In doing this, extra axioms may also be introduced to encode the properties that the new objects are expected to possess in all representations of physical interest.
For example, suppose we want to use our formalism to discuss space-time physics: where does the information about the space-time go? If the subject is classical field theory in a curved space-time, then the topos is , and the space-time manifold is part of the background structure. This makes it natural to have the manifold assumed in the representation; i.e., the information about the space-time is in the representation.
Alternatively, one can add a new ground type symbol, ‘’, to the language, to serve as the linguistic progenitor of ‘space-time’; thus would have the same theoretical status as the symbols and . In this context, we recall the brief discussion in Section 2.2.1 about the use of the real numbers in modelling space and/or time, and the motivation this provides for representing space-time as an object in a topos, and whose sub-objects represent the fundamental ‘regions’.
If ‘’ is added to the language, a function symbol is then the progenitor of a physical field. In a representation, , the object plays the role of ‘space-time’ in the topos , and is the representation of the field.
Of course, the language says nothing about what sort of entity is, except in so far as such information is encoded in extra axioms. For example, if the subject is classical field theory, then , and would be a standard differentiable manifold. On the other hand, if the topos admits ‘infinitesimals’, then could be a manifold according to the language of synthetic differential geometry .
The addition of a ‘time-type’ symbol, , to the language is a prime example of a situation where one might want to add extra axioms. These could involve ordering properties, or algebraic properties like those of an abelian group, and so on. In any topos representation, these properties would then be realised as the corresponding type of object in . Thus abelian group axioms mean that is an abelian-group object in the topos ; total-ordering axioms for the time-type mean that is a totally-ordered object in , and so on.
As an interesting extension of this idea, one could have a space-time ground type symbol , but then add the axioms for a partial ordering. In that case, would be a poset-object in , which could be interpreted physically as the -analogue of a causal set .
Quantum Propositions as Sub-Objects of the Spectral Presheaf
The idea of representing quantum theory in a topos of presheaves stemmed originally from a desire to acquire a new perspective on the Kochen-Specker theorem . It will be helpful at this stage to review some of this older material.
However, standard quantum theory precludes any such naive realist interpretation of the relation between formalism and physical world. And this obstruction comes from the mathematical formalism itself, in the guise of the famous Kochen-Specker theorem which asserts the impossibility of assigning values to all physical quantities whilst, at the same time, preserving the functional relations between them .
In a quantum theory, a physical quantity is represented by a self-adjoint operator on the Hilbert space of the system, and the first thing one has to decide is whether to regard a valuation as a function of the physical quantities themselves, or on the operators that represent them. From a mathematical perspective, the latter strategy is preferable, and we shall therefore define a valuation to be a real-valued function on the set of all bounded, self-adjoint operators, with the properties that : (i) the value of the physical quantity represented by the operator belongs to the spectrum of (the so-called ‘value rule’); and (ii) the functional composition principle (or FUNC for short) holds:
for any pair of self-adjoint operators , such that for some real-valued function . If they existed, such valuations could be used to embed the set of self-adjoint operators in the commutative ring of real-valued functions on an underlying space of microstates, thereby laying the foundations for a hidden-variable interpretation of quantum theory.
Several important results follow from the definition of a valuation. For example, if and commute, it follows from the spectral theorem that there exists an operator and functions and such that and . It then follows from FUNC that
The defining equation (5.30)) for a valuation makes sense whatever the nature of the spectrum of the operator . However, if contains a continuous part, one might doubt the physical meaning of assigning one of its elements as a value. To handle the more general case, we shall view a valuation as primarily giving truth-values to propositions about the values of a physical quantity, rather than assigning a specific value to the quantity itself.
As in Section 3, the propositions concerned are of the type , which (in a realist reading) asserts that the value of the physical quantity lies in the (Borel) subset of the spectrum of the associated operator . This proposition is represented by the spectral projector , which motivates studying the general mathematical problem of assigning truth-values to projection operators.
If is a projection operator, the identity implies that (from (5.32)); and hence, necessarily, or . Thus defines a homomorphism from the Boolean algebra to the ‘false(0)-true(1)’ Boolean algebra . More generally, a valuation induces a homomorphism where is any Boolean sub-algebra of the lattice of projectors on . In particular,
where ‘’ refers to the partial ordering in the lattice , and ‘’ is the ordering in the Boolean algebra .
The Kochen-Specker theorem asserts that no global valuations exist if the dimension of the Hilbert space is greater than two. The obstructions to the existence of such valuations typically arise when trying to assign a single value to an operator that can be written as and as with .
The various interpretations of quantum theory that aspire to use ‘beables’, rather than ‘observables’, are all concerned in one way or another with addressing this issue. Inherent in such schemes is a type of ‘contextuality’ in which a value given to a physical quantity cannot be part of a global assignment of values but must, instead, depend on some context in which is to be considered. In practice, contextuality is endemic in any attempt to ascribe properties to quantities in a quantum theory. For example, as emphasized by Bell , in the situation where , if the value of is construed counterfactually as referring to what would be obtained if a measurement of or of is made—and with the value of then being defined by applying to the result of the measurement the relation , or —then one can claim that the actual value obtained depends on whether the value of is determined by measuring , or by measuring .
In the programme to be discussed here, the idea of a contextual valuation will be developed in a different direction from that of the existing modal interpretations in which ‘reality’ is ascribed to only some commutative subset of physical quantities. In particular, rather than accepting such a limited domain of beables we shall propose a theory of ‘generalised’ valuations that are defined globally on all propositions about values of physical quantities. However, the price of global existence is that any given proposition may have only a ‘generalised’ truth-value. More precisely, (i) the truth-value of a proposition belongs to a logical structure that is larger than ; and (ii) these target-logics, and truth values, are context-dependent.
It is clear that the main task is to formulate mathematically the idea of a contextual, truth-value in such a way that the assignment of generalised truth-values is consistent with an appropriate analogue of the functional composition principle FUNC.
1.2 The Introduction of Coarse-Graining
In the original paper , this task is tackled using a type of ‘coarse-graining’ operation. The key idea is that, although in a given situation in quantum theory it may not be possible to declare a particular proposition to be true (or false), nevertheless there may be (Borel) functions such that the associated propositions “” can be said to be true. This possibility arises for the following reason.
Let denote the spectral algebra of the operator that represents a physical quantity . Thus is the Boolean algebra of projectors that project onto the eigenspaces associated with the Borel subsets of the spectrum of ; physically speaking, represents the proposition . It follows from the spectral theorem that, for all Borel subsets of the spectrum of , the spectral projector for the operator is equal to the spectral projector for . In particular, if is a Borel subset of then, since , we have ; and hence
Physically, the inequality in (5.34) reflects the fact that the proposition “” is generally weaker than the proposition in the sense that the latter implies the former, but not necessarily vice versa. For example, the proposition “” is weaker than the original proposition “” if the function is many-to-one and such that more than one eigenvalue of is mapped to the same eigenvalue of . In general, we shall say that “” is a coarse-graining of .
Now, if the proposition is evaluated as ‘true’ then, from (5.33) and (5.34), it follows that the weaker proposition “” is also evaluated as ‘true’.
This remark provokes the following observation. There may be situations in which, although the proposition cannot be said to be either true or false, the weaker proposition “” can. In particular, if the latter can be given the value ‘true’, then—by virtue of the remark above—it is natural to suppose that any further coarse-graining to give an operator will yield a proposition “” that will also be evaluated as ‘true’. Note that there may be more than one possible choice for the ‘initial’ function , each of which can then be further coarse-grained in this way. This multi-branched picture of coarse-graining is one of the main justifications for our invocation of the topos-theoretic idea of a presheaf.
It transpires that the key remark above is the statement:
This is key because the property thus asserted can be restated by saying that the collection of all functions such that “” is ‘true’ is a sieve; and sieves are closely associated with global elements of the sub-object classifier in a category of presheaves.
This motivates very strongly looking at the topos category, of contravariant Ab initio, we could just as well have looked at covariant functors, but with our definitions the contravariant ones are more natural., set-valued functors on . Then, bearing in mind our discussion of values of physical quantities, it is rather natural to construct the following object in this topos:
The spectral presheaf on is the contravariant functor defined as follows:
On objects: .
On morphisms: If , so that , then is defined by for all .
Note that is well-defined since, if , then is indeed an element of the spectrum of ; indeed, for these discrete-spectrum operators we have .
The key remark now is the following. If is any category, a global element, of a contravariant functor is defined to be a function that assigns to each object in the category an element in such a way that if then (see Appendix 2 for more details).
In the case of the spectral functor , a global element is therefore a function that assigns to each (bounded, discrete spectrum) self-adjoint operator , a real number such that if then . But this is precisely the condition FUNC in Eq. (5.30) for a valuation!
Thus, the Kochen-Specker theorem is equivalent to the statement that, if , the spectral presheaf has no global elements.
It was this observation that motivated the original suggestion by one of us (CJI) and his collaborators that quantum theory should be studied from the perspective of topos theory. However, as it stands, the discussion above works only for operators with a discrete spectrum. This is fine for finite-dimensional Hilbert spaces, but in an infinite-dimensional space operators can have continuous parts in their spectra, and then things get more complicated.
One powerful way of tackling this problem is to replace the category of operators with a category, , whose objects are commutative von Neumann sub-algebras of the algebra of all bounded operators on . There is a close link with the category since each self-adjoint operator generates a commutative von Neumann algebra, but using rather than solves all the problems associated with continuous spectra .
Of course, this particular motivation for introducing is purely mathematical, but there are also very good physics reasons for this step. As we have mentioned earlier, one approach to handling the implications of the Kochen-Specker theorem is to ‘reify’ only a subset of physical variables, as is done in the various ‘modal interpretations’. The topos-theoretic extension of this idea of ‘partial reification’, first proposed in , is to build a structure in which all possible reifiable sets of physical variables are included on an equal footing. This involves constructing a category, , whose objects are collections of quantum observables that can be simultaneously reified because the corresponding self-adjoint operators commute. The application of this type of topos scheme to an actual modal interpretation is discussed in the recent paper by Nakayama
From a physical perspective, the objects in the category can be viewed as contexts (or ‘world-views’, or ‘windows on reality’, or ‘classical snapshots’) from whose perspectives the quantum theory can be displayed. This is the physical motivation for using commutative von Neumann algebras.
In the normal, instrumentalist interpretation of quantum theory, a context is therefore a collection of physical variables that can be measured simultaneously. The physical significance of this contextual logic is discussed at length in and .
1.3 Alternatives to von Neumann Algebras
It should be remarked that is not the only possible choice for the category of concepts. Another possibility is to construct a category whose objects are the Boolean sub-algebras of the non-distributive lattice of projection operators on the Hilbert space; more generally we could consider the Boolean sub-algebras of any non-distributive lattice. This option was discussed in .
Yet another possibility is to consider the abelian -sub-algebras of the algebra of all bounded operators on . More generally, one could consider the abelian sub-algebras of any -algebra; this is the option adopted by Heunen and Spitters in their very interesting recent development of our scheme. One disadvantage of a -algebra is that does not contain projectors, and if one wants to include them it is necessary to move to -algebras, which are the abstract analogue of the concrete von Neumann algebras that we employ. For each of these choices there is a corresponding spectral object, and these different spectral objects are closely related.
It is clear that a similar procedure could be followed for any algebraic quantity that has an ‘interesting’ collection of commutative sub-algebras. We will return to this remark in Section 14.1.4.
2 From Projections to Global Elements of the Outer Presheaf
The fundamental thesis of our work is that in constructing theories of physics one should seek representations of a formal language in a topos that may be other than . We want now to study this idea closely in the context of the ‘toposification’ of standard quantum theory, with particular emphasis on a topos representation of propositions. Most ‘standard’ quantum systems (for example, one-dimensional motion with a Hamiltonian ) are obtained by ‘quantising’ a classical system, and consequently the formal language is the same as it is for the classical system. Our immediate goal is to represent physical propositions with sub-objects of the spectral presheaf .
In this Section we concentrate on the propositional language introduced in Section 3.2. Thus a key task is to find the map , where the primitive propositions in are of the form . As we shall see, this is where the critical concept of daseinisation arises: the procedure whereby a projector is transformed to a sub-object, , of the spectral presheaf, , in the topos (the precise definition of is given in Section 5.3.1).
We are going to consider the projection operator from the perspective of the ‘category of contexts’—a keystone of the topos approach to quantum theory. As we have remarked earlier, there are several possible choices for this category most of which are considered in detail in the original papers . Here we have elected to use the category of unital, abelian sub-algebras of . This partially-ordered set has a category structure in which (i) the objects are the abelian sub-algebras of ; and (ii) there is an arrow , where , We denote by the collection of all objects in the category . if and only if . By definition, the trivial sub-algebra V_{0}=\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1} is not included in the objects of . A context could also be called a ‘world-view’, a ‘classical snap-shot’, a ‘window on reality’, or even a Weltanschauung ‘Weltanschauung’ is a splendid German word. ‘Welt’ means world; ‘schauen’ is a verb and means to look, to view; ‘anschauen’ is to look at; and ‘-ung’ at the end of a word can make a noun from a verb. So it’s Welt-an-schau-ung.; mathematicians often refer to it as a ‘stage of truth’.
The critical question is what can be said about the projector ‘from the perspective’ of a particular context ? If belongs to then a ‘full’ image of is obtained from this view-point, and there is nothing more to say. However, suppose the abelian sub-algebra does not contain : what then?
We need to ‘approximate’ from the perspective of , and an important ingredient in our work is to define this as meaning the ‘smallest’ projection operator, , in that is greater than, or equal to, :
where ‘’ is the usual ordering of projection operators, and where denotes the set of all projection operators in .
To see what this means, let and represent the propositions and respectively with , so that . Since we learn less about the value of from the proposition than from , the former proposition is said to be weaker. Clearly, the weaker proposition is implied by the stronger proposition . The construction of as the smallest projection in greater than or equal to thus gives the strongest proposition expressible in that is implied by (although, if , the projection cannot usually be interpreted as a proposition about ). Note that the definition in (5.35) exploits the fact that the lattice of projection operators in is complete. This is the main reason why we chose von Neumann sub-algebras rather than -algebras: the former contain enough projections, and their projection lattices are complete. Note that if belongs to , then . The mapping was originally introduced by de Groote in , who called it the ‘-support’ of .
The key idea in this part of our scheme is that rather than thinking of a quantum proposition, , as being represented by the single projection operator , instead we consider the entire collection of projection operators, one for each context . As we will see, the link with topos theory is that this collection of projectors is a global element of a certain presheaf.
This ‘certain’ presheaf is in fact the ‘outer’ presheaf, which is defined as follows:
The outer In the original papers by CJI and collaborators, this was called the ‘coarse-graining’ presheaf, and was denoted . The reason for the change of nomenclature will become apparent later. presheaf is defined over the category as follows :
On objects : We have
On morphisms The mapping is given by for all .
With this definition, it is clear that, for each projection operator , the assignment defines a global element of the presheaf . Indeed, for each context , we have the projector , and if , then
and so the elements , , are compatible with the structure of the outer presheaf. Thus we have a mapping
from the projectors in to the global elements, , of the outer presheaf. Vis-a-vis our use of the language a little further on, we should emphasise that the outer presheaf has no linguistic precursor, and in this sense, it has no fundamental status in the theory. In fact, we could avoid the outer presheaf altogether and always work directly with the spectral presheaf, , which, of course, does have a linguistic precursor. However, it is technically convenient to introduce the outer presheaf as an intermediate tool.
2.2 Properties of the Mapping δ:𝒫(ℋ)→Γ\delta:\mathcal{P(H)}\rightarrow\GammaO.
Let us now note some properties of the map that are relevant to our overall scheme.
For all contexts , we have .
The null projector represents all propositions of the form with the property that . These propositions are trivially false.
For all contexts , we have .
The unit operator represents all propositions of the form with the property that . These propositions are trivially true.
There exist global elements of that are not of the form for any projector . This phenomenon will be discussed later. However, if is of the form for some , then
because for all , and for any that contains .
The next result is important as it means that ‘nothing is lost’ in mapping a projection operator to its associated global element, , of the presheaf .
Theorem 5.1 The map is injective.
This simply follows from (5.38): if for two projections , then
2.3 A Logical Structure for Γ\GammaO?
We have seen that the quantities , , are elements of , and if they are to represent quantum propositions, one might expect/hope that (i) these global elements of form a Heyting algebra; and (ii) this algebra is related in some way to the Heyting algebra of sub-objects of . Let us see how far we can go in this direction.
Our first remark is that any two global elements of can be compared at each stage in the sense of logical implication. More precisely, let denote the ’th ‘component’ of , and ditto for . Then we have the following result:
A partial ordering on can be constructed in a ‘local’ way (i.e., ‘local’ with respect to the objects in the category ) by defining
where the ordering on the right hand side of (5.40) is the usual ordering in the lattice of projectors .
It is trivial to check that (5.40) defines a partial ordering on . Thus is a partially ordered set.
Note that if are projection operators, then it follows from (5.40) that
since implies for all contexts . On the other hand, in general, does not imply but only . Thus the mapping respects the partial order.
The next thing is to see if a logical ’-operation can be defined on . Once again, we try a ‘local’ definition:
A ‘’-structure on can be defined locally by
for all , and for all .
Proof. It is not instantly clear that (5.42) defines a global element of . However, a key result in this direction is the following:
For each context , and for all , we have
for all contexts such that .
The proof is a straightforward consequence of the definition of the presheaf .
One immediate consequence is that (5.42) defines a global element The existence of the -operation on can be extended to itself. More precisely, there is an arrow where denotes the product presheaf over , whose objects are . Then the arrow is defined at any context by for all . of . Hence the theorem is proved.
It is also straightforward to show that, for any pair of projectors , we have , for all contexts . This means that, as elements of ,
Thus the mapping preserves the logical ‘’ operation.
However, there is no analogous equation for the logical ‘’-operation. The obvious local definition would be, for each context ,
but this does not define a global element of since, unlike (5.43), for the -operation we have only
for all . As a consequence, for all , we have only the inequality
It is easy to find examples where the inequality is strict. For example, let and . Then and hence , while can be strictly larger than , since and .
2.4 Hyper-Elements of Γ\GammaO.
We have seen that the global elements of , i.e., the elements of , can be equipped with a partial-ordering and a ‘’-operation, but attempts to define a ‘’-operation in the same way fail because of the inequality in (5.47).
However, the form of (5.46–5.47) suggests the following procedure. Let us define a hyper-element of to be an association, for each stage , of an element with the property that
for all . Clearly every element of is a hyper-element, but not conversely.
Now, if and are hyper-elements, we can define the operations ‘’ and ‘’ locally as:
Because of (5.46) we have, for all ,
so that the hyper-element condition (5.49) is preserved.
The occurrence of a logical ‘’ and ’ structure is encouraging, but it is not yet what we want. For one thing, there is no mention of a negation operation; and, anyway, this is not the expected algebra of sub-objects of a ‘state space’ object. To proceed further we must study more carefully the sub-objects of the spectral presheaf.
3 Daseinisation: Heidegger Encounters Physics
The spectral presheaf, , played a central role in the earlier discussions of quantum theory from a topos perspective . Here is the formal definition.
The spectral presheaf, , is defined as the following functor from to :
On objects : is the Gel’fand spectrum of the unital, abelian sub-algebra of ; i.e., the set of all multiplicative linear functionals \lambda:V\rightarrow\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C} such that .
On morphisms : is defined by ; i.e., the restriction of the functional \lambda:V\rightarrow\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C} to the sub-algebra .
One central result of spectral theory is that has a topology that is compact and Hausdorff, and with respect to which the Gel’fand transforms If , the Gel’fand transform, \overline{A}:\underline{\Sigma}_{V}\rightarrow\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}, of is defined by for all . of the elements of are continuous functions from to \mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}. This will be important in what follows .
The spectral presheaf plays a fundamental role in our research programme as applied to quantum theory. For example, it was shown in the earlier work that the Kochen-Specker theorem is equivalent to the statement that has no global elements. However, does have sub-objects, and these are central to our scheme:
A sub-object of the spectral presheaf is a functor such that
is a subset of for all .
If , then is just the restriction (i.e., the same as for ), applied to the elements .
This definition of a sub-object is standard. However, for our purposes we need something slightly different, namely concept of a ‘clopen’ sub-object. This is defined to be a sub-object of such that, for all , the set is a clopen A ‘clopen’ subset of a topological space is one that is both open and closed. subset of the compact, Hausdorff space . We denote by the set of all clopen sub-objects of . We will show later (in the Appendix) that, like , the set is a Heyting algebra. In Section 6.5 we show that there is an object whose global elements are precisely the clopen sub-objects of .
This interest in clopen sets is easy to explain. For, according to the Gel’fand spectral theory, a projection operator corresponds to a unique clopen subset, of the Gel’fand spectrum, . Furthermore, the Gel’fand transform \overline{\alpha}:\underline{\Sigma}_{V}\rightarrow\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C} of takes the values only, since the spectrum of a projection operator is just .
It follows that is the characteristic function of the subset, , of , defined by
The clopen nature of follows from the fact that, by the spectral theory, the function is continuous.
In fact, there is a lattice isomorphism between the lattice of projectors in and the lattice of clopen subsets of , The lattice structure on is defined as follows: if is an arbitrary family of clopen subsets of , then the closure is the maximum. The closure is necessary since the union of infinitely many closed sets need not be closed. The interior is the minimum of the family. One must take the interior since is closed, but not necessarily open. given by
Conversely, given a clopen subset , we get the corresponding projection as the (inverse Gel’fand transform of the) characteristic function of . Hence, each is of the form for some .
For each projection operator , the collection
forms a (clopen) sub-object of the spectral presheaf .
Proof. To see this, let . Then if is some abelian sub-algebra of , we have . Now let . Then , since and . This shows that
However, the left hand side of (5.59) is the subset of the outer-presheaf restriction of elements in to , and the restricted elements all lie in . It follows that the collection of sets
forms a (clopen) sub-object of the spectral presheaf .
By these means we have constructed a mapping
which sends projection operators on to clopen sub-objects of . As a matter of notation, we will denote the clopen subset as . The notation refers to the element (i.e., projection operator) of defined earlier.
3.2 The Definition of Daseinisation
As usual, the projection is regarded as representing a proposition about the quantum system. Thus maps propositions about a quantum system to (clopen) sub-objects of the spectral presheaf. This is strikingly analogous to the situation in classical physics, in which propositions are represented by subsets of the classical state space.
Definition 5.6 The map in (5.61) is a fundamental part of our constructions. We call it the daseinisation of . We shall use the same word to refer to the operation in (5.35) that relates to the outer presheaf.
The expression ‘daseinisation’ comes from the German word Dasein, which plays a central role in Heidegger’s existential philosophy. Dasein translates to ‘existence’ or, in the very literal sense often stressed by Heidegger, to being-there-in-the-world The hyphens are very important.. Thus daseinisation ‘brings-a-quantum-property-into-existence’ The hyphens are very important. by hurling it into the collection of all possible classical snap-shots of the world provided by the category of contexts.
We will summarise here some useful properties of daseinisation.
The null projection is mapped to the empty sub-object of :
The identity projection is mapped to the unit sub-object of :
Since the daseinisation map is injective (see Section 5.2.2), and the mapping is injective (because there is a monic arrow in ; see Section 6.5.2), it follows that the daseinisation map is also injective. Thus no information about the projector is lost when it is daseinised to become .
The reason for daseinising projections is that the set, , of sub-objects of the spectral presheaf forms a Heyting algebra. Thus the idea is to find a map and then extend it to all of using the simple recursion ideas discussed in Section 3.2.2.
In our case, the act of daseinisation gives a map from the projection operators to the clopen sub-objects of , and therefore a map can be defined by
However, to extend this definition to , it is necessary to show that the set of clopen sub-objects, , is a Heyting algebra. This is not completely obvious from the definition alone. However, it is true, and the proof is given in Theorem 16.1 in the Appendix.
In conclusion: daseinisation can be used to give a representation/model of the language in the Heyting algebra . Since the clopen subobjects of correspond bijectively to the global sections of the outer presheaf , it is clear that too is a Heyting algebra.
5 Daseinisation and the Operations of Quantum Logic.
It is interesting to ask to what extent the map respects the lattice structure on . Of course, we know that it cannot be completely preserved since the quantum logic is non-distributive, whereas is a Heyting algebra, and hence distributive.
We saw in Section 5.2.3 that, for the mapping , we have
for all contexts in .
The clopen subset of that corresponds to is . This implies that the daseinisation map is a morphism of -semi-lattices.
On the other hand, corresponds to the subset of . Therefore, since , daseinisation is not a morphism of -semi-lattices. In summary, for all projectors we have
where the logical connectives on the left hand side lie in the quantum logic , and those on the right hand side lie in the Heyting algebra , as do the symbols ‘’ and ‘’.
As remarked above, it is not surprising that (5.68) is not an equality. Indeed, the quantum logic is non-distributive, whereas the Heyting algebra is distributive, and so it would be impossible for both (5.67) and (5.68) to be equalities. The inequality in (5.68) is the price that must be paid for liberating the projection operators from the shackles of quantum logic and transporting them to the existential world of Heyting algebras.
We have the representation in (5.64), , of the primitive propositions , and, as explained in Section 3.2.2, this can be extended to compound sentences by making the obvious definitions:
As a result, we necessarily get a representation of the full language in the Heyting algebra . However, we then find that:
where, (5.75) comes from (5.68), and in (5.76) we have used the property of spectral projectors that . Thus, although by definition, , we only have the inequality
On the other hand, the same line of argument shows that
Thus it would be consistent to add the axiom
Of, course, both axioms are consistent with the representation of in classical physics.
It should be emphasised that there is nothing wrong with this result: indeed, as stated above, it is the necessary price to be paid for forcing a non-distributive algebra to have a ‘representation’ in a Heyting algebra.
5.2 Inner Daseinisation and δ(¬P^)\delta(\neg{\hat{P}}).
that can be added to the classical representation of . Thus something needs to be said about , where is the negation operation in the quantum logic .
To proceed further, we need to introduce another operation:
The inner daseinisation, , of is defined for each context as
This should be contrasted with the definition of outer daseinisation in (5.35).
Thus is the best approximation that can be made to by taking the ‘largest’ projector in that implies .
As with the other daseinisation construction, this operation was first introduced by de Groote in where he called it the core of the projection operator . We prefer to use the phrase ‘inner daseinisation’, and then to refer to (5.35) as the ‘outer daseinisation’ operation on . The existing notation will be replaced with if there is any danger of confusing the two daseinisation operations.
With the aid of inner daseinisation, a new presheaf, , can be constructed as an exact analogue of the outer presheaf, , defined in Section 5.2.1. Specifically:
The inner presheaf is defined over the category as follows:
On objects : We have
On morphisms : The mapping is given by for all .
It is easy to see that the collection of projection operators given by (5.83) is a global element of .
for all projectors in , and for all . It follows from (5.84) that
for all projectors and all contexts .
It is clear from (5.84) that the negation operation on projectors defines a map , ; i.e., for all contexts , we map . Actually, one can go further than this and show that the presheaves and are isomorphic in the category . This means that, in principle, we can always work with one presheaf only. However, for reasons of symmetry it is sometime useful to invoke both presheaves.
As with outer daseinisation, inner daseinisation can also be used to define a mapping from projection operators to sub-objects of the spectral presheaf. Specifically, if is a projection, for each define
It is easy to see that these subsets form a clopen subobject, , of . It follows from (5.85) that .
5.3 Using Boolean Algebras as the Base Category
As we have mentioned several times already, the collection, , of all commutative von Neumann sub-algebras of is not the only possible choice for the base category over which to construct presheaves. In fact, if we are only interested in the propositional language , a somewhat simpler choice is the collection, of all Boolean sub-algebras of the non-distributive lattice, , of projection operators on . More abstractly, for any non-distributive lattice , one could use the category of Boolean sub-algebras of . This possibility was raised in the original paper but has not been used much thereafter. However, it does have some interesting features.
The analogue of the (von Neumann algebra) spectral presheaf, , is the so-called dual presheaf, :
The dual presheaf on is the contravariant functor defined as follows:
On objects in : is the dual of ; i.e., the set of all homomorphisms from the Boolean algebra to the Boolean algebra .
On morphisms in : If then is defined by , where denotes the restriction of to the sub-algebra .
A global element of the functor is then a function that associates to each an element of the dual of such that if then ; thus, for all ,
Since each projection operator, belongs to at least one Boolean algebra (for example, the algebra ) it follows that a global element of the presheaf associates to each projection operator a number which is either or , and is such that, if , then . These types of valuation are often used in the proofs of the Kochen-Specker theorem that focus on the construction of specific counter-examples. In fact, it is easy to see the following:
The Kochen-Specker theorem is equivalent to the statement that, if , the dual presheaf has no global elements.
It is easy to apply the concept of ‘daseinisation’ to the topos . In the case of von Neumann algebras, the outer daseinisation of a projection operator was defined as (see (5.35))
where denotes the collection of all projection operators in the commutative von Neumann algebra . In this form, appears as a global element of the outer presheaf .
When using the base category, , of Boolean sub-algebras of , we define
for each Boolean sub-algebra of projection operators on . Clearly, the (outer) daseinisation, , is now a global element of the obvious -analogue of the outer presheaf . There are parallel remarks for the inner daseinisation and inner presheaf. The existence of these daseinisation operations means that the propositional language can be represented in the topos in a way that is closely analogous to that used above for the topos .
Note that (i) each Boolean algebra of projection operators generates a commutative von Neumann algebra, , (the double commutant); and, conversely, (ii) to each von Neumann algebra there is associated the Boolean algebra of the projection operators in . This implies that the operation
defines a full and faithful functor between the categories and . This functor can be used to pull-back the spectral presheaf, , in to the object in . This pull-back is closely related to the dual presheaf .
6 The Special Nature of Daseinised Projections
We have shown how daseinisation leads to an interpretation/model of the language in the Heyting algebra . In particular, any primitive proposition is represented by the clopen sub-object .
We have seen that, in general, the ‘and’, , of the daseinisation of two projection operators and , is not itself of the form for any projector . The same applies to the negation .
This raises the question of whether the sub-objects of that are of the form can be characterised in a simple way. Rather interestingly, the answer is ‘yes’, as we will now see.
Let be such that . As would be expected, there is a close connection between the restriction , , of the outer presheaf, and the restriction , , of the spectral presheaf. Indeed, if is a projection operator, and is defined as in (5.56), we have the following result:
The proof is given in Theorem 16.2 in the Appendix
This result shows that the sub-objects of are of a very special kind. Namely, they are such that the restrictions
For an arbitrary sub-object of , this will not be the case and only maps into . Indeed, this is essentially the definition of a sub-object of a presheaf. Thus we see that the daseinised projections are optimal in the following sense. As we go ‘down the line’ to smaller and smaller sub-algebras of a context —for example, from to , then to etc.—then the subsets , ,… are as small as they can be; i.e., is the smallest subset of such that , likewise is the smallest subset of such that , and so on.
It is also clear from this result that there are lots of sub-objects of that are not of the form for any projector .
These more general sub-objects of show up explicitly in the representation of the more sophisticated language . This will be discussed thoroughly in Section 8 when we analyse the representation, , of the language in the topos . This involves constructing the quantity-value object (to be denoted ), and then finding the representation of a function symbol in , in the form of a specific arrow in the topos. The generic sub-objects of are then of the form for sub-objects of . This is an illuminating way of studying the sub-objects of that do not come from the propositional language .
Truth Values in Topos Physics
So far we have concentrated on finding a Heyting-algebra representation of the propositions in quantum theory, but of course there is more to physics than that. We also want to know if/when a certain proposition is true: a question which, in physical theories, is normally answered by specifying a (micro) state of the system, or something that can play an analogous role.
In classical physics, the situation is straightforward (see Section 3.2.5). There, a proposition is represented by the subset of the state space ; and then, the proposition is true in a state if and only if ; i.e., if and only if the (micro-) state belongs to the subset, , of that represents the proposition.
Thus, each state assigns to any primitive proposition , a truth value, , which lies in the set (which we identify with ) and is defined as
However, the situation in quantum theory is very different. There, the spectral presheaf —which is the analogue of the classical state space —has no global elements at all. Our expectation is that this will be true in any topos-based theory that goes ‘beyond quantum theory’: i.e., is empty; or, if does have global elements, there are not enough of them to determine as an object in the topos. In this circumstance, a new concept is required to replace the familiar idea of a ‘state of the system’. As we shall see, this involves the concept of a ‘truth object’, or ‘pseudo-state’.
In physics, the propositions of interest are of the form , which refers to the value of a physical quantity. However, in constructing a theory of physics, such physical propositions must first be translated into mathematical propositions. The concept of ‘truth’ is then studied in the context of the latter.
Let us start with set-theory based mathematics, where the most basic proposition is of the form “”, where is a subset of a set , and is an element of . Then the truth value, denoted , of the proposition “” is
Thus the proposition “” is true if, and only if, belongs to . In other words, is the characteristic function of the subset of ; cf. (17.492) in the Appendix.
This remark is the foundation of the assignment of truth values in classical physics. Specifically, if the state is , the truth value, , of the physical proposition is defined to be the truth value of the mathematical proposition “”; or, equivalently, of the mathematical proposition “”. Thus, using (6.95), we get, for all ,
We now consider the analogue of the above in a general topos . Let be an object in , and let be a sub-object of . Then is determined by a characteristic arrow , where is the sub-object classifier; equivalently, we have an arrow .
Now suppose that is a global element of ; i.e., . Then the truth value of the mathematical proposition “” is defined to be
where . Thus is an element of ; i.e., it is a global element of the sub-object classifier .
The connection with the result (6.95) (in the topos ) can be seen by noting that, in (6.95), the characteristic function of the subset is the function such that if , and otherwise. It follows that (6.95) can be rewritten as
where in (6.99), denotes the function that is defined by . The link with (6.97) is clear when one remembers that, in the topos , the terminal object, , is just the singleton set .
In quantum theory, the topos is , and so the objects are all presheaves. In particular, at each stage , the sub-object classifier is the set of sieves on . In this case, if is a sub-object of , and , the explicit form for (6.99) is the sieve
at each stage . In other words, at each stage/context V, the truth value of the mathematical proposition “” is defined to be all those stages ‘down the line’ such that the ‘component’, of at that stage is an element of the component, , of at that stage.
The definitions (6.97) and (6.100) play a central role in constructing truth values in out quantum topos scheme. However, as has no global elements, these truth values cannot be derived from some expression with . Therefore, we must proceed differently, as will become clear by the end of the following Section.
However, before we do so, let us make one final remark concerning (6.95). Namely, in normal set theory the proposition “” is true if, and only if,
i.e., if an only if the set is a subset of . The transition from the proposition “” to the proposition “” is seemingly trivial, but in a topos other than sets it takes on a new significance. In particular, as we shall see shortly, although the spectral presheaf, , has no global elements, it does have certain ‘minimal’ sub-objects that are as ‘close’ as one can get to a global element, and then the topos analogue of (6.101) is very important.
2 Truth Objects
To understand how ‘truth values’ of physical propositions arise we return again to our earlier discussion of local languages. In this Section we will employ the local language rather than the propositional language, , that was used earlier in this article.
Thus, let be the local language for a system . This is a typed language whose minimal set of ground-type symbols is and . In addition, there is a non-empty set, , of function symbols that correspond to the physical quantities of .
in the topos . Then, if is the name of a sub-object, , of the quantity-value object , we get the chain
which is the characteristic arrow of the sub-object of that represents the physical proposition .
Let us first pose this question at a linguistic level. In a representation , an element of is associated with a representation of a term of type with no free variables. Hence the question can be rephrased as asking how a term, , in of type can be ‘converted’ into a term of type ? At this stage, we are happy to have free variables, in which case the desired term will be represented by an arrow in whose co-domain is , but whose domain is other than . This would be an intermediate stage to obtaining a global element of .
In the context of the language there are three obvious ways of ‘converting’ the term of type to a term of type :
Choose a term, , of type ; then the term ‘’ is of type . We will call this the ‘micro-state’ option.
2.2 The Micro-State Option
This is the desired global element of .
This is the procedure that is adopted in classical physics when a truth value is assigned to propositions by specifying a micro-state, , where is the classical state space in the representation of . Specifically, for all , the truth value of the proposition as given by (6.106) is (c.f. (6.94))
2.3 The Truth Object Option.
By hindsight, we know that the option to use global elements of is not available in the quantum case. For there the state object, , is the spectral presheaf, and this has no global elements by virtue of the Kochen-Specker theorem. The absence of global elements of the state object could well be true in many other topos models of physics (particularly those that go ‘beyond quantum theory’), and therefore an alternative general strategy is needed to that employing micro-states .
where is the usual evaluation arrow. In using this expression we need the -representatives:
2.4 The Example of Classical Physics.
3 Truth Objects in Quantum Theory
However, we have to keep in mind the need to restrict to clopen sub-objects of . In particular, we must show that there is a well-defined presheaf such that
There are various examples of the presheaf that are of interest to us. In particular, let for some projector . Then, using the propositional language introduced earlier, the ‘truth’ of the proposition represented by (for example, ) is
for all stages . Here, is the usual expression for the probability that the proposition represented by the projector is true, given that the quantum state is the (normalised) vector .
It is easy to see that the definition of a truth object in (6.124) can be extended to a mixed state with a density-matrix operator : simply replace the definition in (6.124) with
In a sense, this should not surprise us. The analogue of a density matrix in classical physics is a probability measure defined on the classical state space . Individual microstates are in one-to-one correspondence with probability measures of the form defined by if , if .
However, one of the main claims of our programme is that any theory can be made to ‘look like’ classical physics in the appropriate topos. This suggests that, in the topos version of quantum theory, a density matrix should be represented by some sort of measure on the state object in the topos ; and this should relate in some way to an ‘integral’ of ‘vector truth objects’. The recent work by Heunen and Spitters provides the mathematical basis for such a construction . We shall return to some of their ideas later.
4 The Pseudo-state Option
We turn now to the third way mentioned above whereby a term, , of type in can be ‘converted’ to a term of type . Namely, choose a term, , of type and then use ‘’. As we shall see, this idea is easy to implement in the case of quantum theory and leads to an alternative way of thinking about truth objects.
These ordering properties are associated with the following observation. If is any vector state, we can collect together all the projection operators that are ‘larger’ or equal to and define:
It is clear that, for all stages/contexts , we have
The significance of this localisation property is that is a maximal (proper) filter in the non-distributive lattice, , of all projection operators on . Such maximal filters in the projection lattices of von Neumann algebras were extensively discussed by de Groote who called them ‘quasi-points’. In particular, is a, so-called, ‘atomic’ quasi-point in . Every pure state gives rise to an atomic quasi-point, , and vice versa. We will return to these entities in Section 8.4.
4.2 Using Pseudo-States in Lieu of Truth Objects
The equation (6.127) from classical physics suggests that, in the quantum case, we look at the set-valued function on defined by
is of considerable interest. We shall refer to it as a ‘pseudo-state’ for reasons that appear below.
Note that is defined by (6.132) as an element of . However, because of the monic we can also regard as an element of . The corresponding (clopen) sub-object of will be denoted .
for all contexts . From these relations it follows that is is injective.
In terms of sub-objects of , we have if and only if . Hence, (6.134) can be rewritten as
However, the right hand side of (6.136) is just the topos truth value, . It follows that
Thus, if desired, a truth object in quantum theory can be regarded as a sub-object of , rather than a sub-object of . In a sense, these sub-objects, , of are the ‘closest’ we can get to global elements of . This is why we call them ‘pseudo-states’. However, note that a pseudo-state is not a minimal element of the Heyting algebra since these will include stalks that are empty sets, something that is not possible for a pseudo-state. Note that the sub-objects do not have any global elements since any such would give a global element of and, of course, there are none. Thus if one is seeking examples of presheaves with no global elements, the collection , , afford many such.
4.3 Linguistic Implications
Note that, in general, the -representation of such a term is of the form
where the ‘first slot’ on the right hand side of the pairing in (6.138) is a truth-object (in pseudo-state form), and the second correspond to a proposition represented by a sub-object of .
However, this raises the rather obvious question “What is a pseudo-state?”. More precisely, we would like to know a generic set of characteristic properties of those sub-objects of that can be regarded as ‘pseudo-states’. A first step would be to answer this question in the case of quantum theory. In particular, are there any quantum pseudo-states that are not of the form for some vector ?
In this context the localisation property expressed by (6.129) is rather suggestive. In the case that has infinite dimension, de Groote has shown that there exist quasi-points in that are not of the form for some . However, he has also shown that, in an appropriate topology, the set of all atomic quasi-points is dense in the set of all quasi-points. Of course, none of these intriguing structures arise in a finite-dimensional Hilbert space in anything other than a trivial way. So, in that sense, it is unlikely that they will play any fundamental role in explicating the topos representation of quantum theory. If is any such quasi-point, (6.129) suggests strongly that we define an associated presheaf, , by
for all . This construction seems natural enough from a mathematical perspective, but we are not yet clear of the physical significance of the existence of such ‘quasi truth-objects’. The same applies to the associated ‘quasi pseudo-state’, , defined by
4.4 Time-Dependence and the Truth Object.
As emphasised at the end of Section 3.2, the question of time dependence depends on the theory-type being considered. The structure of the language that has been used so far is such that the time variable lies outside the language. In this situation, the time dependence of the system can be implemented in several ways.
It is also possible to construct a ‘Heisenberg picture’ where the truth object is constant but the physical quantities and associated propositions are time dependent. We will return to this in Section 10 when we discuss the use of unitary operators.
5 The Presheaf PclP_{\operatorname{cl}}(Σ\Sigma).
We must now show that there really is a presheaf .
The easiest way of defining is to start with the concrete expression for the normal power object . First, if is any presheaf over , define the restriction of to to be the functor from the category The notation means the partially-ordered set of all sub-algebras . to that assigns to each , the set , and with the obvious induced presheaf maps.
Then, at each stage , is the set of natural transformations from to . These are in one-to-one correspondence with families of maps , with the following commutative diagram for all : Note that any sub-object, of , gives rise to such a natural transformation from to for all stages . Namely, for all , is defined to be the characteristic arrow of the sub-object of .
and the evaluation arrow , has the form, at each stage :
Moreover, in general, given a map , the subset of associated with the corresponding sub-object is , where is the unit (‘truth’) in the Heyting algebra .
This suggests strongly that an object, , in can be defined using the same definition of as above, except that the family of maps must be such that, for all , is a clopen subset of the (extremely disconnected) Hausdorff space . It is straightforward to check that such a restriction is consistent, and that as required.
5.2 The Monic Arrow From O to PclP_{\operatorname{cl}}(Σ\Sigma).
We define , with the power transpose , as follows. First recall that in any topos, there is a bijection , and hence, in particular, (using )
Now let , and let be the clopen subset of that corresponds to the projector via the spectral theorem; see (5.56). Then we define at stage by
for all .
On the other hand, the basic result relating coarse-graining to subsets of is
for all and for all . It follows that
for all . In this form is is clear that is indeed a sieve on ; i.e., an element of .
The next step is to show that the collection of maps defined in (6.146) constitutes a natural transformation from the object to the object in the topos . This involves chasing around a few commutative squares, and we will spare the reader the ordeal. There is some subtlety, since we really want to deal with , not ; but all works in the end.
To prove that is monic, it suffices to show that the map is injective at all stages . This is a straightforward exercise and the details will not be given here.
Finally then, for any given quantum state the basic proposition can be assigned a generalised truth value in , where is the topos of presheaves over . This is defined at each stage/context as
6 Yet Another Perspective on the K-S Theorem
In classical physics, the pseudo-state associated with the microstate is just . This gives the diagram
where and is the canonical map
The singleton is the terminal object in the category, , of sets, and the subset embedding in (6.150) is the categorical pull-back by of the monic .
In the quantum case, the analogue of the diagram (6.150) is
where the arrow has yet to be defined. To proceed further, let us first return to the set-theory map
We can think of (6.153) as the power transpose, , of the map defined by
In our topos case, the obvious definition for the arrow is the power transpose , of the arrow , defined by
With this definition of , the diagram in (6.152) becomes meaningful: in particular the monic is the categorical pull-back by of the monic .
There is, however, a significant difference between (6.152) and its classical analogue (6.150). In the latter case, the function can be ‘lifted’ to a function to give a commutative diagram: i.e., such that
However, in the quantum case there can be no ‘lift’ , as this would correspond to a global element of the spectral presheaf , and of course there are none. Thus, from this perspective, the Kochen-Specker theorem can be understood as asserting the existence of an obstruction to lifting the arrow .
occur in many places in mathematics. A special, but very well-known, example of (6.158) arises when trying to construct cross-sections of a non-trivial principle fiber bundle . In diagrammatic terms we have
A cross-section of this bundle corresponds to a lifting of the map .
The obstructions to lifting through can be studied in various ways. One technique is to decompose the bundle into a series of interpolating fibrations where each fibration has the special property that the fiber is a particular Eilenberg-McLane space (this is known as a ‘Postnikov tower’). One then studies the sequential lifting of the function , i.e., first try to lift it through the fibration ; if that is successful try to lift it through ; and so on. Potential obstructions to performing these liftings appear as elements of the cohomology groups , , where is the fiber of the bundle.
We have long felt that it should possible to describe the non-existence of global elements of (i.e., the Kochen-Specker theorem) in some cohomological way, and the remark above suggests one possibility. Namely, perhaps there is some analogue of a ‘Postnikov factorisation’ for the arrow that could give a cohomological description of the obstructions to a global element of , i.e., to the lifting of a pseudo-state through the arrow to give an arrow .
The de Groote Presheaves of Physical Quantities
Our task now is to consider the representation of the local language, , in the case of quantum theory. We assume that the relevant topos is the same as that used for the propositional language , i.e., , but the emphasis is very different.
From a physics perspective, the key symbols in are (i) the ground-type symbols, and —the linguistic precursors of the state object and the quantity-value object respectively—and (ii) the function symbols , which are the precursors of physical quantities. In the quantum-theory representation, , of , the representation, , of is defined to be the spectral presheaf in the topos .
The critical question is to find the object, (provisionally denoted as a presheaf ), in that represents , and is hence the quantity-value object. One might anticipate that is just the real-number object in the topos , but that turns out to be quite wrong, and the right answer cannot just be guessed. In fact, the correct choice for is found indirectly by considering a related question: namely, how to represent each function symbol , with a concrete arrow in , i.e., with a natural transformation between the presheaves and .
Critical to this task are the daseinisation operations on projection operators that were defined earlier as (5.35) and (5.83), and which are repeated here for convenience:
If is a projection operator, and is any context/stage, we define:
where ‘’ denotes the usual ordering of projection operators, and where is the set of all projection operators in .
Similarly, the ‘inner daseinisation’ operation is defined in the context as (c.f. (5.83))
Thus is the best approximation to in from ‘above’, being the smallest projection in that is larger than or equal to . Similarly, is the best approximation to from ‘below’, being the largest projection in that is smaller than or equal to .
In Section 6.5, we showed that the outer presheaf is a sub-object of the power object (in the category ), and hence that the global element of determines a (clopen) sub-object, , of the spectral presheaf . By these means, the quantum logic of the lattice is mapped into the Heyting algebra of the set, , of clopen sub-objects of .
Our task now is to perform the second stage of the programme: namely (i) identify the quantity-value presheaf, ; and (ii) show that any physical quantity can be represented by an arrow from to .
2 The Daseinisation of an Arbitrary Self-Adjoint Operator
We now want to extend the daseinisation operations from projections to arbitrary (bounded) self-adjoint operators. To this end, consider first a bounded, self-adjoint operator, , whose spectrum is purely discrete. Then the spectral theorem can be used to write where are the eigenvalues of , and are the spectral projection operators onto the corresponding eigenspaces.
A construction that comes immediately to mind is to use the daseinisation operation on projections to define
for each stage . However, this procedure is rather unnatural. For one thing, the projections, , form a complete orthonormal set:
whereas, in general, the collection of daseinised projections, , will not satisfy either of these conditions. In addition, it is hard to see how the expression can be generalised to operators, , with a continuous spectrum.
The answer to this conundrum lies in the work of de Groote. He realised that although it is not useful to daseinise the spectral projections of an operator , it is possible to daseinise the spectral family of .
If then .
The net of projection operators in the lattice is bounded above by , and below by . In fact,
The map is right-continuous: It is a matter of convention whether one chooses right-continuous or left-continuous.
The spectral theorem asserts that for any self-adjoint operator , there exists a spectral family, , such that
It is easy to see that (7.169) defines a genuine partial ordering on (the self-adjoint operators in ). In fact, is a ‘boundedly complete’ lattice with respect to the spectral order, i.e., each bounded set of self-adjoint operators has a minimum and a maximum with respect to this order.
If are projections, then
so the spectral order coincides with the usual partial order on . To ensure this, the ‘reverse’ relation in (7.169) is necessary, since the spectral family of a projection is given by
If are self-adjoint operators such that (i) either or is a projection, or (ii) , then . Here ‘’ denotes the usual ordering on . The ‘usual’ ordering is if for all vectors .
Moreover, if are arbitrary self-adjoint operators, then implies , but not vice versa in general. Thus the spectral order is a partial order on that is coarser than the usual one.
2.2 Daseinisation of Self-Adjoint Operators.
De Groote’s crucial observation was the following. Let be a spectral family in (or, equivalently, a self-adjoint operator ). Then, for each stage , the following maps:
also define spectral families. The reason (7.172) and (7.173) have a different form is that is right continuous whereas is not. On the other hand, the family is right continuous. These spectral families lie in and hence, by the spectral theorem, define self-adjoint operators in . This leads to the definition of the two daseinisations of an arbitrary self-adjoint operator:
Let be an arbitrary self-adjoint operator. Then the outer and daseinisations of are defined at each stage as:
Both outer daseinisation (7.174) and inner daseinisation (7.175) can be used to ‘adapt’ a self-adjoint operator to contexts that do not contain . (On the other hand, if , then .)
2.3 Properties of Daseinisation.
We will now list some useful properties of daseinisation.
1. It is clear that the outer, and inner, daseinisation operations can be extended to situations where the self-adjoint operator does not belong to , or where is not an abelian sub-algebra of . Specifically, let be an arbitrary von Neumann algebra, and let be a proper von Neumann sub-algebra such that . Then outer and inner daseinisation can be defined as the mappings
A particular case is and for two contexts such that . Hence, a self-adjoint operator can be restricted from one context to a sub-context.
For the moment, we will let be an arbitrary von Neumann algebra, with .
where the minimum is taken with respect to the spectral order; i.e., is the smallest self-adjoint operator in that is spectrally larger than (or equal to) . This implies in the usual order. Likewise,
so is the largest self-adjoint operator in spectrally smaller than (or equal to) , which implies .
3. In general, neither nor can be written as Borel functions of the operator , since daseinisation changes the elements of the spectral family, while a function merely ‘shuffles them around’.
for all self-adjoint operators and all von Neumann sub-algebras . Analogous arguments apply to inner daseinisation.
Heuristically, this result implies that the spectrum of the operator is more degenerate than that of ; i.e., the effect of daseinisation is to ‘collapse’ eigenvalues.
5. Outer and inner daseinisation are both non-linear mappings. We will show this for projections explicitly. For example, let . Then , while and in general, so is the sum of two non-orthogonal projections in general (and hence not equal to ). For inner daseinisation, we have and in general, so in general.
6. If , then and . If , then and . This is due the behaviour of spectral families under the mapping .
7. Let be a self-adjoint operator, and let be an element of the spectral family of . From (7.174) we get
where we have used the general result that, for any projection , we have . Then, (7.186) gives
2.4 The de Groote Presheaves
We know that and are global elements of the outer presheaf, , and inner presheaf, , respectively. Using the daseinisation operation for self-adjoint operators, it is straightforward to construct analogous presheaves for which and are global elements. One of these presheaves was briefly considered in . We call these the ‘de Groote presheaves’ in recognition of the importance of de Groote’s work.
The outer de Groote presheaf, \underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm O}}, is defined as follows:
On objects : We define \underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm O}}_{V}:=V_{\rm sa}, the collection of self-adjoint members of .
On morphisms The mapping \underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm O}}(i_{V^{\prime}\,V}):\underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm O}}_{V}\rightarrow\underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm O}}_{V^{\prime}} is given by
for all {\hat{A}}\in\underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm O}}_{V}.
Here we used the fact that the restriction mapping of the inner presheaf is the inner daseinisation of projections .
The inner de Groote presheaf, , is defined as follows:
On objects : We define , the collection of self-adjoint members of .
On morphisms The mapping is given by
for all {\hat{A}}\in\underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm O}}_{V} (where ).
It is now clear that, by construction, is a global element of \underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm O}}, and is a global element of .
De Groote found an example of an element of \Gamma\underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm O}} that is not of the form (as mentioned in ). The same example can be used to show that there are global elements of the outer presheaf that are not of the form for any projection .
from self-adjoint operators in to global sections of the outer de Groote presheaf is injective. Likewise,
Proof. By construction, for all . Since is contained in at least one context, so
where the maximum is taken with respect to the spectral order. If , then we have
Analogously, for all , so
If , then we have
The same argument also holds more generally for arbitrary von Neumann algebras, not just .
Our goal now is to construct a ‘quantity-value’ presheaf with the property that inner and/or outer daseinisation of an self-adjoint operator can be used to define an arrow, i.e., a natural transformation, from to . In fact, we will define several closely related presheaves that can serve as a quantity-value object.
The arrow corresponding to a self-adjoint operator is denoted for now by . At each stage , we need a mapping
and we make the basic assumption that this mapping is given by evaluation. More precisely, is a spectral element A ‘spectral element’, of , is a multiplicative, linear functional \lambda:V\rightarrow\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C} with , see also Def. 5.4. of and hence can be evaluated on operators lying in . And, while will generally not lie in , both the inner daseinisation and the outer daseinisation do.
To answer this we need to see how these operators behave as we go ‘down a chain’ of sub-algebras . The first remark is that if then . When applied to the Gel’fand transforms, this leads to the equation
for all , where denotes the restriction of the spectral element to the sub-algebra . However, the definition of the spectral presheaf is such that , and hence (8.204) can be rewritten as
for all .
Let and be partially ordered sets. A function
is order-preserving if implies for all . It is order-reversing if implies . We denote by the set of order-preserving functions , and by the set of order-reversing functions.
We note that if is order-preserving, then is order-reversing, and vice versa.
Adapting Jackson’s definitions slightly, if is any partially-ordered set, we have the following.
The -valued presheaf, , of order-reversing functions over is defined as follows:
On objects :
where is the set of all von Neumann sub-algebras of .
On morphisms The mapping is given by
where denotes the restriction of the function to .
Jackson uses order-preserving functions with (the non-negative reals), with the usual order .
Clearly, there is an analogous definition of the -valued presheaf, , of order-preserving functions from to . It can be shown that and are isomorphic objects in .
Now let , and let . Then to each there is associated the function
for all . We note that as becomes smaller, becomes larger (or stays the same) in the spectral order, and hence in the usual order on operators. Therefore, is an order-reversing function, for each .
It is worth noting that daseinisation of , i.e., the approximation of the self-adjoint operator in the spectral order, allows to define a function (for each ) with values in the spectrum of , since we have , see (7.182). If we had chosen an approximation in the usual linear order on , then the approximated operators would not have a spectrum that is contained in in general.
The mappings , , are the components of a natural transformation/arrow .
Proof. We only have to prove that, whenever , the diagram
commutes. Here, the vertical arrows are the restrictions of the relevant presheaves from the stage to .
In fact, the commutativity of the diagram follows directly from the definitions. For each , the composition of the upper arrow and the right vertical arrow gives
which is the same function that we get by first restricting from to and then applying .
is injective. Interestingly, these results all carry over to an arbitrary von Neumann algebra . In this way, the formalism is flexible enough to adapt to situations where we have symmetries (which can described mathematically by a von Neumann algebra that has a non-trivial commutant) and super-selection rules (which corresponds to having a non-trivial centre).
Similarly, there is an order-preserving function
that is defined for all by
Since becomes smaller (or stays the same) as gets smaller, indeed is an order-preserving function from to for each . Again, approximation in the spectral order (in this case from below) allows us to define a function with values in , which would not be possible when using the linear order.
It follows from Theorem 7.1 that the mapping from self-adjoint operators to natural transformations is injective.
The functions obtained from inner and outer daseinisation can be combined to give yet another presheaf, and one that will be particularly useful for the physical interpretation of these constructions. The general definition is the following.
Let be a partially-ordered set. The -valued presheaf, , of order-preserving and order-reversing functions on is defined as follows:
(i) On objects :
where is the set of all sub-algebras of . Note that we introduce the condition , i.e., for all we demand .
(ii) On morphisms :
where denotes the restriction of to , and analogously for .
Note that since we have the condition in (i), the presheaf is not simply the product of the presheaves and .
Again from Theorem 7.1, the mapping from self-adjoint operators to natural transformations, , is injective.
Since for all , we can interpret each pair of values as an interval, which gives a first hint at the physical interpretation.
3 Inner and Outer Daseinisation from Functions on Filters
By Zorn’s lemma, every filter is contained in a maximal filter. Obviously, such a maximal filter is also a maximal filter base.
Let , and let be a spectral element of the von Neumann algebra . This means that is a multiplicative state of . For all projections , we have
and so . Moreover, , , and if , then (since ). Hence, for each we have either or . This shows that the family
of projections is an ultrafilter in . Conversely, each is uniquely determined by the set and hence by an ultrafilter in . This shows that there is a bijection between the set of ultrafilters in and the Gel’fand spectrum .
Let be a von Neumann algebra, and let be the set of filters in the projection lattice of . De Groote has shown that to each self-adjoint operator , there corresponds a, so-called, ‘observable function’ . If is abelian, , then is just the Gel’fand transform of . However, it is striking that can be defined even if is non-abelian; for us, the important example is .
It can be shown that each observable function is completely determined by its restriction to the space of maximal filters . Let denote the space of maximal filters in . The sets
form the base of a totally disconnected topology on . Following de Groote, this space is called the Stone spectrum of . If is abelian, , then, upon the identification of maximal filters (which are ultrafilters) in and spectral elements in , the Stone spectrum is the Gel’fand spectrum of .
This shows that for an arbitrary von Neumann algebra , the Stone spectrum is a generalisation of the Gel’fand spectrum (the latter is only defined for abelian algebras). The observable function is a generalisation of the Gel’fand transform of .
We want to show that the observable function of the outer daseinisation of to can be expressed by the observable function of directly. Since this works for all , we obtain a nice encoding of all the functions and hence of the self-adjoint operators . The result (already shown in ) is that, for all stages and all filters in ,
We want to give an elementary proof of this. We need
Let be a von Neumann algebra, a von Neumann sub-algebra of , and let be the inner daseinisation map on projections. Then, for all filters ,
Proof. If , then . Let be such that there is a with , i.e., . Then , which implies , since is a filter in . This shows that . Now let be such that there is no with . Since , there also is no with , so . This shows that .
Let . For all von Neumann sub-algebras and all filters , we have
The second equality is the definition of outer daseinisation (on the level of spectral projections, see (7.173)). In the penultimate step, we used Lemma 8.3.
This clearly implies (8.234). We saw above that to each there corresponds a unique ultrafilter . Since , the observable function is the Gel’fand transform of , and so, upon identifying the ultrafilter with the spectral element , we have
for all and for all . In this sense, the observable function encodes all the outer daseinisations , , of .
There is also a function, , on the filters in that encodes all the inner daseinisations , . This function is given for an arbitrary von Neumann algebra by
and is called the antonymous function of . If is abelian, then is the Gel’fand transform of and coincides with on the space of maximal filters, i.e., ultrafilters in . As functions on , and are different also in the abelian case. For an arbitrary von Neumann algebra , the antonymous function is another generalisation of the Gel’fand transform of .
There is a close relationship between observable and antonymous functions : for all von Neumann algebras and all self-adjoint operators , it holds that
There is a relation analogous to (8.234) for antonymous functions: for all and all filters in ,
Let . For all von Neumann sub-algebras and all filters , we have
where in the penultimate step we used Lemma 8.3.
Let , and let be the corresponding ultrafilter. Since , the antonymous function is the Gel’fand transform of , and we have
for all and all . Thus the antonymous function encodes all the inner daseinisations , , of .
Let be a unit vector in the Hilbert space of the quantum system. The expectation value of a self-adjoint operator in the state is given by
In the discussion of truth objects in section 6, we introduced the maximal filter in , Since is not distributive, is not an ultrafilter; i.e., there are projections such that neither nor . given by (cf. (6.129))
where is the projection onto the one-dimensional subspace of generated by . As shown in , the expectation value can be written as
In an instrumentalist interpretation, Which we avoid in general, of course! one would interpret , resp. , as the smallest, resp. largest, possible result of a measurement of the physical quantity when the state is . If is an eigenstate of , then is an eigenvalue of , and in this case, moreover,
If is not an eigenstate of , then
This allows us to write the expectation value as
These results depend on the fact that we use (inner and outer) daseinisation, i.e., approximations in the spectral, not the linear order.
If is not of the form , for some, then the cone over the ultrafilter corresponding to cannot be identified with a vector in . Nevertheless, the quantity is well-defined, and (8.234) and (8.242) hold. If we go from to a sub-algebra , then and , hence
for all .
For a proper sub-algebra , the spreading is over the (potentially larger) subset
All this is local in the sense that these expressions are defined at a stage and for sub-algebras, , of , where . No similar global construction or interpretation is possible, since the spectral presheaf has no global elements, i.e., no points (while the set does have points).
As we go down to smaller sub-algebras , the spread gets larger. This comes from the fact that has to be adapted more and more as we go to smaller sub-algebras . More precisely, is approximated from below by and from above by . This approximation gets coarser as gets smaller, which basically means that contains less and less projections.
It should be remarked that does not assign actual values to the physical quantity , but rather the possible range of such values; and these are independent of any state . This is analogous to the classical case where physical quantities are represented by real-valued functions on state space. The range of possible values is state-independent, but the actual value possessed by a physical quantity does depend on the state of the system.
5 The value of a physical quantity in a quantum state
We now want to discuss how physical quantities, represented by natural transformations , acquire ‘values’ in a given quantum state. Of course, this is not as straightforward as in the classical case, since from the Kochen-Specker theorem, we know that physical quantities do not have real numbers as their values. As we saw, this is related to the fact that there are no microstates, i.e., the spectral presheaf has no global elements.
We want to mimic this as closely as possible in the quantum case. In order to do so, we take a pseudo-state
(see (6.132)) and consider it as a sub-object of . This means that at each stage , we consider the set
Of course, the sub-object of that we get simply is . Sub-objects of this kind are as close to microstates as we can get, see the discussion in section 6.3 and . We can then form the composition
which is also denoted by . One can think of this arrow as being the ‘value’ of the physical quantity in the state described by .
Let , then
that is, every is given as the restriction of some . This implies that we even obtain the equality
At each stage , we have pairs of order-preserving and order-reversing functions , one function for each . If is an eigenstate of and is an abelian sub-algebra that contains , then . Moreover, contains the single element , which is the pure state that assigns to and to all projections in orthogonal to .
Evaluating at hence gives a pair, consisting of an order-preserving function and an order-reversing function :
The value of both functions at stage is , which is the eigenvalue of in the state . In this sense, we get back the ordinary eigenvalue of when the system is in the eigenstate .
We consider the value of the self-adjoint projection operator , seen as (the representative of) a physical quantity, in the (pseudo-)state . We remark that . By definition,
for all and all . In particular, the function
is given as (see (8.213), for all ,
If , then , see (8.261). Hence, for all , we obtain, for all ,
If we denote the constant function on with value as , then we can write
for all and all . The constant function trivially is an order-reversing function from to . We now consider the function
It is given as (see (8.221), for all ,
If , then, for all , we have . If , then , since and is a projection onto a one-dimensional subspace, so must project onto the zero-dimensional subspace.
Thus we get, for all , for all and all :
Summing up, we have completely described the ‘value’ of the physical quantity described by in the pseudo-state given by .
The behaviour of the order-preserving functions is more complicated than in the case that projects onto a one-dimensional subspace. In general, does not imply for , so the analogue of (8.275) does not hold in general.
for all . Note that if , then and , and so . Likewise, . Thus the definition of in (8.277) makes sense. Obviously, addition is commutative and associative.
However, it is not possible to define ‘’ in this way since the difference between two order-preserving functions may not be order-preserving, nor need the difference of two order-reversing functions be order-reversing. This problem is addressed in Section 9.
7 The Representation of Propositions From Inverse Images
In Section 3.2, we introduced a simple propositional language, , for each system , and discussed its representations for the case of classical physics. Then, in Section 5 we analysed the, far more complicated, quantum-theoretical representation of this language in the set of clopen subsets of the spectral presheaf, , in the topos . This gives a representation of the primitive propositions as sub-objects of :
where ‘’ is the (outer) daseinisation operation, and is the spectral projection corresponding to the subset of the spectrum, , of the self-adjoint operator .
We now want to remark briefly on the nature, and representation, of propositions using the ‘local’ language .
We should consider the analogue of these steps in the representation, , of the same language, , in the topos . In fact, the issues to be discussed apply to a representation in any topos.
We first note that if is a sub-object of , and if , then there is an associated sub-object of , denoted . Specifically, if is the characteristic arrow of the sub-object , then is defined to be the sub-object of whose characteristic arrow is . These sub-objects are analogues of the subsets, , of the classical state space : as such, they can represent propositions. In this spirit, we could denote by the proposition which the sub-object represents, although, of course, it would be a mistake to interpret as asserting that the value of something lies in something else: in a general topos, there are no such values.
To interpret such propositions, note first that in the -propositions , the range ‘’ belongs to the world that is external to the language. Consequently, the meaning of is given independently of . This ‘externally interpreted’ is then inserted into the quantum representation of via the daseinisation of propositions discussed in Section 5.
However, the situation is very different for the -propositions . Here, the quantity ‘’ belongs to the particular topos , and hence it is representation dependent. The implication is that the ‘meaning’ of can only be discussed from ‘within the topos’ using the internal language that is associated with , which, we recall, carries the translation of given by the topos-representation .
From a conceptual perspective, this situation is ‘relational’, with the meanings of the various propositions being determined by their relations to each other as formulated in the internal language of the topos. Concomitantly, the meaning of ‘truth’ cannot be understood using the correspondence theory (much favoured by instrumentalists) for there is nothing external to which a proposition can ‘correspond’. Instead, what is needed is more like a coherence theory of truth in which a whole body of propositions is considered together . This is a fascinating subject, but further discussion must be deferred to later work.
8 The relation between the formal languages ℒ(S)\mathcal{L}({S}) and 𝒫ℒ(S){\cal PL}(S)
In the propositional language , we have symbols representing primitive propositions. In the quantum case, such a primitive proposition is represented by the outer daseinisation of the projection corresponding to the proposition. (The spectral theorem gives the link between propositions and projections.)
We start with the case that , i.e., is the projection onto a one-dimensional subspace.
The inverse image is .
Proof. Let . In any case, we have
Let us assume that the inclusion is proper. Then there exists some such that
which is equivalent to the existence of some such that
see (8.271). This implies that for all , we must have , which contradicts (8.285). Hence there cannot be a proper inclusion , and we rather have equality
for all .
The proof is based on the fact that the order-reversing functions of the form , where and , are constant functions . The remark at the end of Section 8.5 shows that this holds more generally for arbitrary non-zero projections . Hence we obtain:
The inverse image is .
Extending the Quantity-Value Presheaf to an Abelian Group Object
Similarly, is given at each stage by a pair of functions , and for all , we write
We define, for all , and all , (c.f. (8.280))
It is clear that is a pair consisting of an order-preserving and an order-reversing function for all and all , so that is well defined. To avoid confusion we should emphasise that, in general, the sum is not equal to .
Since the whole -construction is quite complicated and is not used in this paper beyond the present section, we have decided to put all the relevant definitions into the Appendix where it can be read at leisure by anyone who is interested.
We can show that uniquely determines as follows: Let
3 Algebraic properties of the potential quantity-value presheaves
This is well-defined since if is order-preserving, then is order-reversing, and if is order-reversing, then is order-preserving. For , we obtain
The Role of Unitary Operators
Unitary operators play an important role in the formulation of quantum theory, and we need to understand the analogue of this in our topos formalism.
Unitary operators arise in the context of both ‘covariance’ and ‘invariance’. In elementary quantum theory, the ‘covariance’ aspect comes the fact that if we have made the associations
then the same physical predictions will be obtained if the following associations are used instead
for any unitary operator . Thus the mathematical representatives of physical quantities are defined only up to arbitrary transformations of the type above. In non-relativistic quantum theory, this leads to the canonical commutation relations; the angular-momentum commutator algebra; and the unitary time displacement operator. Similar considerations in relativistic quantum theory involve the Poincaré group.
The ‘invariance’ aspect of unitary operators arises when the operator commutes with the Hamiltonian, giving rise to conserved quantities.
As a side remark, we first consider the question if daseinisation can be applied to a unitary operator . The answer is clearly ‘yes’, via the spectral representation:
where is the spectral family for . Then, in analogy with (7.174–7.175) we have the following:
The outer daseinisation, , resp. the inner daseinisation, , of a unitary operator are defined as follows:
The outer, unitary de Groote presheaf, \underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm U}}, is defined by:
On objects : \underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm U}}_{V}:=V_{\rm un}, the collection of unitary operators in .
On morphisms The mapping \underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm U}}(i_{V^{\prime}\,V}):\underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm U}}_{V}\rightarrow\underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm U}}_{V^{\prime}} is given by
for all \hat{\alpha}\in\underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm U}}_{V}.
Clearly, (i) there is an analogous definition of an ‘inner’, unitary de Groote presheaf; and (ii) the map defines a global element of \underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm U}}.
This definition has the interesting consequence that, at each stage ,
A particular example of this construction is the one-parameter family of unitary operators, , where is the Hamiltonian of the system.
Of course, in our case everything commutes. Thus suppose is a representation of a Lie group on the Hilbert space . Then these operators can be daseinised to give the map , but generally this is not a representation of (or of its Lie algebra) since, at each stage we have
for all . Clearly, there is an analogous result for inner daseinisation.
2 Unitary Operators and Arrows in 𝐒𝐞𝐭𝐬𝒱(ℋ)op{\bf Sets}^{{{\cal V}({\cal H})}^{\rm op}}.
In classical physics, the analogue of unitary operators are ‘canonical transformations’; i.e., symplectic diffeomorphisms from the state space to itself. This suggests that should try to associate arrows in with each unitary operator .
Thus we want to see if unitary operators can act on the objects in . In fact, if denotes the group of all unitary operators in , we would like to find a realisation of in the topos .
As a first step, if and is an abelian von Neumann sub-algebra of , let us define
Clearly, for all ,
If , then, for all ,
We recall that if is any projection, then the (outer) daseinisation, , of at stage is ((5.35))
where we have resorted once more to using the propositional language . Thus
where we used the fact that the map is weakly continuous.
for all unitary operators , and for all stages . There is an analogous result for inner daseinisation.
Equation (10.322) can be applied to the de Groote presheaf \underline{\mkern 1.0mu\raise 2.5pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm O}} to give
for unitary operators , and all stages .
This can be viewed as the topos analogue of the statement in (10.307) about the invariance of the results of quantum theory under the transformations , . Of course, there is a pseudo-state analogue of all these expressions involving the sub-objects , .
2.3 The U^{\hat{U}}-twisted Presheaf
Note that, if then, for all presheaves ,
Since this is true for all functors in , we deduce that
For each , there is a natural isomorphism as given in the following diagram
for all , and all .
The proof, which just involves chasing round the diagram above using the basic definitions, is not included here.
We have the following commutative diagram:
It is interesting to reflect on the analogue of the above constructions for a general topos. It soon becomes clear that, once again, we encounter the antithetical concepts of ‘internal’ and ‘external’.
For example, in the discussion above, the unitary operators and the group lie outside the topos and enter directly from the underlying, standard quantum formalism. As such, they are external to both the languages and . We anticipate that notions of ‘covariance’ and ‘symmetry’ have applications well beyond those in classical physics and quantum physics. However, at the very least, in a general topos one would presumably replace the external with an internal group object in the topos concerned. And, of course, the notion of ‘symmetry’ is closely related to the concept to time, and time development, which opens up a Pandora’s box of possible speculation. These issues are important, and await further development.
The Category of Systems
We now return to the more general aspects of our theory, and study its application to a collection of systems, each one of which may be associated with a different topos. For example, if is a pair of systems, with associated topoi and , and if is a sub-system of , then we wish to consider how is related to . Similarly, if a composite system is formed from a pair of systems , what relations are there between the topos of the composite system and the topoi of the constituent parts?
Of course, in one sense, there is only one true ‘system’, and that is the universe as a whole. Concomitantly, there is just one local language, and one topos. However, in practice, the science community divides the universe conceptually into portions that are sufficiently simple to be amenable to theoretical and/or empirical discussion. Of course, this division is not unique, but it must be such that the coupling between portions is weak enough that, to a good approximation, their theoretical models can be studied in isolation from each other. Such an essentially isolated The ideal monad has no windows. portion of the universe is called a ‘sub-system’. By an abuse of language, sub-systems of the universe are usually called ‘systems’ (so that the universe as a whole is one super-system), and then we can talk about ‘sub-systems’ of these systems; or ‘composites’ of them; or sub-systems of the composite systems, and so on.
In practice, references by physicists to systems and sub-systems The word ‘sub-system’ does not only mean a collection of objects that is spatially localised. One could also consider sub-systems of field systems by focussing on a just a few modes of the fields as is done, for example, in the Robertson-Walker model for cosmology. Another possibility would be to use fields localised in some fixed space, or space-time region provided that this is consistent with the dynamics. do not generally signify actual sub-systems of the real universe but rather idealisations of possible systems. This is what a physics lecturer means when he or she starts a lecture by saying “Consider a point particle moving in three dimensions…..”.
To develop these ideas further we need mathematical control over the systems of interest, and their interrelations. To this end, we start by focussing on some collection, , of physical systems to which a particular theory-type is deemed to be applicable. For example, we could consider a collection of systems that are to be discussed using the methodology of classical physics; or systems to be discussed using standard quantum theory; or whatever. For completeness, we require that every sub-system of a system in is itself a member of , as is every composite of members of .
We shall assume that the systems in are all associated with local languages of the type discussed earlier, and that they all have the same set of ground symbols which, for the purposes of the present discussion, we take to be just and . It follows that the languages , , differ from each other only in the set of function symbols ; i.e., the set of physical quantities.
As a simple example of the system-dependence of the set of function symbols let system be a point particle moving in one dimension, and let the set of physical quantities be . In the language , these function-symbols represent the position, momentum, and energy of the system respectively. On the other hand, if is a particle moving in three dimensions, then in the language we could have to allow for three-dimensional position and momentum. Or, we could decide to add angular momentum as well, to give the set .
2 The Category 𝐒𝐲𝐬{\bf Sys}
The use of local languages is central to our overall topos scheme, and therefore we need to understand, in particular, (i) the relation between the languages and if is a sub-system of ; and (ii) the relation between , and , where denotes the composite of systems and .
These discussions can be made more precise by regarding as a category whose objects are the systems. To control the size of we assume that the collection of objects/systems is a set rather than a more general class. The arrows in need to cover two basic types of relation: (i) that between and if is a ‘sub-system’ of ; and (ii) that between a composite system, , and its constituent systems, and .
This may seem straightforward but, in fact, care is needed since although the idea of a ‘sub-system’ seems intuitively clear, it is hard to give a physically acceptable definition that is universal. However, some insight into this idea can be gained by considering its meaning in classical physics. This is very relevant for the general scheme since one of our main goals is to make all theories ‘look’ like classical physics in the appropriate topos.
To this end, let and be classical systems whose state spaces are the symplectic manifolds and respectively. If is deemed to be a sub-system of , it is natural to require that is a sub-manifold of , i.e., . However, this condition cannot be used as a definition of a ‘sub-system’ since the converse may not be true: i.e., if , this does not necessarily mean that, from a physical perspective, could, or would, be said to be a sub-system of . For example, consider the diagonal sub-manifold of the symplectic manifold that represents the composite of two copies of a system . Evidently, the states in correspond to the situation in which both copies of ‘march together’. It is doubtful if this would be recognised physically as a sub-system.
On the other hand, there are situations where being a sub-manifold clearly does imply being a physical sub-system. For example, suppose the state space of a system is a disconnected manifold with two components and , so that is the disjoint union, , of the sub-manifolds and . Then it seems physically appropriate to say that the system itself is disconnected, and to write where the symplectic manifolds that represent the sub-systems and are and respectively.
One reason why it is reasonable to call and ‘sub-systems’ in this particular situation is that any continuous dynamical evolution of a state point in will always lie in either one component or the other. This suggests that perhaps, in general, a necessary condition for a sub-manifold to represent a physical sub-system is that the dynamics of the system must be such that is mapped into itself under the dynamical evolution on ; in other words, is a dynamically-invariant sub-manifold of . This correlates with the idea mentioned earlier that sub-systems are weakly-coupled with each other.
However, such a dynamical restriction is not something that should be coded into the languages, and : rather, the dynamics is to be associated with the representation of these languages in the appropriate topoi.
Still, this caveat does not apply to the disjoint sum of two systems , and we will assume that, in general, (i.e., not just in classical physics) it is legitimate to think of and as being sub-systems of ; something that we indicate by defining arrows , and in .
To proceed further it is important to understand the connection between the putative arrows in the category , and the ‘translations’ of the associated languages. The first step is to consider what can be said about the relation between , and and . All three languages share the same ground-type symbols, and so what we are concerned with is the relation between the function symbols of signature in these languages.
By considering what is meant intuitively by the disjoint sum, it seems plausible that each physical quantity for the system produces a physical quantity for , and another one for . Conversely, specifying a pair of physical quantities—one for and one for —gives a physical quantity for . In other words,
However, it is important not to be too dogmatic about statements of this type since in non-classical theories new possibilities can arise that are counter to intuition.
Associated with (11.338) are the maps and , defined as the projection maps of the product. In the theory of local languages, these transformations are essentially translations of in and respectively; a situation that we denote , and .
To be more precise, these operations are translations if, taking as the explanatory example, the map is supplemented with the following map from the ground symbols of to those of :
Such a translation map is then extended to all type symbols using the definitions
for all finite and all type symbols . This, in turn, can be extended inductively to all terms in the language. Thus, in our case, the translations act trivially on all the type symbols.
Motivated by this argument we now turn everything around and, in general, define an arrow in the category to mean that there is some physically meaningful way of transforming the physical quantities in to physical quantities in . If, for any pair of systems there is more than one such transformation, then there will be more than one arrow from to .
To make this more precise, let denote the collection of all (small This means that the collection of symbols is a set, not a more general class.) local languages. This is a category whose objects are the local languages, and whose arrows are translations between languages. Then our basic assumption is that the association is a covariant functor from to , which we denote as .
Note that the combination of a pair of arrows in exists in so far as the associated translations can be combined.
2.2 The Arrows and Translations for the Composite System S1⋄S2S_{1}\diamond S_{2}.
Let us now consider the composition of a pair of systems. In the case of classical physics, if and are the symplectic manifolds that represent the systems and respectively, then the manifold that represents the composite system is the cartesian product . This is distinguished by the existence of the two projection functions and .
It seems reasonable to impose the same type of structure on : i.e., to require there to be arrows and in . However, bearing in mind the definition above, these arrows exist if, and only if, there are corresponding translations , and . But there are such translations: for if is a physical quantity for system , then can be defined as that same physical quantity, but now regarded as pertaining to the combined system ; and analogously for system . For example, if is the energy of particle , then we can talk about this energy in the combination of a pair of particles. Of course, in—for example—classical physics there is no reason why the energy of particle should be conserved in the composite system, but that, dynamical, question is a different matter. We shall denote this translated quantity, , by .
Note that we do not postulate any simple relation between and and ; i.e., there is no analogue of (11.338) for combinations of systems.
The definitions above of the basic arrows suggest that we might also want to impose the following conditions:
The arrows , and are monic in .
The arrows and are epic arrows in .
However, we do not require that and are the co-product and product, respectively, of and in the category .
2.3 The Concept of ‘Isomorphic’ Systems.
We also need to decide what it means to say that two systems and are isomorphic, to be denoted . As with the concept of sub-system, the notion of isomorphism is to some extent a matter of definition rather than obvious physical structure, albeit with the expectation that isomorphic systems in will correspond to isomorphic local languages, and be represented by isomorphic mathematical objects in any concrete realisation of the axioms: for example, by isomorphic symplectic manifolds in classical physics.
To a considerable extent, the physical meaning of ‘isomorphism’ depends on whether one is dealing with actual physical systems, or idealisations of them. For example, an electron confined in a box in Cambridge is presumably isomorphic to one confined in the same type of box in London, although they are not the same physical system. On the other hand, when a lecturer says “Consider an electron trapped in a box….”, he/she is referring to an idealised system.
One could, perhaps, say that an idealised system is an equivalence class (under isomorphisms) of real systems, but even working only with idealisations does not entirely remove the need for the concept of isomorphism.
For example, in classical mechanics, consider the (idealised) system of a point particle moving in a box, and let denote the ‘trivial system’ that consists of just a single point with no internal or external degrees of freedom. Now consider the system . In classical mechanics this is represented by the symplectic manifold , where is a single point, regarded as a zero-dimensional manifold. However, is isomorphic to the manifold and it is clear physically that the system is isomorphic to the system . On the other hand, one cannot say that is literally equal to , so the concept of ‘isomorphism’ needs to be maintained.
One thing that is clear is that if then , and if any other non-empty sets of function symbols are present, then they too must be isomorphic.
For the moment then, we will say that the trivial system has just a single physical quantity, which in classical physics translates to the number . More generally, for the language we specify that , i.e., has just a single element, , say. Furthermore, we add the axiom
If desired, an ‘empty’ system, , can be added too, with . This, so called, ‘pure language’, , is an initial object in the category .
2.4 An Axiomatic Formulation of the Category 𝐒𝐲𝐬{\bf Sys}
Let us now summarise, and clarify, our list of axioms for a category :
The collection is a small category whose objects are the systems of interest (or, if desired, isomorphism classes of such systems) and whose arrows are defined as above.
Thus the fundamental property of an arrow in is that it induces, and is essentially defined by, a translation . Physically, this corresponds to the physical quantities for system being ‘pulled-back’ to give physical quantities for system .
Arrows of particular interest are those associated with ‘sub-systems’ and ‘composite systems’, as discussed above.
The axioms for a category are satisfied because:
Physically, the ability to form composites of arrows follows from the concept of ‘pulling-back’ physical quantities. From a mathematical perspective, if and , then the translations give functions and . Then clearly , and this can thought of as the translation corresponding to the arrow .
The associativity of the law of arrow combination can be proved in a similar way.
We add by hand a special arrow which is defined to correspond to the translation that is given by the identity map on . Clearly, acts an an identity morphism should.
For any pair of systems , there is a disjoint sum, denoted . The disjoint sum has the following properties:
For all systems in :
For all systems in :
that are associated with translations in the sense discussed in Section 11.2.1. These are associated with the decomposition
We assume that if belong to , then also contains .
For any given pair of systems , there is a composite system in , denoted The product operation in a monoidal category is often written ‘’. However, a different symbol has been used here to avoid confusion with existing usages in physics of the tensor product sign ‘’. , with the following properties:
For all systems in :
For all systems in :
that are associated with translations in the sense discussed in Section 11.2.2.
We assume that if belong to , then also contains the composite system .
It seems physically reasonable to add the axiom
for all systems . However, physical intuition can be a dangerous thing, and so, as with most of these axioms, we are not dogmatic, and feel free to change them as new insights emerge.
There is a trivial system, , such that for all systems , we have
It may be convenient to postulate an ‘empty system’, , with the properties
Within the meaning given to arrows in , is a terminal object in . This is because the empty set of function symbols of signature is a subset of any other set of function symbols of this signature.
It might seem tempting to postulate that composition laws are well-behaved with respect to arrows. Namely, if , then, for any , there is an arrow and an arrow . A more accurate way of capturing this idea is to say that the operation in which (11.357) is a bi-functor from to . Ditto for the operation in which .
In the case of the disjoint sum, such an arrow can be easily constructed using (11.349). First split the function symbols in into and the function symbols in into . Since there is an arrow , there is a translation , given by a mapping . Of course, then there is also a mapping , i.e., a translation between and . Since we assume that there is an arrow in whenever there is a translation (in the opposite direction), there is indeed an arrow .
In the case of the composition, however, this would require a translation , and this cannot be done in general since we have no prima facie information about the set of function symbols . However, if we restrict the arrows in to be those associated with sub-systems, combination of systems, and compositions of such arrows, then it is easy to see that the required translations exist (the proof of this makes essential use of (11.353)).
If we make this restriction of arrows, then the axioms (11.351), (11.354–11.357), mean that, essentially, has the structure of a symmetric monoidal In the actual definition of a monoidal category the two isomorphisms in (11.354) are separated from each other, whereas we have identified them. Further more, these isomorphism are required to be natural. This seems a correct thing to require in our case, too. category in which the monoidal product operation is ‘’, and the left and right unit object is . There is also a monoidal structure associated with the disjoint sum ‘’, with as the unit object.
We say ‘essentially’ because in order to comply with all the axioms of a monoidal category, must satisfy certain additional, so-called, ‘coherence’ axioms. However, from a physical perspective these are very plausible statements about (i) how the unit object intertwines with the -operation; how the null object intertwines with the -operation; and (iii) certain properties of quadruple products (and disjoint sums) of systems.
It might be helpful at this point to give a simple example of a category . To that end, let denote a point particle that moves in three dimensions, and let us suppose that has no sub-systems other than the trivial system . Then is defined to be a pair of particles moving in three dimensions, and so on. Thus the objects in our category are , , , , where the ‘’ operation is formed any finite number of times.
At this stage, the only arrows are those that are associated with the constituents of a composite system. However, we could contemplate adding to the systems the disjoint sum which is a system that is either one particle or two particles (but, of course, not both at the same time). And, clearly, we could extend this to , and so on. Each of these disjoint sums comes with its own arrows, as explained above.
Note that this particular category of systems has the property that it can be treated using either classical physics or quantum theory.
3 Representations of 𝐒𝐲𝐬{\bf Sys} in Topoi
We assume that all the systems in are to be treated with the same theory type. We also assume that systems in with the same language are to be represented in the same topos. Then we define: As emphasised already, the association is generally not one-to-one: i.e., many systems may share the same language. Thus, when we come discuss the representation of the language in a topos, the extra information about the system is used in fixing the representation.
A topos realisation of is an association, , to each system in , of a triple where:
is the topos in which the theory-type applied to system is to be realised.
is the local language in that is associated with . This is not dependent on the realisation .
is a representation of the local language in the topos . As a more descriptive piece of notation we write . The key part of this representation is the map
where and are the state object and quantity-value object, respectively, of the representation in the topos . As a convenient piece of notation we write for all .
This definition is only partial; the possibility of extending it will be discussed shortly.
Now, if is an arrow in , then there is a translation arrow . Thus we have the beginnings of a commutative diagram
However, to be useful, the arrow on the right hand side of this diagram should refer to some relation between (i) the topoi and ; and (ii) the realisations and : this is the significance of the two ‘?’ symbols in the arrow written ‘’.
Indeed, as things stand, Definition 11.1 says nothing about relations between the topoi representations of different systems in . We are particularly interested in the situation where there are two different systems and with an arrow in .
We know that the arrow is associated with a translation , and an attractive possibility, therefore, would be to seek, or postulate, a ‘covering’ map to be construed as a topos representation of the translation , and hence of the arrow in .
This raises the questions of what properties these ‘translation representations’ should possess in order to justify saying that they ‘cover’ the translations. A minimal requirement is that if and , then the map factorises as
The conditions (11.360) and (11.361) seem eminently plausible, and they are not particularly strong. A far more restrictive axiom would be to require the following diagram to commute:
At first sight, this requirement seems very appealing. However, caution is needed when postulating ‘axioms’ for a theoretical structure in physics. It is easy to get captivated by the underlying mathematics and to assume, erroneously, that what is mathematically elegant is necessarily true in the physical theory.
The translation maps an arrow from to to an arrow from to . Intuitively, if is a ‘much larger’ object than (although since they lie in different topoi, no direct comparison is available), the translation can only be ‘faithful’ on some part of that can be identified with (the ‘image’ of) . A concrete example of this will show up in the treatment of composite quantum systems, see Subsection 13.3. As one might expect, a form of entanglement plays a role here.
4 Classical Physics in This Form
Constructing maps is likely to be complicated when and are different topoi, and so we begin with the example of classical physics, where the topos is always .
In general, we are interested in the relation(s) between the representations and that is associated with an arrow in . In classical physics, we only have to study the relation between the representations and .
Let us summarise what we have said so far (with denoting the -realisation of classical physics):
For any system in , a representation consists of the following ingredients.
The ground symbol is represented by a symplectic manifold, , that serves as the classical state space.
The trivial system is mapped to a singleton set (viewed as a zero-dimensional symplectic manifold):
The empty system is represented by the empty set:
Propositions about the system are represented by (Borel) subsets of the state space .
The composite system is represented by the Cartesian product ; i.e.,
The disjoint sum is represented by the disjoint union ;i.e.,
Let be an arrow in . Then
There is a translation map .
There is a symplectic function from the symplectic manifold to the symplectic manifold .
The existence of this function follows directly from the properties of sub-systems and composite systems in classical physics. It is discussed in detail below in Section (11.4.2). As we shall see, it underpins the classical realisation of our axioms.
4.2 Details of the Translation Representation.
from the components , to the disjoint sum . The systems , and have symplectic manifolds , and . We write .
maps (or ‘translates’) the topos representative of the function symbol to a real-valued function on . This function is clearly equal to .
We now consider arrows in of the form
from the composite classical system to the constituent systems and . Here, signals that is a constituent of the composite system , likewise . The systems , and have symplectic manifolds , and , respectively; i.e., the state space of the composite system is the cartesian product of the state spaces of the components. For typographical simplicity in what follows we denote .
for all .
This natural translation representation is based on the fact that, for the symplectic manifold , each point can be identified with a pair, , of points and . This is possible since the cartesian product is a product in the categorial sense and hence has projections . Then the translation representation of functions is constructed in a straightforward manner. Thus, let
is such that, for all ,
Clearly, can be seen as the representation of the function symbol .
Theories of Physics in a General Topos
Motivated by the above, let us try now to see what can be said about the scheme in general. Basically, what is involved is the topos representation of translations of languages. To be more precise, let be an arrow in , so that there is a translation defined by the translation function . Now suppose that the systems and are represented in the topoi and respectively. Then, in these representations, the function symbols of signature in and are represented by elements of and respectively.
that can be construed as the topos representation of the translation , and hence of the arrow in . We are particularly interested in seeing if can be chosen so that the following diagram, (see (11.362)) commutes:
However, as has been emphasised already, it is not clear that one should expect to find a function with this property. The existence and/or properties of such a function will be dependent on the theory-type, and it seems unlikely that much can be said in general about the diagram (12.384). Nevertheless, let us see how far we can get in discussing the existence of such a function in general.
Thus, if , the critical question is if there is some ‘natural’ way whereby this arrow can be ‘pulled-back’ to give an element .
The first pertinent remark is that is an arrow in the topos , whereas the sought-for pull-back will be an arrow in the topos , and so we need a mechanism for getting from one topos to the other (this problem, of course, does not arise in classical physics since the topos of every representation is always ).
The obvious way of implementing this change of topos is via some functor, from to . Indeed, given such a functor, an arrow in is transformed to the arrow
To convert this to an arrow from to , we need to supplement (12.385) with a pair of arrows in to get the diagram:
The pull-back, , with respect to these choices can then be defined as
It follows that a key part of the construction of a topos representation, , of will be to specify the functor from to , and the arrows and in the topos . These need to be defined in such a way as to be consistent with a chain of arrows .
When applied to the representative of a physical quantity , the diagram (12.386) becomes (augmented with the upper half)
The commutativity of (12.384) would then require
where both the left hand side and the right hand side of (12.390) are mappings from to .
Note that the analogous diagram in classical physics is simply
and the commutativity/pull-back condition (12.389) becomes
which is satisfied by virtue of (11.370).
It is clear from the above that the arrow can be viewed as the topos analogue of the map that arises in classical physics whenever there is an arrow .
1.2 The Pull-Back of Propositions.
More insight can be gained into the nature of the triple by considering the analogous operation for propositions. First, consider an arrow in in classical physics. Associated with this there is (i) a translation ; (ii) an associated translation mapping ; and (iii) a symplectic function .
Let be a (Borel) subset of the state space, ; hence represents a proposition about the system . Then is a subset of and, as such, represents a proposition about the system . We say that is the pull-back to of the -proposition represented by . The existence of such pull-backs is part of the consistency of the representation of propositions in classical mechanics, and it is important to understand what the analogue of this is in our topos scheme.
Consider the general case with the two systems as above. Then let be a proposition, represented as a sub-object of , with a monic arrow . The question now is if the triple can be used to pull back to give a proposition in , i.e., a sub-object of ?
The first requirement is that the functor should preserve monics. In this case, the monic arrow in is transformed to the monic arrow
in ; thus is a sub-object of in . It is a property of a topos that the pull-back of a monic arrow is monic ; i.e., if is monic, and if , then is a sub-object of . Therefore, in the case of interest, the monic arrow can be pulled back along (see diagram (12.388)) to give the monic . This is a candidate for the pull-back of the proposition represented by the sub-object .
In conclusion, propositions can be pulled-back provided that the functor preserves monics. A sufficient way of satisfying this requirement is for to be left-exact. However, this raises the question of “where do left-exact functors come from?”.
1.3 The Idea of a Geometric Morphism.
It transpires that there is a natural source of left-exact functors, via the idea of a geometric morphism. This fundamental concept in topos theory is defined as follows .
A geometric morphism between topoi and is a pair of functors and such that
, i.e., is left adjoint to ;
is left exact, i.e., it preserves all finite limits.
The morphism is called the inverse image part of the geometric morphism ; is called the direct image part.
Geometric morphisms are very important because they are the topos equivalent of continuous functions. More precisely, if and are topological spaces, then any continuous function induces a geometric morphism between the topoi and of sheaves on and respectively. In practice, just as the arrows in the category of topological spaces are continuous functions, so in any category whose objects are topoi, the arrows are normally defined to be geometric morphisms. In our case, as we shall shortly see, all the examples of left-exact functors that arise in the quantum case do, in fact, come from geometric morphisms. For these reasons, from now on we will postulate that any arrows between our topoi arise from geometric morphisms.
One central property of a geometric morphism is that it preserves expressions written in terms of geometric logic. This greatly enhances the attractiveness of assuming from the outset that the internal logic of the system languages, , is restricted to the sub-logic afforded by geometric logic.
En passant, another key result for us is the following theorem ( p359):
If is a functor between categories and , then it induces a geometric morphism (also denoted )
for which the functor takes a functor to the functor
In addition, has a left adjoint ; i.e., .
We will use this important theorem in several crucial places.
2 The Topos Rules for Theories of Physics
We will now present our general rules for using topos theory in the mathematical representation of physical systems and their theories.
The category is the following:
The objects of are the topoi that are to be used in representing the systems in .
The arrows from to are defined to be the geometric morphisms from to . Thus the inverse part, , of an arrow is a left-exact functor from to .
The rules for using topos theory are as follows:
A topos realisation, , of in is an assignment, to each system in , of a triple where:
is the topos in in which the physical theory of system is to be realised.
is the local language that is associated with . This is independent of the realisation, , of in .
is a representation of the local language in the topos .
In addition, for each arrow in there is a triple ,, that interpolates between and ; for details see below.
The representations, and , of the ground symbols and in are denoted and , respectively. They are known as the ‘state object’ and ‘quantity-value object’ in .
The representation by of each function symbol of the system is an arrow, in ; we will usually denote this arrow as .
Propositions about the system are represented by sub-objects of . These will typically be of the form , where is a sub-object of . Here, denotes the sub-object of whose characteristic arrow is , where is the characteristic arrow of the sub-object .
Generally, there are no ‘microstates’ for the system ; i.e., no global elements (arrows ) of the state object ; or, if there are any, they may not be enough to determine as an object in .
See Section 6.2 for full information on the idea of a ‘truth object’. Alternatively, one may use pseudo-states rather than truth objects, in which case the relevant truth values are of the form .
There is a ‘unit object’ in such that if denotes the trivial system in then, for all topos realisations ,
Motivated by the results for quantum theory (see Section 13.2), we postulate that the unit object in is the category of sets:
To each arrow in , we have the following:
There is a translation . This is specified by a map between function symbols: .
With the translation there is associated a corresponding function
These may, or may not, fit together in the commutative diagram:
The function is built from the following ingredients. For each topos realisation , there is a triple where:
is a geometric morphism; i.e., an arrow in the category (thus is left exact).
N.B. To simplify the notation a little we will denote by . This is sensible in so far as, for the most part, only the inverse part of will be used in our constructions.
is an arrow in the topos .
is an arrow in the topos .
The arrows and should behave appropriately under composition of arrows in .
The commutativity of the diagram (12.400) is equivalent to the relation
for all . As we keep emphasising, the satisfaction or otherwise of this relation will depend on the theory-type and, possibly, the representation .
If a proposition in is represented by the monic arrow, , the ‘pull-back’ of this proposition to is defined to be .
If is a sub-system of , with an associated arrow in then, in the diagram in (12.401), the arrow is a monic arrow in .
In other words, is a sub-object of , which is denoted
Another possible conjecture is the following: if is an epic arrow in , then, in the diagram in (12.401), the arrow is an epic arrow in .
In particular, for the epic arrow , the arrow is an epic arrow in the topos .
One should not read Rule 2. above as implying that the choice of the state object and quantity-value object are unique for any given system . These objects would at best be selected only up to isomorphism in the topos . Such morphisms in the Care is needed not to confuse morphisms in the topos with morphisms in the category of topoi. An arrow from the object to itself in the category is a geometric morphism in the topos . However, not every arrow in need arise in this way, and an important role can be expected to be played by arrows of this second type. A good example is when is the category of sets, . Typically, is the identity, but there are many morphisms from an object in to itself: they are just the functions from to . can be expected to play a key role in developing the topos analogue of the important idea of a symmetry, or covariance transformation of the theory.
In the example of classical physics, for all systems we have and is a symplectic manifold, and the collection of all symplectic manifolds is a category. It would be elegant if we could assert that, in general, for a given theory-type the possible state objects in a given topos form the objects of an internal category in . However, to make such a statement would require a general theory of state-objects and, at the moment, we do not have such a thing.
From a more conceptual viewpoint we note that the ‘similarity’ of our axioms to those of standard classical physics is reflected in the fact that (i) physical quantities are represented by arrows ; (ii) propositions are represented by sub-objects of ; and (iii) propositions are assigned truth values. Thus any theory satisfying these axioms ‘looks’ like classical physics, and has an associated neo-realist interpretation.
The General Scheme applied to Quantum Theory
We now want to study the extent to which our ‘rules’ apply to the topos representation of quantum theory.
For a quantum system with (separable) Hilbert space , the appropriate topos (what we earlier called ) is : the category of presheaves over the category (actually, partially-ordered set) of unital, abelian von Neumann sub-algebras of the algebra, , of bounded operators on .
A particularly important object in is the spectral presheaf , where, for each , is defined to be the Gel’fand spectrum of the abelian algebra . The sub-objects of can be identified as the topos representations of propositions, just as the subsets of represent propositions in classical physics.
2 The Translation Representation for a Disjoint Sum of Quantum Systems
Let be a category whose objects are systems that can be treated using quantum theory. Let be the local language of a system in whose quantum Hilbert space is denoted . We assume that to each function symbol, , in there is associated a self-adjoint operator More specifically, one could postulate that the elements of are associated with self-adjoint operators in some unital von Neumann sub-algebra of . and that the map
is injective (but not necessarily surjective, as we will see in the case of a disjoint sum of quantum systems).
from the components , to a disjoint sum ; for convenience we write . The systems , and have the Hilbert spaces , and , respectively.
As always, the translation goes in the opposite direction to the arrow , so
Then our first step is find an ‘operator translation’ from the relevant self-adjoint operators in to those in ,
To do this, let be a function symbol in . In Section 11.2.1, we argued that (as in (11.338)), and hence we introduce the notation , where and . It is then natural to assume that the quantisation scheme is such that the operator, , on can be decomposed as , where the operators and are defined on and respectively, and correspond to the function symbols and . It should be noted that our scheme does not use all the self-adjoint operators on the direct sum : only the ‘block diagonal’ operators of the form arise. Then the obvious operator translation is .
We now consider the general rules in the Definition 12.3 and see to what extent they apply in the example of quantum theory.
1. As we have stated several times, the topos associated with a quantum system is
Thus (i) the objects of the category are topoi of the form , ; and (ii) the arrows between two topoi are defined to be geometric morphisms. In particular, to each arrow in there must correspond a geometric morphism with associated left-exact functor . Of course, the existence of these functors in the quantum case has yet to be shown.
2. The realisation of the language in the topos is given as follows. First, we define the state object to be the spectral presheaf, , over , the context category of . To keep the notation brief, we will denote Presheaves are always denoted by symbols that are underlined. as .
4. Let {\cal H}=\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C} be the one-dimensional Hilbert space, corresponding to the trivial quantum system . There is exactly one abelian sub-algebra of \mathcal{B}(\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C})\simeq\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}, namely \mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C} itself. This leads to
where the right hand side (13.412) denotes an order-reversing function.
Let , and let such that and . Then
Proof. Every projection is of the form for unique projections and . Let be of the form such that and . The largest projection in smaller than or equal to , i.e., the inner daseinisation of to , is
where is the largest projection in smaller than or equal to , and is the largest projection in smaller than or equal to , so
This implies , since (outer) daseinisation of a self-adjoint operator just means inner daseinisation of the projections in its spectral family, and all the projections in the spectral family of are of the form .
The presheaves and lie in different topoi, and in order to ‘transform’ between them we need we need a (left-exact) functor from the topos to the topos : this is the functor in (12.401). One natural place to look for such a functor is as the inverse-image part of a geometric morphism from to . According to Theorem 12.1, one source of such a geometric morphism, , is a functor
for all . This function from to is clearly order preserving, and hence is a genuine functor.
Let denote the geometric morphism induced by . The inverse-image functor of is given by
This means that, for all , we have
For example, for the spectral presheaf we get
This is the functor that is denoted in (12.401).
We next need to find an arrow that is the analogue of the arrow in (12.401).
for all V\oplus\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{2}}\in{\rm Ob(\mathcal{V}({\cal H}_{1}\oplus{\cal H}_{2}))}. This makes clear how the arrow
is defined. Our conjectured pull-back/translation representation is
Using the definitions of and , it becomes clear that
Hence, the commutativity condition in (12.402) is satisfied for arrows in of the form .
3 The Translation Representation for Composite Quantum Systems
We now consider arrows in of the form
where the quantum systems , and have the Hilbert spaces , and , respectively. As usual, the composite system has as its Hilbert space the tensor product of the Hilbert spaces of the components.
The canonical translation As discussed in Section 11.2.2, this translation, , transforms a physical quantity of system into a physical quantity , which is the ‘same’ physical quantity but now seen as a part of the composite system . The symbol is the trivial physical quantity: it is represented by the operator . between the languages and (see Section 11.2.2) is such that if is a function symbol in , then the corresponding operator will be ‘translated’ to the operator . By assumption, this corresponds to the function symbol in .
We should be cautious about what to expect from this translation when we represent a physical quantity in by an arrow between presheaves, since there are no canonical projections
from the spectral presheaf of the composite system to the spectral presheaves of the components. On the other hand, in the classical case, there are canonical projections (13.430) because the symplectic manifold that represents the composite system is the cartesian product , which is a product in the categorial sense and hence comes with canonical projections.
This is the point where a form of entanglement enters the picture. The spectral presheaf is a presheaf over the context category of . Clearly, the context category can be embedded into by the mapping V_{1}\mapsto V_{1}\otimes\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{2}}, and likewise can be embedded into . But not every is of the form .
This comes from the fact that not all vectors in are of the form , hence not all projections in are of the form , which in turn implies that not all are of the form . There are more contexts, or world-views, available in than those coming from and . We call this ‘operator entanglement’.
3.2 A Geometrical Morphism and a Possible Translation
The most natural approach to a translation is the following. Let , and define to be the largest sub-algebra of such that V_{W}\otimes\,\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{2}} is a sub-algebra of . Depending on , may, or may not, be the trivial sub-algebra \mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{1}}. We note that if , then
but only implies .
The trivial algebra \mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{1}} is not an object in the category . This is why we introduce the ‘augmented context category’ , whose objects are those of united with \mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{1}}, and with the obvious morphisms (\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{1}} is a sub-algebra of all ).
Then there is a functor , defined as follows. On objects,
and if is an arrow in , we define (an arrow in ); if , then is the identity arrow .
denote the geometric morphism induced by . Then the (left-exact) inverse-image functor
acts on a presheaf in the following way. For all , we have
for all arrows in the category . We remark, although will not prove it here, that the inverse-image presheaf coincides with the direct image presheaf of constructed from the geometric morphism induced by the functor \displaystyle V\otimes\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{2}}. (13.437) Of course, the inverse image presheaf is much easier to construct.
In particular, for all , we have
In terms of our earlier notation, the functor is , and the arrow in (13.440) is the arrow in (12.401) with being replaced by , which is the arrow in whose translation representation we are trying to construct.
We also need an arrow in from to , where is defined in (13.438). This is the arrow denoted in (12.401).
The obvious choice is to restrict to the sub-algebra V_{W}\otimes\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{2}}\subseteq W, and to identify V_{W}\otimes\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{1}}\simeq V_{W}\otimes\hat{1}_{{\cal H}_{1}}\simeq V_{W} as von Neumann algebras, which gives \underline{\Sigma}^{{\cal H}_{1}\otimes{\cal H}_{2}}_{V_{W}\otimes\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{2}}}\simeq\underline{\Sigma}^{{\cal H}_{1}}_{V_{W}}. Let
is a natural transformation which is defined for all and all by
Note that, by construction, for each , the arrow corresponds to the self-adjoint operator , since
for all .
This is about as far as we can get with the arrows associated with the composite of two quantum systems. The results above can be summarised in the equation
If is not of the form W=V_{1}\otimes\mkern 1.0mu\raise 2.2pt\hbox{\scriptscriptstyle|}{\mkern-7.0mu\rm C}\hat{1}_{{\cal H}_{2}}, then it is relatively easy to show that
whereas, intuitively, one might have expected equality. Thus the ‘commutativity’ condition (12.389) is not satisfied.
Clearly, both technical and interpretational work remain to be done.
A major motivation for our work is the desire to find mathematical structures with whose aid genuinely new types of theory can be constructed. Consequently, however fascinating the ‘toposification’ of quantum theory may be, this particular theory should not be allowed to divert us too much from the main goal. However, it is also important to see if any general lessons can be learnt from what has been done so far. This is likely to be crucial in the construction of new theories.
In developing the topos version of quantum theory we have constructed concrete objects in the topos to function as the state object and quantity-value object. We have also seen how each quantum vector state gives a precise truth object, or ‘pseudo-state’.
Let us start with the state object . In classical physics, this is a symplectic manifold; in quantum theory it is the spectral presheaf in the topos . Does this suggest any properties for in general?
One possibility is that the state-object, , has some sort of ‘symplectic structure’. If taken literally, this phrase suggests synthetic differential geometry (SDG): a theory that is based on the existence in certain topoi (not ) of genuine ‘infinitesimals’. However, this seems unlikely for the quantum topoi and we would probably need to extend these topoi considerably in order to incorporate SDG. Thus when we say “…some sort of symplectic structure”, the phrase ‘some sort’ has to be construed rather broadly.
We suspect that, with this caveat, the state object may have such a structure, particularly for those quantum systems that come from quantising a given classical system. However, at the moment this is still a conjecture. We are currently studying systems whose classical state space is the cotangent bundle, , of a configuration space . We think that the quantum analogue of this space is a certain presheaf, , that is associated with the maximal commutative sub-algebra, , generated by the smooth, real-valued functions on . This is currently work in progress.
But even if the quantum state-object does have a remnant ‘symplectic structure’, it is debatable if this should be axiomatised in general. Symplectic structures arise in classical physics because the underlying equations of motion are second-order in the ‘configuration’ variables , and hence first-order in the pair , where are the ‘momentum variables’.
However if, say, Newton’s equations of gravity had been third-order in , this would lead to triples ( are ‘acceleration’ variables) and symplectic structure would not be appropriate.
1.2 Σϕ\Sigma_{\phi} as a Spectral Object: the Work of Heunen and Spitters
Another way of understanding the state object is suggested by the recent work of Heunen and Spitters . They start with a non-commutative -algebra, , of observables in some ‘ambient topos’, —in our case, this is —and then proceed with the following steps:
Construct the poset category This notation has been chosen to suggest more clearly the analogues with our topos constructions that use the base category . It is not that used by Heunen and Spitters. of commutative sub-algebras of .
Construct the topos, of covariant functors (i.e., co-presheaves) on the category/poset . They affirm that the operation defines a functor from the category of -algebras in to the category of elementary topoi and geometric morphisms.
Construct the ‘tautological’ co-presheaf in which for each commutative sub-algebra, , of . Then if , the associated arrow is just the inclusion map of in .
Using a recent, very important, result of Banacheswski and Mulvey , Heunen and Spitters show that the spectrum, , of the commutative algebra can be computed internally, and that it has the structure of an internal locale in .
They then show that, in the case of quantum theory, is essentially our spectral object, , but viewed as a co-presheaf of locales, rather than as a presheaf of topological spaces.
Thus Heunen and Spitters differ from us in that (i) they work in a general ambient topos , whereas we use ; (ii) they use -algebras rather than von Neumann algebras One problem with -algebras is that they very often do not contain enough projectors; and, of course, these are the entities that represent propositions. This obliges Heunen and Spitters to move from a -algebra to a -algebra, which is just an abstract version of a von Neumann algebra.; and (iii) they use covariant rather than contravariant functors.
The fact that they recover what is (essentially) our spectral presheaf is striking. Amongst other things, it suggests a possible axiomatisation of the state object, . Namely, we could require that in any topos representation, , the state object is (i) the spectrum of some internal, commutative (pre) -algebra; and (ii) the spectrum has the structure of an internal locale in the topos .
It is not currently clear whether or not it makes physical sense to always require to be the spectrum of an internal algebra. However, even in the contrary case it still makes sense to explore the possibility that has the ‘topological’ property of being an internal locale. This opens up many possibilities, including that of constructing the (internal) topos, , of sheaves over .
1.3 Using Boolean Algebras as the Base Category
As remarked earlier, there are several possible choices for the base category over which the set-valued functors are defined. Most of our work has been based on the category, , of commutative von Neumann sub-algebras of . As indicated above, the Heunen-Spitters constructions use the category of commutative -algebras. More abstractly, one can start with any -algebra or -algebra.
However, as discussed briefly in Section 5.5.3, another possible choice is the category, , of all Boolean sub-algebras of the lattice of projection operators on . The ensuing topos, , or , is interesting in its own right, but particularly so when combined with the ideas of Heunen and Spitters. As applied to the category , their work suggests that we first construct the tautological co-presheaf which associates to each , the Boolean algebra . Viewed internally in the topos , this co-presheaf is a Boolean-algebra object. We conjecture that the spectrum of can be obtained in a constructive way using the internal logic of . If so, it seems clear that, after using the locale trick of , this spectrum will essentially be the same as our dual presheaf .
Thus, in this approach, the state object is the spectrum of an internal Boolean-algebra, and daseinisation maps the projection operators in into elements of this algebra. This reinforces still further our claim that quantum theory looks like classical physics in an appropriate topos. This raises some fascinating possibilities. For example, we make the following:
Conjecture: The subject of quantum computation is equivalent to the study of ‘classical’ computation in the quantum topos .
1.4 Application to Other Branches of Algebra
It is clear that the scheme discussed above could fruitfully be extended to various branches of algebra. Thus, if is any algebraic structure We are assuming that the ambient topos is , but other choices could be considered., we can consider the category whose objects are the commutative sub-algebras of , and whose arrows are algebra embeddings (or, slightly more generally, monomorphisms). One can then consider the topos, , of all set-valued, contravariant functors on ; alternatively, one might look at the topos, , of covariant functors.
For this structure to be mathematically interesting it is necessary that the abelian sub-objects of have a well-defined spectral structure. For example, let be any locally-compact topological group. Then the spectrum of any commutative (locally-compact) subgroup is just the Pontryagin dual of , which is itself a locally-compact, commutative group. The spectral presheaf of can then be defined as the object, , in that is constructed in the obvious way (i.e., analogous to the way in which was constructed) from this collection of Pontryagin duals.
We conjecture that a careful analysis would show that, for at least some structures of this type:
There is a ‘tautological’ object, , in the topos that is associated with the category .
Viewed internally, this tautological object is a commutative algebra.
This object has a spectrum that can be constructed internally, and is essentially the spectral presheaf, , of .
It seems clear that, in general, the spectral presheaf, , is a potential candidate as the basis for non-commutative spectral theory.
1.5 The Partial Existence of Points of Σϕ\Sigma_{\phi}
One of the many intriguing features of topos theory is that it makes sense to talk about entities that only ‘partially exist’. One can only speculate on what would have been Heidegger’s reaction had he been told that the answer to “What is a thing?” is “Something that partially exists”. However, in the realm of topos theory the notion of ‘partial existence’ lies easily with the concept of propositions that are only ‘partly true’.
A particularly interesting example is the existence, or otherwise, of ‘points’ (i.e., global elements) of the state object . If has no global elements (as is the case for the quantum spectral presheaf, ) it may still have ‘partial elements’. A partial element is defined to be an arrow where the object in the topos is a sub-object of the terminal object . Thus there is a monic with the property that the arrow cannot be extended to an arrow . Studying the obstruction to such extensions could be another route to finding a cohomological expression of the Kochen-Specker theorem.
However, local cross-sections do exist, these being defined as sections of the bundle restricted to any open subset of the base space . In fact, this bundle is locally trivial; i.e., each point has a neighbourhood such that the restriction of the bundle to is trivial, and hence sections of the bundle restricted to exist.
There is an analogue of local triviality in the topos quantum theory where . Thus, let be any object in and define . Then is like a ‘neighbourhood’ of ; indeed, that is precisely what it is if the poset is equipped with the topology generated by the lower sets. Furthermore, given any presheaf in , the restriction, , to , can be defined as in Section 6.5. It is easy to see that, for all stages , the presheaf does have global elements. In this sense, every presheaf in is ‘locally trivial’. Furthermore, to each there is associated a sub-object of such that each global element of corresponds to a partial element of .
Thus, for the topos , there is a precise sense in which the spectral presheaf has ‘local elements’, or ‘points that partially exist’. However, it is not clear to what extent such an assertion can, or should, be made for a general topos . Certainly, for any presheaf topos, , one can talk about ‘localising’ with respect to the objects in the base category , but the situation for a more general topos is less clear.
1.6 The Work of Corbett et al
Another interesting question is whether these different ways of seeing the state-object relate at all to the work of Corbett and his collaborators .
For some time Corbett has been studying what he calls ‘quantum’ real numbers, or ‘qr-numbers’, as another way of a obtaining a ‘realist’ interpretation of quantum theory. The first step is to take the space of states, , of a quantum system (where a state is viewed as a positive linear functional on an appropriate -algebra, ) and equip it with the weakest topology such that the functions are continuous for all states . Then a ‘qr-number’ is defined as a global element of the sheaf of germs of continuous real-valued functions on . Put another way, a qr-number is a (Dedekind) real number in the topos, , of sheaves over . The fundamental physical postulate is then:
The ‘numerical values’ of a physical quantity, , are given by the qr-numbers where is an open subset of .
Every physical quantity has a qr-number value at all times.
Every physical quantity has an open subset of associated with it at all times. This is the extent to which the quantity can be said to ‘exist’.
Evidently this theory also is ‘contextual’, with the contexts now being identified with the open sets of .
It would be interesting to see if there is any relation between the real numbers in the Corbett presheaf, and the interval domains in the Heunen-Spitters presheaf . Roughly speaking, we can say that Corbett et al assign exact values to physical quantities by making the state ‘fuzzy’, whereas we (and Heunen & Spitter) keep the state sharp, but ascribe ‘fuzzy’ values to physical quantities. Clearly, there are some interesting questions here for further research.
2 The Quantity-Value Object ℛϕ{\cal R}_{\phi}
Let us turn now to the quantity-value object . This plays a key role in the representation of any physical quantity, , by an arrow . In so far as a ‘thing’ is a bundle of properties, these properties refer to values of physical quantities, and so the nature of these ‘values’ is of central importance.
We anticipate that has many global elements , and these can be interpreted as the possible ‘values’ for physical quantities. If also has global elements/microstates , then these combine with any arrow to give global elements of . It seems reasonable to refer to the element, as the ‘value’ of when the microstate is . However, our expectation is that, in general, may well have no global elements, in which case the interpretation of in terms of values is somewhat subtler. This has to be done internally using the language associated with the topos : the overall logical structure is a nice example of a ‘coherence’ theory of truth .
As far as axiomatic properties of are concerned, the minimal requirement is presumably that it should have some ordering property that arises in all topos representations of the system . This universal property could be coded into the internal language, , of . This implements our intuitive feeling that, in so far as the concept of ‘value’ has any meaning, it must be possible to say that the value of one quantity is ‘larger’ (or ‘smaller’) than that of another. It seems reasonable to expect this relation to be transitive, but that is about all. In particular, we see no reason to suppose that this relation will always correspond to a total ordering: perhaps there are pairs of physical quantities whose ‘values’ simply cannot be compared at all. Thus, tentatively, we can augment with the axioms for a poset structure on .
A more general perspective is given by the work of Heunen and Spitters . We recall that their starting point is a (non-commutative) -algebra, in an ambient topos . Then they construct the topos of co-presheaves, , and show that is an internal, pre -algebra in this topos. Finally, they construct the spectrum, , of and show that it is an internal locale.
This approach might be a useful tool when looking for ways of axiomatising . Thus, if in any topos representation , we assume that the state object is an internal local in , we can construct the internal topos and consider its interval-domain number object. It remains to be seen if this has any generic use in practice.
An attractive possibility is that there is a general analogue of (6.152) in the form
Conclusion
In this long article we have developed the idea that, for any given theory-type (classical physics, quantum physics, DI-physics,…) the theory of a particular physical system, , is to be constructed in the framework of a certain, system-dependent, topos. The central idea is that a local language, , is attached to each system , and that the application of a given theory-type to is equivalent to finding a representation, , of in a topos ; this is equivalent to finding a translation of into the internal language associated with ; or a functor to from the topos associated with .
Physical quantities are represented by arrows in the topos from the state object to the quantity-value object , and propositions are represented by sub-objects of the state object. The idea of a ‘truth sub-object’ of (or a ‘pseudo-state’ sub-object of ) then leads to a neo-realist interpretation of propositions in which each proposition is assigned a truth value that is a global element of the sub-object classifier . In general, neo-realist statements about the world/system are to be expressed in the internal language of the topos . Underlying this is the intuitionistic, deductive logic provided by the local language .
These axioms are based on ideas from the topos representation of quantum theory, which we have discussed in depth. Here, the topos involved is : the topos of presheaves over the base category of commutative, von Neumann sub-algebras of the algebra, , of all bounded operators on the quantum Hilbert space, . Each such sub-algebra can be viewed as a context in which the theory can be viewed from a classical perspective. Thus a context can be described as a ‘classical snap-shot’, or ‘window on reality’, or ‘world-view’/weltanschauung. Mathematically, a context is a ‘stage of truth’: a concept that goes back to Kripke’s use of a presheaf topos as a model of his intuitionistic view of time and process.
Every classical system uses the same topos, . However, in general, the topos will be system dependent as, for example, is the case with the quantum topoi of the form , where is the Hilbert space of the system. This leads to the problem of understanding how the topoi for a class of systems behave under the action of taking a sub-system, or combining a pair of systems to give a single composite system. We have presented a set of axioms that capture the general ideas we are trying to develop. Of course, these axioms are not cast in stone, and are still partly ‘experimental’ in nature. However, we have shown that classical physics exactly fits our suggested scheme, and that quantum physics ‘almost’ does: ‘almost’ because of the issues concerning the translation representation of the arrows associated with compositions of systems that were discussed in Section 13.3.
An important challenge for future work is to show that our general topos scheme can be used to develop genuinely new theories of physics, not just to rewrite old ones in a new language. Of particular interest is the problem with which we motivated the scheme in the first place: namely, to find tools for constructing theories that go beyond quantum theory and which do not use Hilbert spaces, path integrals, or any of the other familiar entities in which the continuum real and/or complex numbers play a fundamental role.
As we have discussed, the topoi for quantum systems are of the form , and hence embody contextual logic in a fundamental way. One way of going ‘beyond’ quantum theory, while escaping the a priori imposition of continuum concepts, is to use presheaves over a more general ‘category of contexts’, , i.e., develop the theory in the topos . Such a structure embodies contextual, multi-valued logic in an intrinsic way, and in that sense might be said to encapsulate one of the fundamental insights of quantum theory. However, and unlike in quantum theory, there is no obligation to use the real or complex numbers in the construction of the category .
Indeed, early on in this work we noted that real numbers arise in theories of physics in three different (but related) ways: (i) as the values of physical quantities; (ii) as the values of probabilities; and (iii) as a fundamental ingredient in models of space and time. The first of these is now subsumed into the quantity-value object , and which now has no a priori relation to the real number object in . The second source of real numbers has gone completely since we no longer have probabilities of propositions but rather generalised truth values whose values lie in . The third source is also no long binding since models of space and time in a topos could depend on many things: for example, infinitesimals.
Of course, although true, these remarks do not of themselves give a concrete example of a theory that is ‘beyond quantum theory’. On the other hand, these ideas certainly point in a novel direction, and one at which, almost certainly, we would not have arrived if the challenge to ‘go beyond quantum theory’ had been construed only in terms of trying to generalise Hilbert spaces, path integrals, and the like.
From a more general perspective, other types of topoi are possible realms for the construction of physical theories. One simple, but mathematically rich example arises from the theory of -sets. Here, is a monoid and, like all monoids, can be viewed as a category with a single object, and whose arrows are the elements of . Thought of as a category, a monoid is ‘complementary’ to a partially-ordered set. In a monoid, there is only one object, but plenty of arrows from that object to itself; whereas in a partially-ordered set there are plenty of objects, but at most one arrow between any pair of objects. Thus a partially-ordered set is the most economical category with which to capture the concept of ‘contextual logic’. On the other hand, the logic associated with a monoid is non-contextual as there is only one object in the category.
It is easy to see that a functor from to is just an ‘-set’: i.e., a set on which acts as a monoid of transformations. An arrow between two such -sets is an equivariant map between them. In physicists’ language, one would say that the topos —usually denoted — is the category of the ‘non-linear realisations’ of .
The sub-object classifier, , in is the collection of left ideals in ; hence, many of the important constructions in the topos can be handled using the language of algebra. The topos is one of the simplest to define and work with and, for that reason, it is a popular source of examples in texts on topos theory. It would be intriguing to experiment with constructing model theories of physics using one of these simple topoi. One possible use of -sets is discussed in in the context of reduction of the state vector, but there will surely be others.
It is clear that there are many other topics for future research. A question that is of particular interest is if there is a single topos within which all systems of a given theory-type can be discussed. For example, in the case of quantum theory the relevant topoi are of the form , where is a Hilbert space, and the question is whether all such topoi can be gathered together to form a single topos (what Grothendieck termed ‘un gros topos’) within which all quantum systems can be discussed.
There are well-known examples of such constructions in the mathematical literature. For example, the category, , of sheaves on a topological space is a topos, and there are collections of topological spaces which form a Grothendieck site, so that the topos can be constructed. A particular object in will then be a sheaf over whose stalk over any object in will be the topos .
For our purposes, the ideal situation would be if the various categories of systems, , can be chosen in such a way that is a site. Then the topos of sheaves, over this site would provide a common topos in which all systems of this theory type—i.e., the objects of —can be discussed. We do not know if this is possible, and it is a natural subject for future study.
At a conceptual level, one motivating desire for the entire research programme was to find a formalism that would always give some sort of ‘realist’ interpretation, even in the case of quantum theory which is normally presented in an instrumentalist way. But this particular example raises an interesting point because the neo-realist interpretation takes place in the topos , whereas the instrumentalist interpretation works in the familiar topos of sets, and one might wonder how universal is the use of a pair of topoi in this way.
In general, if we have an example of our axioms working neo-realistically in a topos , one might wonder if there is an ‘instrumentalist’ interpretation of the same theory in a different topos, , say? Of course, the word ‘instrumentalism’ is used metaphorically here, and any serious consideration of such a pair would require a lot of very careful thought.
However, if a pair does exist, the question then arises of whether there is a categorial way of linking the neo-realist and instrumentalist interpretations: for example, via a functor . If so, is this related to some analogue of the daseinisation operation that produced the representation of the -propositions, in quantum theory? Care is needed in discussing such issues since informal set theory is used as a meta-language in constructing a topos, and one has to be careful not to confuse this with the existence, or otherwise, of an ‘instrumentalist’ interpretation of any given representation.
If such a functor, , did exist then one could speculate on the possibility of finding an ‘interpolating chain’ of functors
which could be interpreted conceptually as corresponding to an interpolation between the philosophical views of realism and instrumentalism!
Even more speculatively one might wonder if “one person’s realism is another person’s instrumentalism”. More precisely, given a pair in the sense above, could there be cases in which the topos carries a neo-realist interpretation of a theory with respect to an instrumentalist interpretation in , whilst being the carrier of an instrumentalist interpretation with respect to the neo-realism of a ‘higher’ topos; and so on? For example, is there some theory whose ‘instrumentalist manifestation‘ takes place in the topos ?
On the other hand, one might want to say that ‘instrumentalist’ interpretations always take place in the world of classical set theory, so that should always be chosen to be . In any event, it would be interesting to study the quantum case more closely to see if there are any categorial relations between the formulation in and the instrumentalism interpretation in . It can be anticipated that the action of daseinisation will play an important role here.
All this is, perhaps To be honest, the ‘perhaps’ should really be replaced by ‘highly’., rather speculative but there is a more obvious situation in which a double-topos structure will be necessary, irrespective of philosophical musings on instrumentalism. This is if one wants to discuss the ‘classical limit’ of some topos theory. In this case this limit will exist in the topos, , and this must be used in addition to the topos of the basic theory. A good example of this, of course, is the topos of quantum theory discussed in this article. If, pace Landsmann, one thinks of ‘quantisation’ as a functor from to , then the classical limit will perhaps involve a functor going in the opposite direction.
A serious claim stemming from our work is that a successful theory of quantum gravity should be constructed in some topos —the ‘topos of the universe’—that is not the topos of sets. All entities of physical interest will be represented in this topos, including models for space-time (if there are any at a fundamental level in quantum gravity) and, if relevant, loops, membranes etc. as well as incorporating the anticipated generalisation of quantum theory.
Such a theory of quantum gravity will have a neo-realist interpretation in the topos , and hence would be particularly useful in the context of quantum cosmology. However, in practice, physicists divide the world up into smaller, more easily handled, chunks, and each of them would correspond to what earlier we have called a ‘system’ and, correspondingly, would have its own topos. Thus is something like the ‘gros topos’ of the theory, and would combine together the individual ‘sub-systems’ in a categorial way. Of course, it is most unlikely that there is any preferred way of dividing the universe up into bite-sized chunks, but this is not problematic as the ensuing relativism is naturally incorporated into the idea of a Grothendieck site.
Appendix 1: Some Theorems and Constructions Used in the Main Text
The collection, , of all clopen sub-objects of is a Heyting algebra.
Proof. First recall how a Heyting algebra structure is placed on the set, , of all sub-objects of .
Let be two sub-objects of . Then the ‘’ and ‘’ operations are defined by
for all contexts . It is easy to see that if and are clopen sub-objects of , then so are and .
The zero element in the Heyting algebra is the empty sub-object , where is the empty subset of . The unit element in is . It is clear that both and are clopen sub-objects of .
The most interesting part is the definition of the implication . For all , it is given by
Since , the expression for negation follows from the above as
where denotes the complement of in . Clearly, is clopen in since is clopen. Since the restriction is continuous and surjective See proof of Theorem 16.2 below., it is easy to see that the inverse image is clopen in . Clearly,
The problem is that we want to be a clopen subset of . Now the right hand side of (16.460) is the intersection of a family, parameterised by of clopen sets. Such an intersection is always closed, but it is only guaranteed to be open if is a finite set, which of course may not be the case.
If and , then . Indeed, if we had , then by the definition of a sub-object, so we would have a contradiction. This implies , and hence the right hand side of (16.460) is a decreasing net of clopen subsets of which converges to something, which we take as the subset of that is to be .
Here we have used the fact that the set of clopen subsets of is a complete lattice, where the minimum of a family of clopen subsets is defined as the interior of . This leads us to define
as the negation in . This modified definition guarantees that is a clopen sub-object. A straightforward extension of this method gives a consistent definition of .
The following theorem shows the relation between the restriction mappings of the outer presheaf and those of the spectral presheaf . We basically follow de Groote’s proof of Prop. 3.22 in and show that this result, which uses quite a different terminology, actually gives the desired relation.
Let such that . Then
Proof. First of all, to simplify notation, we can replace by (which amounts to the assumption that . This does not play a rôle for the current argument). By definition, , so we have to show that holds.
If , then , which implies for all . In particular, , so . This shows that .
To show the converse inclusion, let , which means that . We have . Let
As shown in section 8.3, is an ultrafilter in the projection lattice . In general, each ultrafilter in the projection lattice of an abelian von Neumann algebra corresponds to a unique element of the Gel’fand spectrum of . The ultrafilter is the collection of all those projections that are mapped to by , i.e., . The idea is to show that is a filter base in that can be extended to an ultrafilter, which corresponds to an element of the Gel’fand spectrum of .
Let us assume that is not a filter base in . Then there exists some such that
which implies , so
By Zorn’s lemma, the filter base is contained in some (not necessarily unique) maximal filter base in . Such a maximal filter base is an ultrafilter and thus corresponds to an element of the Gel’fand spectrum of . Since is contained in the ultrafilter, we have , so . By construction, , the element of we started from. This shows that , and we obtain
It is well-known that every state is of the form for some . This implies
Note that on the right hand side, (and not , which is a smaller set in general) shows up.
De Groote has shown in that for any unital abelian von Neumann algebra , the clopen sets , , form a base of the Gel’fand topology on . Formulas (16.468) and (16.469) hence show that the restriction mappings
of the spectral presheaf are open and continuous. Using continuity, it is easy to see that is also closed: let be a closed subset. Since is compact, is compact, and since is continuous, is compact, too. However, is Hausdorff, and so is closed in .
2 The Grothendieck kk-Construction for an Abelian Monoid
Let us briefly review the Grothendieck construction for an abelian monoid .
A group completion of is an abelian group together with a monoid map that is universal. Namely, given any monoid morphism , where is an abelian group, there exists a unique group morphism such that factors through ; i.e., we have the commutative diagram
It is easy to see that any such is unique up to isomorphism.
To prove existence, first take the set of all pairs , each of which is to be thought of heuristically as . Then, note that if inverses existed in , we would have if and only if . This suggests defining an equivalence relation on in the following way:
The Grothendieck completion of an abelian monoid is the pair defined as follows:
is the set of equivalence classes , where the equivalence relation is defined in (16.470). A group law on is defined by
where is the unit in the abelian monoid .
The map is defined by
It is straightforward to show that (i) these definitions are independent of the representative elements in the equivalence classes; (ii) the axioms for a group are satisfied; and (iii) the map is universal in the sense mentioned above.
It is also clear that is a functor from the category of abelian monoids to the category of abelian groups. For, if is a morphism between abelian monoids, define by for all .
which is well-defined on equivalence classes.
of proper subsets, and define the variation of on this chain to be
where we set . Now take the supremum of for all such chains . If this is finite, we say that has a bounded variation and define
Then it is clear that (i) is an order-preserving function on ; (ii) is an order-reversing function on ; and (iii) is an order-reversing function on . Thus, any function, , of bounded variation can be written as
This suggests the following strategy. First, define functions and by
Clearly, for all . Also, for all , , and hence
However, (i) the function is order-reversing; and (ii) the function is order-preserving. But then is order-reversing. Hence, by rewriting (16.485) as
On objects :
where denotes the -equivalence class of .
5.3 The square of an arrow [δ˘o(A^)][\breve{\delta}^{o}(\hat{A})].
Appendix 2: A Short Introduction to the Relevant Parts of Topos Theory
It is impossible to give here more than the briefest of introductions to topos theory. At the danger of being highly imprecise, we restrict ourselves to mentioning some aspects of this well-developed mathematical theory and give a number of pointers to the literature. The aim merely is to give a very rough idea of the structure and internal logic of a topos.
There are a number of excellent textbooks on topos theory, and the reader should consult at least one of them. We found the following books useful: .
Topos theory is a remarkably rich branch of mathematics which can be approached from a variety of different viewpoints. The basic area of mathematics is category theory; where, we recall, a category consists of a collection of objects and a collection of morphisms (or arrows).
In the special case of the category of sets, the objects are sets, and a morphism is a function between a pair of sets. In general, each morphism in a category is associated with a pair of objects The collection of all objects in category, , is denoted . The collection of arrows from to is denoted . We will only be interested in ‘small’ categories in which both these collections are sets (rather than the, more general, classes.), known as its ‘domain’ and ‘codomain’, and is written as where and are the domain and codomain respectively. Note that this arrow notation is used even if is not a function in the normal set-theoretic sense. A key ingredient in the definition of a category is that if and (i.e., the codomain of is equal to the domain of ) then and can be ‘composed’ to give an arrow ; in the case of the category of sets, this is just the usual composition of functions.
A simple example of a category is given by any partially-ordered set (‘poset’) : (i) the objects are defined to be the elements of ; and (ii) if , a morphism from to is defined to exist if, and only if, in the poset structure. Thus, in a poset regarded as a category, there is at most one morphism between any pair of objects ; if it exists, we shall write this morphism as . This example is important for us in form of the ‘category of contexts’, , in quantum theory. The objects in are the commutative, unital ‘Unital’ means that all these algebras contain the identity operator . von Neumann sub-algebras of the algebra, , of all bounded operators on the Hilbert space .
Every (elementary) topos can be seen as a mathematical universe. As a category, a topos possesses a number of structures that generalise constructions that are possible in the category, , of sets and functions. More precisely, small sets and functions between them. Small means that we do not have proper classes. One must take care in these foundational issues to avoid problems like Russell’s paradox. Namely, in , we can construct new sets from given ones in several ways. Specifically, let be two sets, then we can form the cartesian product , the disjoint union and the exponential —the set of all functions from to .
These constructions turn out to be fundamental, and they can all be phrased in an abstract, categorical manner, where they are called the ‘product’, ‘co-product’ and ‘exponential’, respectively. By definition, in a topos , these operations always exist. The first and second of these properties are called ‘finite completeness’ and ‘finite co-completeness’, respectively.
One consequence of the existence of finite limits is that each topos, , has a terminal object, denoted by . This is characterised by the property that for any object in the topos , there exists exactly one arrow from to . In , any one-element set is terminal. Like many categorical constructions, the terminal object is fixed only up to isomorphism: all one-element sets are isomorphic to each other, and any of them can serve as a terminal object. Nonetheless, one speaks of the terminal object.
Of course, is a topos, too, and it is precisely the topos which usually plays the rôle of our mathematical universe, since we construct our mathematical objects starting from sets and functions between them. As a slogan, we have: a topos is a category with ‘certain crucial’ properties that are similar to those in . A very nice and gentle introduction to these aspects of topos theory is the book . Other good sources are .
In order to ‘do mathematics’, one must also have a logic, including a deductive system. Each topos comes equipped with an internal logic, which is of intuitionistic type. We will now very briefly sketch the main characteristics of intuitionistic logic and the mathematical structures in a topos that realise this logic.
Let be a set, and let be the power set of ; i.e., the set of subsets of . Given a subset , one can ask for each point whether or not it lies in . Thus there is the characteristic function of , which is defined as
for all ; cf. (6.95). The two-element set plays the rôle of a set of truth-values for propositions (of the form “”). Clearly, corresponds to ‘true’, corresponds to ‘false’, and there are no other possibilities. This is an argument about sets, so it takes place in, and uses the logic of, the topos of sets and functions. is a Boolean topos, in which the familiar two-valued logic and the axiom () hold. (This does not contradict the fact that the internal logic of topoi is intuitionistic, since Boolean logic is a special case of intuitionistic logic.)
In an arbitrary topos, , there is a special object , called the sub-object classifier, that takes the rôle of the set of truth-values. Let be an object in the topos, and let be a sub-object of . This means that there is a monic , A monic is the categorical version of an injective function. In the topos , monics exactly are injective functions. (this is the categorical generalisation of the inclusion of a subset into a larger set ). As in the case of , we can also characterise as a sub-object of by an arrow from to the sub-object classifier ; in , this arrow is the characteristic function of (17.492). Intuitively, this ‘characteristic arrow’ from to describes how ‘lies in’ . The textbook definition is:
In a category with finite limits, a sub-object classifier is an object , together with a monic , such that to every monic in there is a unique arrow which, with the given monic, forms a pullback square
In , the arrow is given by . In general, the sub-object classifier, , need not be a set, since it is an object in the topos , and the objects of need not be sets. Nonetheless, there is an abstract notion of elements (or points) in category theory that we can use. Then the elements of are the truth-values available in the internal logic of our topos , just like ‘false’ and ‘true’, the elements of , are the truth-values available in the topos .
To understand the abstract notion of elements, let us consider sets for a moment. Let be a one-element set, the terminal object in . Let be a set and consider an arrow from to . Clearly, (i) is an element of ; and (ii) the set of all functions from to corresponds exactly to the set of all elements of .
This idea can be generalised to any category that has a terminal object . More precisely, an element of an object is defined to be an arrow from to in the category. For example, in the definition of the sub-object classifier the arrow ‘’ is an element of . It may happen that an object has no elements, i.e., there are no arrows . It is common to consider arrows from subobjects of to as generalised elements.
As mentioned above, the elements of the sub-object classifier, understood as the arrows , are the truth-values. Moreover, the set of these arrows forms a Heyting algebra (see, for example, section 8.3 in ). This is how (the algebraic representation of) intuitionistic logic manifests itself in a topos. Another, closely related fact is that the set, , of sub-objects of any object in a topos forms a Heyting algebra.
Let us pull together these various remarks and list the most important properties of a topos, , for our purposes:
There is a terminal object in . Thus, given any object in the topos, there is a unique arrow .
For any object in the topos, an arrow is called a global element of . The set of all global elements of is denoted .
Given , there is a product in . In fact, a topos always has pull-backs, and the product is just a special case of this. The conditions in 1. above are equivalent to saying that is finitely complete.
There is an initial object in . This means that given any object in the topos, there is a unique arrow .
Given , there is a co-product in . In fact, a topos always has push-outs, and the co-product is just a special case of this. The conditions in 2. above are equivalent to saying that is finitely co-complete.
There is exponentiation: i.e., given objects in we can form an object , which is the topos analogue of the set of functions from to in set theory. The definitive property of exponentiation is that, given any object , there is an isomorphism
that is natural in and ; i.e., it is ‘well-behaved’ under morphisms of the objects involved.
There is a sub-object classifier .
2 Presheaves on a Poset
To illustrate the main ideas, we will first give a few definitions from the theory of presheaves on a partially ordered set (or ‘poset’); in the case of quantum theory, this poset is the space of ‘contexts’ in which propositions are asserted. We shall then use these ideas to motivate the definition of a presheaf on a general category. Only the briefest of treatments is given here, and the reader is referred to the standard literature for more information .
A presheaf (also known as a varying set) on a poset is a function that assigns to each , a set ; and to each pair (i.e., ), a map such that (i) is the identity map on , and (ii) whenever , the composite map is equal to , i.e.,
The notation is shorthand for the more cumbersome ; see below in the definition of a functor.
An arrow, or natural transformation between two presheaves on is a family of maps , , that satisfy the intertwining conditions
whenever . This is equivalent to the commutative diagram
It follows from these basic definitions, that a sub-object of a presheaf is a presheaf , with an arrow such that (i) for all ; and (ii) for all , the map is the restriction of to the subset . This is shown in the commutative diagram
where the vertical arrows are subset inclusions.
The collection of all presheaves on a poset forms a category, denoted . The arrows/morphisms between presheaves in this category the arrows (natural transformations) defined above.
3 Presheaves on a General Category
The ideas sketched above admit an immediate generalization to the theory of presheaves on an arbitrary ‘small’ category (the qualification ‘small’ means that the collection of objects is a genuine set, as is the collection of all arrows/morphisms between any pair of objects). To make the necessary definition we first need the idea of a ‘functor’:
A central concept is that of a ‘functor’ between a pair of categories and . Broadly speaking, this is an arrow-preserving function from one category to the other. The precise definition is as follows.
A covariant functor from a category to a category is a function that assigns
to each -object , a -object ;
to each -morphism , a -morphism such that ; and, if , and then
A contravariant functor from a category to a category is a function that assigns
to each -object , a -object ;
to each -morphism , a -morphism such that ; and, if , and then
The connection with the idea of a presheaf on a poset is straightforward. As mentioned above, a poset can be regarded as a category in its own right, and it is clear that a presheaf on the poset is the same thing as a contravariant functor from the category to the category of normal sets. Equivalently, it is a covariant functor from the ‘opposite’ category The ‘opposite’ of a category is a category, denoted , whose objects are the same as those of , and whose morphisms are defined to be the opposite of those of ; i.e., a morphism in is said to exist if, and only if, there is a morphism in . to . Clearly, (17.494) corresponds to the contravariant condition (17.499). Note that mathematicians usually call the objects in ‘stages of truth’, or just ‘stages’. For us they are ‘contexts’, ‘classical snap-shops’, or ‘world views’.
These remarks motivate the definition of a presheaf on an arbitrary small category : namely, a presheaf on is a covariant functor Throughout this series of papers, a presheaf is indicated by a letter that is underlined. from to the category of sets. Equivalently, a presheaf is a contravariant functor from to the category of sets.
We want to make the collection of presheaves on into a category, and therefore we need to define what is meant by a ‘morphism’ between two presheaves and . The intuitive idea is that such a morphism from to must give a ‘picture’ of within . Formally, such a morphism is defined to be a natural transformation , by which is meant a family of maps (called the components of ) , , such that if is a morphism in , then the composite map is equal to . In other words, we have the commutative diagram
of which (17.496) is clearly a special case. The category of presheaves on equipped with these morphisms is denoted .
The idea of a sub-object generalizes in an obvious way. Thus we say that is a sub-object of if there is a morphism in the category of presheaves (i.e., a natural transformation) with the property that, for each , the component map is a subset embedding, i.e., . Thus, if is any morphism in , we get the analogue of the commutative diagram (17.497):
where, once again, the vertical arrows are subset inclusions.
The category of presheaves on , , forms a topos. We do not need the full definition of a topos; but we do need the idea, mentioned in Section 17.2, that a topos has a sub-object classifier , to which we now turn.
Among the key concepts in presheaf theory is that of a ‘sieve’, which plays a central role in the construction of the sub-object classifier in the topos of presheaves on a category .
A sieve on an object in is defined to be a collection of morphisms in with the property that if belongs to , and if is any morphism with co-domain , then also belongs to . In the simple case where is a poset, a sieve on is any subset of such that if then (i) , and (ii) for all ; in other words, a sieve is nothing but a lower set in the poset.
The presheaf is now defined as follows. If is an object in , then is defined to be the set of all sieves on ; and if , then is defined as
for all ; the sieve is often written as , and is known as the pull-back to of the sieve on by the morphism .
It should be noted that if is a sieve on , and if belongs to , then from the defining property of a sieve we have
where denotes the principal sieve on , defined to be the set of all morphisms in whose codomain is .
If is a poset, the pull-back operation corresponds to a family of maps (where denotes the set of all sieves/lower sets on in the poset) defined by if (i.e., ). It is straightforward to check that if , then
where .
A crucial property of sieves is that the set of sieves on has the structure of a Heyting algebra. Specifically, the unit element in is the principal sieve , and the null element is the empty sieve . The partial ordering in is defined by if, and only if, ; and the logical connectives are defined as:
As in any Heyting algebra, the negation of an element (called the pseudo-complement of ) is defined as ; so that
It can be shown that the presheaf is a sub-object classifier for the topos . That is to say, sub-objects of any object in this topos (i.e., any presheaf on ) are in one-to-one correspondence with morphisms . This works as follows. First, let be a sub-object of with an associated characteristic arrow . Then, at any stage in , the ‘components’ of this arrow, , are defined as
for all . That the right hand side of (17.509) actually is a sieve on follows from the defining properties of a sub-object.
Thus, in each ‘branch’ of the category going ‘down’ from the stage , picks out the first member in that branch for which lies in the subset , and the commutative diagram (17.501) then guarantees that will lie in for all . Thus each stage in serves as a possible context for an assignment to each of a generalised truth value—a sieve belonging to the Heyting algebra . This is the sense in which contextual, generalised truth values arise naturally in a topos of presheaves.
There is a converse to (17.509): namely, each morphism (i.e., a natural transformation between the presheaves and ) defines a sub-object of via
For the category of presheaves on , a terminal object can be defined by at all stages in ; if is a morphism in then is defined to be the map . This is indeed a terminal object since, for any presheaf , we can define a unique natural transformation whose components are the constant maps for all .
As a morphism in the topos , a global element corresponds to a choice of an element for each stage in , such that, if , the ‘matching condition’
Acknowledgements
This research was supported by grant RFP1-06-04 from The Foundational Questions Institute (fqxi.org). AD gratefully acknowledges financial support from the DAAD.
This work is also supported in part by the EC Marie Curie Research and Training Network “ENRAGE” (European Network on Random GEometry) MRTN-CT-2004-005616.
We are both very grateful to Professor Hans de Groote for his detailed and insightful comments on our work.
CJI expresses his gratitude to Jeremy Butterfield for the lengthy, and most enjoyable, collaboration in which were formulated the early ideas about using topoi to study quantum theory.