Categories for the practising physicist

Prologue: cooking with vegetables

Consider a ‘raw potato’. Conveniently, we refer to it as $A$. Raw potato $A$ admits several states e.g. ‘dirty’, ‘clean’, ‘skinned’, … Since raw potatoes don’t digest well we need to process $A$ into ‘cooked potato’ $B$. We refer to $A$ and $B$ as kinds or types of food. Also $B$ admits several states e.g. ‘boiled’, ‘fried’, ‘baked with skin’, ‘baked without skin’, … Correspondingly, there are several ways to turn raw potato $A$ into cooked potato $B$ e.g. ‘boiling’, ‘frying’, ‘baking’, to which we respectively refer as $f$, $f^{\prime}$ and $f^{\prime\prime}$. We make the fact that each of these cooking processes applies to raw potato $A$ and produces cooked potato $B$ explicit via labelled arrows:

Sequential composition. A plain cooked potato tastes dull so we’d like to process it into ‘spiced cooked potato’ $C$. We refer to the composite process that consists of first ‘boiling’ $A\rTo^{f}B$ and then ‘salting’ $B\rTo^{g}C$ as

We refer to the trivial process of ‘doing nothing to vegetable $X$’ as

Clearly we have $1_{Y}\circ\xi=\xi\circ 1_{X}=\xi$ for all processes $X\rTo^{\xi}Y$. Note that there is a slight subtlety here: we need to specify what we mean by equality of cooking processes. We will conceive two cooking processes $X\rTo^{\xi}Y$ and $X\rTo^{\zeta}Y$ as equal, and write $\xi=\zeta$, if the resulting effect on each of the states which $X$ admits is the same. A stronger notion of equality arises when we also want some additional details of the processes to coincide e.g. the brand of the cooking pan that we use.

Let $D$ be a ‘raw carrot’. Note that it is indeed very important to explicitly distinguish our potato and our carrot and any other vegetable such as ‘lettuce’ $L$ in terms of their respective names $A$, $D$ and $L$, since each admits distinct ways of processing. And also a cooked potato admits different ways of processing than a raw one, for example, while we can mash cooked potatoes, we can’t mash raw ones. We denote all processes which turn raw potato $A$ into cooked potato $B$ by ${\bf C}(A,B)$. Consequently, we can repackage composition of cooking processes as a function

Parallel composition. We want to turn ‘raw potato’ $A$ and ‘raw carrot’ $D$ into ‘carrot-potato mash’ $M$. We refer to the fact that this requires (or consumes) both $A$ and $D$ as $A\otimes D$. Refer to ‘frying the carrot’ as $D\rTo^{h}E$. Then, by

we mean ‘boiling potato $A$’ while ‘frying carrot $D$’ and by

we mean ‘mashing spiced cooked potato $C$ and spiced cooked carrot $F$’.

Laws. The whole process from raw components $A$ and $D$ to ‘meal’ $M$ is

where ‘peppering the carrot’ is referred to as $E\rTo^{k}F$. We refer to the list of the operations that we apply, i.e. ($f$ while $h$, $g$ while $k$, $x$), as a recipe. Distinct recipes can yield the same meal. The reason for this is that the two operations ‘and then’ (i.e. $-\circ-$) and ‘while’ (i.e. $-\otimes-$) which we have at our disposal are not totally independent but interact in a certain way. This is exemplified by the equality

on cooking processes, which states that it makes no difference whether ‘we first boil the potato and then fry the carrot’, or, ‘first fry the carrot and then boil the potato’.

Eq.(1) is in fact a generally valid equational law for cooking processes, which does not depend on specific properties of $A,B,D,E,f$ nor $h$.

Of course, chefs do not perform computations involving eq.(1), since their ‘intuition’ accounts for its content. But, if we were to teach an android how to become a chef, which would require it/him/her to reason about recipes, then we would need to teach it/him/her the laws governing these recipes.

In fact, there is a more general law governing cooking processes from which eq.(1) can be derived, namely,

That is, ‘boiling the potato and then salting it, while, frying the carrot and then peppering it’, is equal to ‘boiling the potato while frying the carrot, and then, salting the potato while peppering the carrot’. In the light of the previous footnote, note here that this law applies to any reasonable notion of equality for processes. A proof of the fact that eq.(1) can be derived from eq.(2) is in Proposition 2 below.

Logic. Eq.(2) is indeed a logical statement. In particular, note the remarkable similarity, but at the same time also the essential difference, of eq.(2) with the well-known distributive law of classical logic, which states that

For simple situations, if one possesses enough brainpower, ‘intuition’ again accounts for this distributive law. Ot the other hand, it needs to be explicitly taught to androids, since this distributive law is key to the resolution method which is the standard implementation of artificial reasoning in AI and robotics resolution_method. Also for complicated sentences we ourselves will need to rely on this method too.

The $(\circ,\otimes)$-logic is a logic of interaction. It applies to cooking processes, physical processes, biological processes, logical processes (i.e. proofs), or computer processes (i.e. programs). The theory of monoidal categories, the subject of this chapter, is the mathematical framework that accounts for the common structure of each of these theories of processes. The framework of monoidal categories moreover enables modeling and axiomatising (or ‘classify’) the extra structure which certain families of processes may have. For example, how cooking processes differ from physical processes, and how quantum processes differ from classical processes.

Pictures. We mentioned that our intuition accounts for $(\circ,\otimes)$-logic. Wouldn’t it be nice if there would be mathematical structures which also ‘automatically’ (or ‘implicitly’) account for the logical mechanisms which we intuitively perform? Well, these mathematical structures do exist. While they are only a fairly recent development, they are becoming more and more prominent in mathematics, including in important ‘Fields Medal awarding areas’ such as algebraic topology and representation theory — see for example (Morrison, and references therein). Rather than being symbolic, these mathematical structures are purely graphical. Indeed, by far ***the*** coolest thing about monoidal categories is that they admit a purely pictorial calculus, and these pictures automatically account for the logical mechanisms which we intuitively perform. As pictures, both sides of eq.(2) become:

Hence eq.(2) becomes an implicit salient feature of the graphical calculus and needs no explicit attention anymore. This, as we will see below, substantially simplifies many computations. To better understand in which manner these pictures simplify computations note that the differences between the two sides of eq.(2) can be recovered by introducing ‘artificial’ brackets within the two pictures:

A detailed account on this graphical calculus is in Section 2.2.

In the remainder of this chapter we provide a formal tutorial on several kinds of monoidal categories that are relevant to physics. If you’d rather stick to the informal story of this prologue you might want to first take a bite of C2005c; C2005d. Paper C2005c provided a conceptual template for setting up the content of this paper. However, here we go in more detail and provide more examples. Section 1 introduces categories and Section 2 introduces tensor structure. Section 3 studies quantum-like tensors and Section 4 studies classical-like tensors. Section 5 introduces mappings between monoidal categories (= monoidal functors), and natural transformations between these, which enable to concisely define topological quantum field theories. Section 6 suggests further reading.

The 1D case: New arrows for your quiver

The bulk of the previous section discussed the two manners in which we can compose processes, namely sequentially and in parallel, or more physically put, in time and in space. These are indeed the situations we truly care about in this chapter. Historically however, category theoreticians cared mostly about one-dimensional fragments of the two-dimensional monoidal categories. These one-dimensional fragments are (ordinary) categories, hence the name category theory. Some people will get rebuked by the terminology and particular syntactic language used in category theory — which can sound and look like unintelligible jargon — resulting in its unfortunate label of generalised abstract nonsense. The reader should realise that initially category theory was crafted as ‘a theory of mathematical structures’. Hence substantial effort was made to avoid any reference to the underlying concrete models, resulting in its seemingly idiosyncratic format. The personalities involved in crafting category theory, however brilliant minds they had, also did not always help the cause of making category theory accessible to a broader community.

But this ‘theory of mathematical structures’ view is not the only way to conceive category theory. As we argued above, and as is witnessed by its important use in computer science, in proof theory, and more recently also in quantum informatics and in quantum foundations, category theory is a theory which brings the notions of (type of ) system and process to the forefront, two notions which are hard to cast within traditional monolithic mathematical structures.

We profoundly believe that the fact that the mainstream physics community has not yet acquired this (type of) systems/process structure as a primal part of its theories is merely accidental, and temporary, … and will soon change.

We will use the following syntax to denote a function:

where $X$ is the set of arguments, $Y$ the set of possible values, and

means that argument $x$ is mapped on value $y$.

A family Typically, ‘family’ will mean a class rather than a set. While for many constructions the size of $|{\bf C}|$ is important, it will not play a key role in this paper. $|{\bf C}|$ of objects ;

For any $A,B\in|{\bf C}|$, a set ${\bf C}(A,B)$ of morphisms, the hom-set ;

For any $A,B,C\in|{\bf C}|$, and any $f\in{\bf C}(A,B)$ and $g\in{\bf C}(B,C)$, a composite $g\circ f\in{\bf C}(A,C)$, i.e., for all $A,B,C\in|{\bf C}|$ there is a composition operation

and this composition operation is associative and has units, that is,

for any $f\in{\bf C}(A,B)$, $g\in{\bf C}(B,C)$ and $h\in{\bf C}(C,D)$ we have

for any $A\in|{\bf C}|$, there exists a morphism $1_{A}\in{\bf C}(A,A)$, called the identity, which is such that for any $f\in{\bf C}(A,B)$ we have

A shorthand for $f\in{\bf C}(A,B)$ is $A\rTo^{f}B$. As already mentioned above, this definition was proposed by Samuel Eilenberg and Saunders Mac Lane in 1945 as part of a framework which intended to unify a variety of mathematical constructions within different areas of mathematics EilenbergMacLane. Consequently, most of the examples of categories that one encounters in the literature encode mathematical structures: the objects will be examples of this mathematical structure and the morphisms will be the structure-preserving maps between these. This kind of categories is usually referred to as concrete categories Adamek. We will also call them concrete categorical models.

2 Concrete categories

Traditionally, mathematical structures are defined as a set equipped with some operations and some axioms, for instance:

A group is a set $G$ with an associative binary operation $-\bullet-:G\times G\rightarrow G$ and with a two-sided identity $1\in G$, relative to which each element is invertible, that is, for all $g\in G$ there exists $g^{-1}\in G$ such that $g\bullet g^{-1}=g^{-1}\bullet g=1$.

Similarly we define rings and fields. Slightly more involved but in the same spirit:

It is to these operations and axioms that one usually refers to as structure. Functions on the underlying sets which preserve (at least part of) this structure are called structure preserving maps. Here are some examples of structure preserving maps:

group homomorphisms, i.e. functions which preserve the group multiplication, from which it then also follows that the unit and inverses are preserved;

linear maps, i.e. functions from a vector space to a vector space which preserve linear combinations of vectors.

Let ${\bf Set}$ be the concrete category with:

all functions between sets as morphisms, that is, more precisely, if $X$ and $Y$ are sets and $f:X\to Y$ is a function between these sets, then $f\in{\bf Set}(X,Y)$,

ordinary composition of functions, that is, for $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ we have $(g\circ f)(x):=g(f(x))$ for the composite $g\circ f:X\to Z$, and,

the obvious identities i.e. $1_{X}(x):=x$.

function composition is associative, and,

for any function $f:X\rightarrow Y$ we have $(1_{Y}\circ f)(x)=f(x)=(f\circ 1_{X})(x)$ .

all linear maps between these vectors spaces as morphisms, and

ordinary composition of the underlying functions, and,

the composite of two linear maps is again a linear map, and,

${\bf Grp}$ is the concrete category with:

group homomorphisms between these groups as morphisms, and,

the composite of two group homomorphisms is a group homomorphism, and,

identity functions are group homomorphisms.

Above we explained that mathematical structures such as groups typically consist of a set with additional structure. In the case of a category we have a collection of objects, and for each pair of objects a set of morphisms. The ‘structure of a category’ then consists of the composition operation on morphisms and the identities on objects. So there is no reference to what the individual objects actually are (e.g. a set, a vector space, or a group). Consequently, one would expect that when passing from a mathematical structure (cf. group) to the corresponding concrete category with these mathematical structures as objects (cf. ${\bf Grp}$), one looses the object’s ‘own’ structure. But fortunately, this happens not to be the case. The fact that we consider structure preserving maps as morphisms will allow us to recover the mathematical structures that we started from. In particular, by only relying on categorical concepts we are still able to identify the ‘elements’ of the objects.

For the set $X\in|{\bf Set}|$ and some chosen element $x\in X$ the function

where $\{*\}$ is any one-element set, maps the unique element of $\{*\}$ onto the chosen element $x$. If $X$ contains $n$ elements, then there are $n$ such functions each corresponding to the element on which $*$ is mapped. Hence the elements of the set $X$ are now encoded as the set ${\bf Set}(\{*\},X)$.

${\bf Pos}$ is the concrete category with:

partially ordered sets, that is, a set together with a reflexive, anti-symmetric and transitive relation, as objects,

order preserving maps, i.e. $x\leq y\Rightarrow f(x)\leq f(y)$, as morphisms, and,

ordinary function composition, and identity functions.

An extended version of this category is ${\bf Pre}$ where we consider arbitrary pre-ordered sets, that is, a set together with a reflexive and transitive relation.

${\bf Cat}$ is the concrete category with: In order to conceive ${\bf Cat}$ as a concrete category, the family of objects should be restricted to the so-called “small” categories i.e., categories for which the family of objects is a set.

so-called functors between these as morphisms (see Section 1.6), and,

functor composition, and identity functors.

3 Real world categories

But viewing category theory as some kind of metatheory about mathematical structure is not necessarily the most useful perspective for the sort of applications that we have in mind. Indeed, here are a few examples of the kind of categories we truly care about, and which are not categories with mathematical structures as objects and structure preserving maps as morphisms.

all physical systems $A,B,C,\ldots$ as objects,

all physical processes which take a physical system of type $A$ into a physical system of type $B$ as the morphisms of type $A\rTo B$ (these processes typically require some finite amount of time to be completed), and,

sequential composition of these physical processes as composition, and the process which leaves system $A$ invariant as the identity $1_{A}$.

Note that in this case associativity of composition admits a physical interpretation: if we first have process $f$, then process $g$, and then process $h$, it doesn’t matter whether we either consider $(g\circ f)$ as a single entity after which we apply $h$, or whether we consider $(h\circ g)$ as a single entity which we apply after $f$. Hence brackets constitute superfluous data that can be omitted i.e.

The category $\mathbf{PhysOpp}$ is an operational variant of the above where, rather than general physical systems such as stars, we focus on systems which can be manipulated in the lab, and rather than general processes, we consider the operations which the practising experimenter performs on these systems, for example, applying force-fields, performing measurements etc.

The category $\mathbf{QuantOpp}$ is a restriction of the above where we restrict ourselves to quantum systems and operations thereon. Special processes in $\mathbf{QuantOpp}$ are preparation procedures, or states. If $Q$ denotes a qubit, then the type of a preparation procedure would be ${\rm I}\rTo Q$ where ${\rm I}$ stands for ‘unspecified’. Indeed, the point of a preparation procedure is to provide a qubit in a certain state, and the resources which we use to produce that state are typically not of relevance for the remainder of the experimental procedure. We can further specialise to either pure (or closed) quantum systems or mixed (or open) quantum systems, categories to which we respectively refer as $\mathbf{PurQuantOpp}$ and $\mathbf{MixQuantOpp}$.

Obviously, Example 9 is related to the concrete category which has Hilbert spaces as objects and certain types of linear mappings (e.g. completely positive maps) as morphisms. The preparation procedures discussed above then correspond to ‘categorical elements’ in the sense of Example 4. We discuss this correspondence below.

While to the sceptical reader the above examples still might not seem very useful yet, the next two ones, which are very similar, have become really important for Computer Science and Logic. They are the reason that, for example, University of Oxford Computing Laboratory offers category theory to its undergraduates.

all data types, e.g. Booleans, integers, reals, as objects,

all programs which take data of type $A$ as their input and produce data of type $B$ as their output as the morphisms of type $A\rTo B$, and,

sequential composition of programs as composition, and the programs which output their input unaltered as identities.

all proofs which conclude from proposition $A$ that proposition $B$ holds as the morphisms of type $A\rTo B$, and,

concatenation (or chaining) of proofs as composition, and the tautologies ‘from $A$ follows $A$’ as identities.

Computer scientists particularly like category theory because it explicitly introduces the notion of type: an arrow $A\rTo^{f}B$ has type $A\rTo B$. These types prevent silly mistakes when writing programs, e.g. the composition $g\circ f$ makes no sense for $C\rTo^{g}D$ because the output — called the codomain — of $f$ doesn’t match the input — called the domain — of $g$. Computer scientists would say:

Similar categories ${\bf BioProc}$ and ${\bf ChemProc}$ can be build for organisms and biological processes, chemicals and chemical reactions, etc. The first time the 1st author heard about categories was in a Philosophy of Science course, given by a biologist specialised in population dynamics, who discussed the importance of category theory in the influential work of Robert Rosen Rosen. The recipe for producing these categories is obvious:

Composition boils down to ‘first $f$ and then $g$ happens’ and identities are just ‘nothing happens’. Somewhat more operationally put, composition is ‘first do $f$ and then do $g$’ and identities are just ‘doing nothing’. The reason for providing both the ‘objectivist’ (= passive) and ‘instrumentalist’ (= active) perspective is that we both want to appeal to members of the theoretical physics community and members of the quantum information community. The first community typically doesn’t like instrumentalism since it just doesn’t seem to make sense in the context of theories such as cosmology; on the other hand, instrumentalism is as important to quantum informatics as it is to ordinary informatics. We leave it up to the reader to decide whether it should play a role in the interpretation of quantum theory.

4 Abstract categorical structures and properties

One can treat categories as mathematical structures in their own right, just as groups and vector spaces are mathematical structures. In contrast with concrete categories, abstract categorical structures then arise by either endowing categories with more structure or by requiring them to satisfy certain properties.

We are of course aware that this is not a formal definition. Our sheepish excuse is that physicists rarely provide precise definitions. There is however a formal definition which can be found in Adamek. We do provide one below in Example 24.

A monoid $(M,\bullet,1)$ is a set together with a binary associative operation

which admits a unit — i.e. a ‘group without inverses’. Equivalently, we can define a monoid as a category ${\bf M}$ with a single object $*$. Indeed, it suffices to identify

the elements of the hom-set ${\bf M}(*,*)$ with those of $M$,

with the associative monoid multiplication $\bullet$, and

the identity $1_{*}:*\rightarrow*$ with the unit $1$.

Dually, in any category ${\bf C}$, for any $A\in|{\bf C}|$, the set ${\bf C}(A,A)$ is always a monoid.

Two objects $A,B\in|{\bf C}|$ are isomorphic if there exists morphisms $f\in{\bf C}(A,B)$ and $g\in{\bf C}(B,A)$ such that $g\circ f=1_{A}$ and $f\circ g=1_{B}$. The morphism $f$ is called an isomorphism and $f^{-1}:=g$ is called the inverse to $f$.

The notion of isomorphism known to the reader is the set-theoretical one, namely that of a bijection. We now show that in the concrete category ${\bf Set}$ the category-theoretical notion of isomorphism coincides with the notion of bijection. Given functions $f:X\to Y$ and $g:Y\to X$ satisfying $g(f(x))=x$ for all $x\in X$ and $f(g(y))=y$ for all $y\in Y$ we have:

$f(x_{1})=f(x_{2})\Rightarrow g(f(x_{1}))=g(f(x_{2}))\Rightarrow x_{1}=x_{2}$ so $f$ is injective, and,

for all $y\in Y$, setting $x:=g(y)$, we have $f(x)=y$ so $f$ is surjective,

so $f$ is indeed a bijection. We leave it to the reader to verify that the converse also holds. For the other concrete categories mentioned above the categorical notion of isomorphism also coincides with the usual one.

Since a group $(G,\bullet,1)$ is a monoid with inverses it can now be equivalently defined as a category with one object in which each morphism is an isomorphism. More generally, a groupoid is a category in which each morphism has an inverse. For instance, the category ${\bf Bijec}$ which has sets as objects and bijections as morphisms is such a groupoid. So is ${\bf FdUnit}$ which has finite dimensional Hilbert spaces as objects and unitary operators as morphisms. Groupoids have important applications in mathematics, for example, in algebraic topology Brown.

From this, we see that any group is an example of an abstract categorical structure. At the same time, all groups together, with structure preserving maps between them, constitute a concrete category. Still following? That categories allow several ways of representing mathematical structures might seem confusing at first, but it is a token of their versatility. While monoids correspond to categories with only one object, with groups as a special case, similarly, pre-orders are categories with very few morphisms, with partially ordered sets as a special case.

Any preordered set $(P,\leq)$ can be seen as a category ${\bf P}$:

The elements of $P$ are the objects of ${\bf P}$,

Whenever $a\leq b$ for $a,b\in P$ then there is a single morphism of type $a\rTo b$, that is, ${\bf P}(a,b)$ is a singleton, and whenever $a\not\leq b$ then there is no morphism of type $a\rTo b$, that is, ${\bf P}(a,b)$ is empty.

Whenever there is pair of morphisms of types $a\rTo b$ and $b\rTo c$, that is, whenever $a\leq b$ and $b\leq c$, then transitivity of $\leq$ guarantees the existence of a unique morphism of type $a\rTo c$, which we take to be the composite of the morphisms of type $a\rTo b$ and $b\rTo c$.

Reflexivity guarantees the existence of a unique morphism of type $a\rTo a$, which we take to be the identity on the object $a$.

Conversely, a category ${\bf C}$ of which the objects constitute a set, and in which there is at most one morphism of any type i.e., hom-sets are either singletons or empty, is in fact a preordered set. Concretely:

The set $|{\bf C}|$ are the elements of the preordered set,

We set $A\leq B$ if and only if ${\bf C}(A,B)$ is non-empty,

Since ${\bf C}$ is a category, whenever there exist morphisms $f\in{\bf C}(A,B)$ and $g\in{\bf C}(B,C)$, that is, whenever both ${\bf C}(A,B)$ and ${\bf C}(B,C)$ are non-empty, then there exist a morphism $g\circ f\in{\bf C}(A,C)$, so ${\bf C}(A,C)$ is also non-empty. Hence, $A\leq B$ and $B\leq C$ yields $A\leq C$, so $\leq$ is transitive.

Since $1_{A}\in{\bf C}(A,A)$ we also have $A\leq A$, so $\leq$ is reflexive.

Hence, preordered sets indeed constitute an abstract category: its defining property is that every hom-set contains at most one morphism. Such categories are sometimes called thin categories. Conversely, categories with non-trivial hom-sets are called thick. Partially ordered sets also constitute an abstract category, namely one in which:

every hom-set contains at most one morphism ;

whenever two objects are isomorphic then they must be equal .

This second condition imposes anti-symmetry on the partial order.

Let $\{*\}$ and $\emptyset$ denote a singleton set and the empty set respectively. Then for any set $A\in|{\bf Set}|$, the set ${\bf Set}(A,\{*\})$ of all functions of type $A\rightarrow\{*\}$ is itself a singleton, since there is only one function which maps all $a\in A$ on $*$, the single element of $\{*\}$. This concept can be dualised. The set ${\bf Set}(\emptyset,A)$ of functions of type $\emptyset\rightarrow A$ is again a singleton consisting of the ‘empty function’. Due to these special properties, we call $\{*\}$ and $\emptyset$ respectively the terminal object and the initial object in ${\bf Set}$. All this can be generalised to arbitrary categories as follows:

An object $\top\in|{\bf C}|$ is terminal in ${\bf C}$ if, for any $A\in|{\bf C}|$, there is only one morphism of type $A\rTo\top$. Dually, an object $\bot\in|{\bf C}|$ is initial in ${\bf C}$ if, for any $A\in|{\bf C}|$, there is only one morphism of type $\bot\rTo A$.

If a category ${\bf C}$ has two initial objects then they are isomorphic. The same property holds for terminal objects.

Indeed. Let $\bot$ and $\bot^{\prime}$ both be initial objects in ${\bf C}$. Since $\bot$ is initial, there is a unique morphism $f$ such that ${\bf C}(\bot,\bot^{\prime})=\{f\}$. Analogously, there is a unique morphism $g$ such that ${\bf C}(\bot^{\prime},\bot)=\{g\}$. Now, since ${\bf C}$ is a category and relying again on the fact that $\bot$ is initial, it follows that $g\circ f\in{\bf C}(\bot,\bot)=\{1_{\bot}\}$. Similarly, $g\circ f\in{\bf C}(\bot^{\prime},\bot^{\prime})=\{1_{\bot^{\prime}}\}$. Hence, $\bot\simeq\bot^{\prime}$ as claimed. Similarly we show that $\top\simeq\top^{\prime}$.

A partially ordered set $P$ is bounded if there exist two elements $\top$ and $\bot$ such that for all $a\in P$ we have $\bot\leq a\leq\top$. Hence, when $P$ is viewed as a category, this means that it has both a terminal and an initial object.

The next example of an abstract categorical structure is the most important one in this paper. Therefore, we state it as a definition. Among many (more important) things, it axiomatises ‘cooking with vegetables’.

A strict monoidal category is a category for which:

objects come with monoid structure $(|{\bf C}|,\otimes,{\rm I})$ i.e, for all $A,B,C\in|{\bf C}|$,

for all objects $A,B,C,D\in|{\bf C}|$ there exists an operation

which is associative and has $1_{\rm I}$ as its unit, that is, Note that this operation on morphisms is a typed variant of the notion of monoid.

for all morphisms $f,g,h,k$ with matching types we have

for all objects $A,B\in|{\bf C}|$ we have

As we will see in Section 5.1, the two equational constraints eq.(4) and eq.(5) can be conceived as a single principle.

The symbol $\otimes$ is sometimes called the tensor. We will also use this terminology, since ‘tensor’ is shorter than ‘monoidal product’. However, the reader should not deduce from this that the above definition necessitates $\otimes$ to be anything like a tensor product, since this is not at all the case.

The categories of systems and processes discussed in Section 1.3 are all examples of strict monoidal categories. We already explained in Section 0 what $-\otimes-$ stands for: it enables dealing with situations where several systems are involved. To a certain extent $-\otimes-$ can be interpreted as a logical conjunction:

There is however considerable care required with this view: while

This is where the so-called linear logic Girard; Seely kicks in, which is discussed in substantial detail in AbramskyT.

since it is the unit for the monoid. Hence, it refers to a system which leaves any system invariant when adjoined to it. In short, it stands for ‘unspecified’, for ‘no system’, or even for ‘nothing’. We already made reference to it in Example 9 when discussing preparation procedures. Similarly, $1_{\rm I}$ is the operation which ‘does nothing to nothing’. The system ${\rm I}$ will allow us to encode a notion of state within arbitrary monoidal categories, and also a notion of number and probabilistic weight — see Example 27 below.

Now, a monoid $(M,\bullet,1)$ can also be conceived as a strict monoidal category in which all morphisms are identities. Indeed, take $M$ to be the objects, $\bullet$ to be the tensor and $1$ to be the unit for the tensor. By taking identities to be the only morphisms, we can equip these with the same monoid structure as the monoid structure on the objects. Hence it satisfies eq.(5). By

5 Categories in physics

In the previous section, we saw how groups and partial orders, both of massive importance for physics, are themselves abstract categorical structures.

While there is no need to argue for the importance of group theory to physics here, it is worth mentioning that John Slater (cf. Slater determinant in quantum chemistry) referred to Weyl, Wigner and others’ use of group theory in quantum physics as der Gruppenpest, what translates as the ‘plague of groups’. Even in 1975 he wrote: As soon as [my] paper became known, it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as ‘Slater has slain the Gruppenpest’. I believe that no other piece of work I have done was so universally popular.” Similarly, we may wonder whether it are the category theoreticians or their opponents which are the true aliens.

Partial orders model spatio-temporal causal structure Penrose72; Sorkin91. Roughly speaking, if $a\leq b$ then events $a$ and $b$ are causally related, if $a<b$ then they are time-like separated, and if $a$ and $b$ don’t compare then they are space-like separated. This theme is discussed in great detail in MartinPanangaden.

The degree of bipartite quantum entanglement gives rise to a preorder on bipartite quantum states Nielsen. The relevant preorder is Muirheads’ majorization order Muirhead. However, multipartite quantum entanglement and mixed state quantum entanglement are not well understood yet. We strongly believe that category theory provides the key to the solution, in the following sense:

We also acknowledge the use of category theory in several involved subjects in mathematical physics ranging from topological quantum field theories (TQFTs) to proposals for a theory of quantum gravity; here the motivation to use category theory is of a mathematical nature. We discuss one such topic, namely TQFT, in Section 5.5.

But the particular perspective which we would like to promote here is categories as physical theories. Above we discussed three kinds of categories:

Concrete categories have mathematical structures as objects, and structure preserving maps between these as morphisms.

Real world categories have some notion of system as objects, and corresponding processes thereoff as morphisms.

Abstract categorical structures are mathematical structures in their own right; they are defined in terms of additional structure and/or certain properties.

The real world categories constitute the area of our focus (e.g. quantum physics, proof theory, computation, organic chemistry, …), the concrete categories constitute the formal mathematical models for these (e.g., in the case of quantum physics, Hilbert spaces as objects, certain types of linear maps as morphisms, and the tensor product as the monoidal structure), while the abstract categorical structures constitute axiomatisations of these.

The latter is the obvious place to start when one is interested in comparing theories. We can study which axioms and/or structural properties give rise to certain physical phenomena, for example, which tensor structures give rise to teleportation (e.g. AC2004), or to non-local quantum-like behavior BES. Or, we can study which structural features distinguish classical from quantum theories (e.g. CPav2006; CPP2008).

Quantum theory is subject to the so-called No-Cloning, No-Deleting and No-Broadcasting theorems Broadcast; Pati; WZ, which impose key constraints on our capabilities to process quantum states. Expressing these clearly requires a formalism that allows to vary types from a single to multiple systems, as well as one which explicitly accommodates processes (cf. copying/deleting process). Monoidal categories provide the appropriate mathematical arena for this on-the-nose.

Why does a tiger have stripes and a lion doesn’t? One might expect that the explanation is written within the fundamental building blocks which these animals are made up from, so one could take a big knife and open the lion’s and the tiger’s bellies. One finds intestines, but these are the same for both animals. So maybe the answer is hidden in even smaller constituents. With a tiny knife we keep cutting and identify a smaller kind of building block, namely the cell. Again, there is no obvious difference between tigers and lions at this level. So we need to go even smaller. After a century of advancing ‘small knife technology’ we discover DNA and this constituent truly reveals the difference. So yes, now we know why tigers have stripes and lions don’t! Do we really? No, of course not. Following in the footsteps of Charles Darwin, your favorite nature channel would tell you that the explanation is given by a process of type

which represents the successful challenge of a predator, operating within some environment, on some prey. Key to the success of such a challenge is the predator’s camouflage. Sandy savanna is the lion’s habitat while forests constitute the tiger’s habitat, so their respective coat blends them within their natural habitat. Any (neo-)Darwinist biologist will tell you that the fact that this is encoded in the animal’s DNA is not a cause, but rather a consequence, via the process of natural selection.

This example illustrates how monoidal categories enable to shift the focus from an atomistic or reductionist attitude to one where systems are studied in terms of their interactions with other systems, rather than in terms of their constituents. Clearly, in recent history, physics has solely focused on chopping down things into smaller things. Focussing on interactions might provide us with a complementary understanding of the fundamental theories of nature.

6 Structure preserving maps for categories

The notion of structure preserving map between categories — which we referred to in Example 6 — wasn’t made explicit yet. These ‘maps which preserve categorical structure’, the so-called functors, must preserve the structure of a category, that is, composition and identities. An example of a functor that might be known to the reader because of its applications in physics, is the linear representation of a group. A representation of a group $G$ on a vector space $V$ is a group homomorphism from $G$ to $\mbox{GL}(V)$, the general linear group on $V$, i.e., a map $\rho:G\rightarrow\mbox{GL}(V)$ such that

Secondly, we need to specify to which linear map in

is mapped. This defines a map from a hom-set to a hom-set, namely

The fact that $\rho$ is a group homomorphism implies in our category-theoretic context that $R_{\rho}$ preserves composition of morphisms as well as identities, that is, $R_{\rho}$ preserves the categorical structure.

Having this example in mind, we infer that a functor must consist not of a single but of two kinds of mappings: one map on the objects, and a family of maps on the hom-sets which preserve identities and composition.

Let ${\bf C}$ and ${\bf D}$ be categories. A functor

which preserves identities and composition, i.e.,

for any $f\in{\bf C}(A,B)$ and $g\in{\bf C}(B,C)$ we have

Typically one drops the parentheses unless they are necessary. For instance, $F(A)$ and $F(f)$ will be denoted simply as $FA$ and $Ff$.

Consider the category ${\bf PhysProc}$ of Example 7 and a concrete category ${\bf Mod}$ in which we wish to model these mathematically by assigning to each process a morphism in the concrete category ${\bf Mod}$. Functoriality of

means that sequential composition of physical processes is mapped on composition of morphisms in ${\bf Mod}$, and that void processes are mapped on the identity morphisms. From this, we see that functoriality is an obvious requirement when designing mathematical models for physical processes.

matrix composition, and identity matrices.

This example is closely related to Example 2. However, it strongly emphasizes that objects are but labels with no internal structure. Strictly speaking this is not a concrete category in the sense of Section 1.2. However, for all practical purposes, it can serve as well as a model as any other concrete category. Therefore, we can relax our conception of concrete categories to accommodate such models.

We now introduce the concept of duality which we already hinted at above. Simply put, it means reversal of the arrows in a given category ${\bf C}$. We illustrate this notion in term of an example. Transposition of matrices, just like a functor, is a mapping on both objects and morphisms which:

reverses the direction of the morphisms since when the matrix $M$ has type $n\rTo m$, then the matrix $M^{T}$ has type $m\rTo n$, and

preserves the composition ‘up to this reversal of the arrows’, i.e. for any pair of matrices $N$ and $M$ for which types match we have

So transposition is a functor up to reversal of the arrows.

A contravariant functor $F:{\bf C}\rightarrow{\bf D}$ consists of the same data as a functor, it also preserves identities, but reverses composition that is:

In contrast to contravariant functors, ordinary functors are often referred to as covariant functors.

The opposite category ${\bf C}^{\scriptsize\mbox{op}}$ of a category ${\bf C}$ is the category with

in which morphisms are ‘reversed’, that is,

where to avoid confusion from now on we denote $f\in{\bf C}^{\scriptsize\mbox{op}}(B,A)$ by $f^{\scriptsize\mbox{op}}$,

identities in ${\bf C}^{\scriptsize\mbox{op}}$ are those of ${\bf C}$, and

Contravariant functors of type ${\bf C}\rightarrow{\bf D}$ can now be defined as functors of type ${\bf C}^{\scriptsize\mbox{op}}\rightarrow{\bf D}$. Of course, the operation $(-)^{\scriptsize\mbox{op}}$ on categories is involutive: reversing the arrows twice is the same as doing nothing. The process of reversing the arrow is sometimes indicated by the prefix ‘co’, indicating that the defining equations for those structures are the same as the defining equations for the original structure, but with arrows reversed.

The transpose is the involutive contravariant functor

which maps each vector space on the corresponding dual vector space, and which maps each linear map $f$ on its transpose $f^{T}$.

Let ${\bf FdHilb}$ be the category with finite dimensional Hilbert spaces as objects and with linear maps as morphisms. Of course, one could define other categories with Hilbert spaces as objects, for example, the groupoid ${\bf FdUnit}$ of Example 13. But as we will see below in Section 2.3, the category ${\bf FdHilb}$ as defined here comes with enough extra structure to extract all unitary maps from it. Hence, ${\bf FdHilb}$ subsumes ${\bf FdUnit}$. This extra structure comes as a functor, whose action is taking the adjoint or hermitian transpose. This is the contravariant functor

and assigns morphisms to their adjoints, that is,

Since for $f\in{\bf FdHilb}({\cal H},{\cal K})$ and $g\in{\bf FdHilb}({\cal K},{\cal L})$ we have:

we indeed obtain an identity-on-object contravariant functor. This functor is moreover involutive, that is, for all morphisms $f$ we have

While the morphisms of ${\bf FdHilb}$ do not reflect the inner-product structure, the latter is required to specify the adjoint. In turn, this adjoint will allow us to recover the inner-product in purely category-theoretic terms, as we shall see in Section 2.3.

Define the category ${\bf Funct}_{{\bf C},{\bf D}}$ with

all functors from ${\bf C}$ to ${\bf D}$ as objects,

natural transformations between these as morphisms (cf. Section 5.2), and,

composition of natural transformations and corresponding identities.

The defining equations of strict monoidal categories, that is,

to which we from now on refer as bifunctoriality, is nothing but functoriality of a certain functor. We will discuss this in detail in Section 5.1.

A concrete category, or even better, a ${\bf Set}$-concrete category, is a category ${\bf C}$ together with a functor $U:{\bf C}\longrightarrow{\bf Set}$. The way in which we construct this functor for categories with mathematical structures as objects is by sending each object to the underlying set, and morphisms to the underlying functions. So we forget the extra structure the object has. Therefore the functor $U$ is typically called forgetful. For example, the category ${\bf Grp}$ is a concrete category for the functor

which ‘forgets’ the group’s multiplication and unit, and morphisms are mapped on their underlying functions. More generally, a ${\bf D}$-concrete category is a category ${\bf C}$ with a functor $U:{\bf C}\longrightarrow{\bf D}$.

The TQFTs of Section 5.5 are special kinds of functors.

The 2D case: Muscle power

We now genuinely start to study the interaction of the parallel and the sequential modes of composing systems, and operations thereon.

The starting point of this Section is the notion of a strict monoidal category as given in Definition 4. Such categories enable us to give formal meaning to physical processes which involve several types, e.g. classical and quantum as the following example clearly demonstrates.

Define ${\bf CQOpp}$ to be the strict monoidal category containing both classical and quantum systems, with operations thereon as morphisms, and with the obvious notion of monoidal tensor, that is, a physical analogue of the tensor for vegetables that we saw in the prologue. Concretely, by $A\otimes B$ we mean that we have both $A$ and $B$ available to operate on. Note in particular that at this stage of the discussion there are no Hilbert spaces involved, so $\otimes$ cannot stand for the tensor product, but this does not exclude that we may want to model it by the tensor product at a later stage. In this category, non-destructive (projective) measurements have type

where $Q$ is a quantum system and $X$ is the classical data produced by the measurement. Obviously, the hom-sets

have a very different structure since ${\bf CQOpp}(Q,Q)$ stands for the operations we can perform on a quantum system while ${\bf CQOpp}(X,X)$ stands for the classical operations (e.g. classical computations) which we can perform on classical systems. But all of these now live within a single mathematical entity ${\bf CQOpp}$.

The structure of a strict monoidal category does not yet capture certain important properties of cooking with vegetables. Denote the strict monoidal category constructed in the Prologue by ${\bf Cook}$.

Clearly ‘boil the potato while fry the carrot’ is very much the same thing as ‘fry the carrot while boil the potato’. But we cannot just bluntly say that in the category ${\bf Cook}$ the equality

holds. By plain set theory, for this equality to be meaningful, the two morphisms $h\otimes f$ and $f\otimes h$ need to live in the same set. That is, respecting the structure of a category, within the same hom-set. So

need to have the same type, which implies that

must hold. But this completely blurs the distinction between a carrot and a potato. For example, we cannot distinguish anymore between ‘boil the potato while fry the carrot’, which we denoted by

and ‘fry the potato while boil the carrot’, which given eqs.(7), we can write as

So we basically threw out the child with the bath water.

The solution to this problem is to introduce an operation

which swaps the role of the potato and the carrot relative to the monoidal tensor. The fact that ‘boil the potato while fry the carrot’ is essentially the same thing as ‘fry the carrot while boil the potato’ can now be expressed as

In our ‘real world example’ of cooking this operation can be interpreted as physically swapping the vegetables C2005d. An equational law governing ‘swapping’ is:

A strict symmetric monoidal category is a strict monoidal category ${\bf C}$ which moreover comes with a family of isomorphisms

called symmetries, and which are such that:

for all $A,B\in|{\bf C}|$ we have $\sigma_{A,B}^{-1}=\sigma_{B,A}$, and

for all $A,B,C,D\in|{\bf C}|$ and all $f,g$ of appropriate type we have

All Examples of Section 1.3 are strict symmetric monoidal categories for the obvious notion of symmetry in terms of ‘swapping’.

We can rewrite eq.(8) in a form which makes the types explicit:

This representation is referred to as commutative diagrams.

Indeed, relying on bifunctoriality we have:

The reader can easily verify that, given a connective $-\otimes-$ defined both on objects and morphisms as in items 1 & 2 of Definition 4, the four equations

when varying over all objects $A,B,C,D\in|{\bf C}|$ and all morphisms $f$ and $g$ of appropriate type, are equivalent to the single equation

when varying over $f,g,h,k$. Eqs.(12,13) together with

is usually referred to as $-\otimes-$ being functorial in both arguments. They are indeed equivalent to the mappings on objects and morphisms

both being functors, for all objects $A,B\in|{\bf C}|$ — their action on objects is

Hence, functoriality in both arguments is strictly weaker than bifunctoriality (cf. Example 23), since the latter also requires eqs(11).

2 Graphical calculus for symmetric monoidal categories

The most attractive, and at the same time, also the most powerful feature of strict symmetric monoidal categories, is that they admit a purely diagrammatic calculus. Such a graphical language is subject to the following characteristics:

The symbolic ingredients in the definition of strict symmetric monoidal structure, e.g. $\otimes$, $\circ$, $A$, ${\rm I}$, $f$ etc., or any other abstract categorical structure which refines it, all have a purely diagrammatic counterpart ;

The corresponding axioms become very intuitive graphical manipulations ;

And crucially, an equational statement is derivable in the graphical language if and only if it is symbolically derivable from the axioms of the theory.

For a more formal presentation of what we precisely mean by a graphical calculus we refer the reader to Peter Selinger’s marvelous paper Selinger in these volumes.

These diagrammatic calculi trace back to Penrose’s work in the early 1970s, and have been given rigorous formal treatments in FY; JS; JSV; SelingerPre. Some examples of possible elaborations and corresponding applications of the graphical language presented in this paper are in CP2006; CPP2008; CoeckeDuncan; Lauda; Selinger; StreetBook; Vicary; YetterBook.

The graphical counterparts to the axioms are typically much simpler then their formal counterparts. For example, in the Prologue we mentioned that bifunctoriality becomes a tautology in this context. Therefore such a graphical language radically simplifies algebraic manipulations, and in many cases trivialises something very complicated. Also the physical interpretation of the axioms, something which is dear to the authors of this paper, becomes very direct.

The graphical counterparts to strict symmetric monoidal structure are:

The identity $1_{\rm I}$ is the empty picture (= it is not depicted).

The identity $1_{A}$ for and object $A$ different of ${\rm I}$ is depicted as

The composition of morphisms $f:A\rTo B$ and $g:B\rTo C$ is depicted by locating $g$ above $f$ and by connecting the output of $f$ to the input of $g$, i.e.

The tensor product of morphisms $f:A\rTo B$ and $g:C\rTo D$ is depicted by aligning the graphical representation of $f$ and $g$ side by side in the order they occur within the expression $f\otimes g$, i.e.

The diamond shape of the morphisms of type ${\rm I}\rTo{\rm I}$ indicates that they arise when composing two triangles:

In the category ${\bf QuantOpp}$ the triangles of respective types ${\rm I}\rTo A$ and $A\rTo{\rm I}$ represent states and effects, and the diamonds of type ${\rm I}\rTo{\rm I}$ can be interpreted as probabilistic weights: they give the likeliness of a certain effect to occur when the system is in a certain state. In the usual quantum formalism these values are obtained when computing the Born rule or Luders’ rule. In appropriate categories, we find these exact values back as one of these diamonds, by composing a state and an effect deLL; Selinger.

established in Proposition 2 is depicted as:

In words: we can ‘slide’ boxes along their wires.

The first defining equation of symmetry, i.e. eq.(9), depicts as:

i.e., we can still ‘slide’ boxes along crossings of wires. The equation

which when varying $A,B\in|{\bf C}|$ states that

Suppose now that for any three arbitrary morphisms

in any strict symmetric monoidal category, one intends to prove that

always holds. Then, the typical textbook proof proceeds by diagram chasing:

One needs to read this ‘dragon’ as follows. The two outer paths both going from the left-upper-corner to the right-lower-corner represent the two sides of the equality we want to prove. Then, we do what category-theoreticians call diagram chasing, that is, ‘pasting’ together several commutative diagrams, which connect one of the outer paths to the other. For example, the triangle at the top of the diagram expresses that

that is, an instance of bifunctoriality. Using properties of strict symmetric monoidal categories, namely bifunctoriality and eq.(9) expressed as commutative diagrams, we can pass from the outer path at the top and the right to the outer path on the left and the bottom. This is clearly a very tedious task and getting these diagrams into LaTeX becomes a time-consuming activity.

On the other hand, when using the graphical calculus, one immediately sees that

must hold. We pass from one picture to the other by sliding the boxes along wires and then by rearranging these wires. In terms of the underlying equations of strict symmetric monoidal structure, ‘sliding the boxes along wires’ uses eq.(9) and eq.(15), while ‘rearranging these wires’ means that we used eq.(9) as follows:

Indeed, since symmetry is a morphism it can be conceived as a box, and hence we can ‘slide it along wires’.

In a broader historical perspective, we are somewhat unfair here. Writing equational reasoning down in terms of these commutative diagrams rather than long lists of equalities was an important step towards a better geometrical understanding of the structure of proofs.

3 Extended Dirac notation

A strict dagger monoidal category ${\bf C}$ is a strict monoidal category equipped with an involutive identity-on-objects contravariant functor

$A^{\dagger}=A$ for all $A\in|{\bf C}|$, and

$f^{\dagger\dagger}=f$ for all morphisms $f$,

and this functor preserves the tensor, that is,

We will refer to $B\rTo^{f^{\dagger}}A$ as the adjoint to $A\rTo^{f}B$. A strict dagger symmetric monoidal category ${\bf C}$ is both a strict dagger monoidal category and a strict symmetric monoidal category such that

AC2004 A morphism $U:A\rTo B$ in a strict dagger monoidal category ${\bf C}$ is called unitary if its inverse and its adjoint coincide, that is, if

Let $\psi,\phi:{\rm I}\rTo A$ be ‘elements’ in ${\bf C}$. Their inner-product is the ‘scalar’

So in any strict monoidal category we refer to morphisms of type

as elements (cf. Example 4), to those of type

as scalars. As already discussed in Example 27 in the category ${\bf QuantOpp}$ these corresponds respectively to states, effects and probabilistic weights.

Even at this abstract level, many familiar things follow from Definition 10. For example, we recover the defining property of adjoints for any dagger functor:

From this it follows that unitary morphisms preserve the inner-product:

Importantly, the graphical calculus of the previous section extends to strict dagger symmetric monoidal categories. Following Selinger Selinger, we introduce an asymmetry in the graphical notation of the morphisms $A\rTo{f}B$ as follows:

Then we depict the adjoint $B\rTo{f^{\dagger}}A$ of $A\rTo{f}B$ as follows:

that is, we turn the box representing $f$ upside-down. All this enables interpreting Dirac notation Dirac in terms of strict dagger symmetric monoidal categories, and in particular, in terms of the corresponding graphical calculus:

The latter notation merely requires closing the bra’s and ket’s and performing a $90^{\circ}$ rotation. This $90^{\circ}$ rotation is merely a consequence of our convention to read pictures from bottom-to-top. Other authors obey different conventions e.g. top-to-bottom or left-to-right. Summarising we now have:

In particular, note that in the language of strict dagger symmetric monoidal categories both a bra-ket and a ket-bra are compositions of morphisms, namely $\phi^{\dagger}\circ\psi$ and $\psi\circ\phi^{\dagger}$ respectively. What the diagrammatic calculus adds to standard Dirac notation is a second dimension to accommodate the monoidal composition:

The advantages of this have already been made clear in the previous section and will even become clearer in Section 3.1.

since by linearity the image of $1$ fully specifies this map.

However, by making explicit reference to ${\bf FdHilb}$ and hence also by having matrices (morphisms in ${\bf FdHilb}$ expressed relative to some bases) in the above table, we are actually cheating. The fact that Hilbert spaces and linear maps are set-theoretic based mathematical structures has non-trivial ‘unpleasant’ implications. In particular, while the $\otimes$-notation for the monoidal structure of strict monoidal categories insinuates that the tensor product would turn ${\bf FdHilb}$ into a strict symmetric monoidal category, this turns out not to be true in the ‘strict’ sense of the word true.

4 The set-theoretic verdict on strictness

As outlined in Section 1.5, we ‘model’ real world categories in terms of concrete categories. While the real world categories are indeed strict monoidal categories, their corresponding models typically aren’t.

What goes wrong is the following: for set-theory based mathematical structures such as groups, topological spaces, partial orders and vector spaces, neither

hold. This is due to the fact that for the underlying sets $X,Y,Z$ we have that

hold. We do have something very closely related to this, namely

That is, we have isomorphisms rather than strict equations. But these isomorphisms are not just ordinary isomorphisms but so-called natural isomorphisms. They are an instance of the more general natural transformations which we will discuss in Section 5.2. Naturality is one of the most important concepts of formal category theory. In fact, in the founding paper EilenbergMacLane Eilenberg and MacLane argue that their main motivation for introducing the notion of a category is to introduce the notion of a functor, and that their main motivation for introducing the notion of a functor is to introduce the notion of a natural transformation. Meanwhile we introduce a restricted version of this general notion of natural transformation, one which comes with a clear interpretation.

Consider a category ${\bf C}$ that comes with an operation on objects

and with for all objects $A,B,C,D\in|{\bf C}|$ we also have an operation on hom-sets

be two well-formed expressions built from:

and constants $C_{1},\ldots,C_{m}\in|{\bf C}|$.

Then a natural transformation is a family

of morphisms which are such that for all objects $A_{1},\ldots,A_{n},B_{1},\ldots,B_{n}\in|{\bf C}|$ and all morphisms $A_{1}\rTo^{f_{1}}B_{1}\,,\,\ldots,\,A_{n}\rTo^{f_{n}}B_{n}$ we have:

A natural transformation is a natural isomorphism if, in addition, all these morphisms $\xi_{A_{1},\ldots\!,A_{n}}$ are isomorphisms in the sense of Definition 2.

Examples of such well-formed expressions are

and the corresponding constraint on the morphims is

If diagram (20) commutes for all $A,B,C,A^{\prime},B^{\prime},C^{\prime},f,g,h$ and the morphisms

are all isomorphisms, then this natural isomorphism is called associativity. Its name refers to the fact that this natural isomorphism embodies a weaker form of the strict associative law $A\otimes(B\otimes C)=(A\otimes B)\otimes C$. A better name would actually be re-bracketing, since that is what it truly does: it is a morphism —which we like to think of as a process— which transforms type $A\otimes(B\otimes C)$ into type $(A\otimes B)\otimes C$. In other words, it provides a formal witness to the actual processes of re-bracketing a mathematical expression. The naturality condition in diagram (20) formally states that re-bracketing commutes with any triple of operations $f,g,h$ we apply to the systems, and hence it tells us that the process of re-bracketing does not interfere with any non-trivial processes $f,g,h$ — almost as if it wasn’t there.

Other important pairs of well-formed formal expressions are

and, if ${\rm I}$ is taken to be the constant object, the corresponding naturality constraint is

The natural isomorphisms $\lambda$ and $\rho$ in diagrams (21) are called left- and right unit. In this case, a better name would have been left- and right introduction since they correspond to the process of introducing a new object relative to an existing one.

We encountered a fourth important example in Definition 8, namely

for which diagram (9) is the naturality condition. The isomorphism $\sigma$ is called symmetry but a better name could have been exchange or swapping.

The category ${\bf Set}$ has associativity, left- and right unit, and symmetry natural isomorphisms relative to the Cartesian product, with the singleton set $\{*\}$ as the monoidal unit. Explicitly, setting

for $f:X\to Y$ and $f^{\prime}:X^{\prime}\to Y^{\prime}$, these natural isomorphisms are

The reader can easily verify that diagrams (9), (20) and (21) all commute. Showing that bifunctoriality holds is somewhat more tedious.

A monoidal category consists of the following data:

a bifunctor $-\otimes-$, that is, an operation both on objects and on morphisms as in prescriptions (18) and (19) above, which moreover satisfies

for all $A,B\in|{\bf C}|$ and all morphisms $f,g,h,k$ of appropriate type , and

hence satisfying eq.(20) and eq.(21), and such that the Mac Lane pentagon

commutes for all $A,B,C,D\in|{\bf C}|$, that also

commutes for all $A,B\in|{\bf C}|$, and that

A monoidal category is moreover symmetric if there is a fourth natural isomorphism

commutes for all $A\in|{\bf C}|$, and that

The set-theoretic verdict on strictness is very hard! The punishment is grave: a definition which stretches over two pages, since we need to carry along associativity and unit natural isomorphisms, which, on top of that, are subject to a formal overdose of coherence conditions, that is, eqs.(22,23,24,25,27). They embody rules which should be obeyed when natural ismorphisms interact with each other, in addition to the naturality conditions which state how natural isomorphisms interact with other morphisms in the category. For example, eq.(26) tells us that if we introduce ${\rm I}$ on the left of $A$, and then swap ${\rm I}$ and $A$, that this should be the same as introducing ${\rm I}$ on the right of $A$. Eq.(26) tells us that the two ways of re-bracketing the four variable expressions involved should be the same.

The idea behind coherence conditions is as follows: if for formal expressions $\Lambda(A_{1},\mbox{\scriptsize\ldots},A_{n},C_{1},\mbox{\scriptsize\ldots},C_{m})$ and $\Xi(A_{1},\mbox{\scriptsize\ldots},A_{n},C_{1},\mbox{\scriptsize\ldots},C_{m})$ there are two morphisms

which are obtained by composing the natural isomorphisms $\alpha$, $\sigma$, $\lambda$, $\rho$ and $1$ both with $-\otimes-$ and $-\circ-$, then $f=g$ – identities are indeed natural isomorphisms, for the formal expressions $\Lambda(A)=\Xi(A)=A$. That eqs.(22,23,24,25,27) suffice for this purpose is in itself remarkable. This is a the consequence of MacLane’s highly non-trivial coherence theorem for symmetric monoidal categories SML, which states that from this set of equations we can derive any other one.

If it wasn’t for this theorem, things could have been even worse, potentially involving equations with an unbounded number of symbols.

Any monoidal category ${\bf C}$ is categorically equivalent, via a pair of strong monoidal functors $G:{\bf C}\longrightarrow{\bf D}$ and $F:{\bf D}\longrightarrow{\bf C}$, to a strict monoidal category ${\bf D}$.

The definitions of categorical equivalence and strong monoidal functor can be found below in Section 5.3. In words, what this means is that for practical purposes, arbitrary monoidal categories behave the same as strict monoidal categories. In particular, the connection between diagrammatic reasoning (incl. Dirac notation) and axiomatic reasoning for strict monoidal categories extends to arbitrary monoidal categories. The essence of the above theorem is that the unit and associativity isomorphims are so well-behaved that they don’t affect this correspondence. In the graphical calculus, the associativity natural isomorphisms becomes implicit when we write

The absence of any brackets means that we can interpret this picture either as

That is, it does not matter whether in first order we want to associate $f$ with $g$, and then in second order this pair as a whole with $h$, or whether in first order we want to associate $g$ with $h$, and then in second order this pair as a whole with $f$.

So things turn out not to be as bad as they looked at first sight!

The category ${\bf Set}$ admits two important symmetric monoidal structures. We discussed the Cartesian product in Example 28. The other one is the disjoint union. Given two sets $X$ and $Y$ their disjoint union is the set

This set can be thought of as the set of all elements both of $X$ and $Y$, but where the elements of $X$ are “coloured” with 1 while those of $Y$ are “coloured” with 2. This guarantees that, when the same element occurs both in $X$ and $Y$, it is twice accounted for in $X+Y$ since the “colours” 1 and 2 recall whether the elements in $X+Y$ either originated in $X$ or in $Y$. As a consequence, the intersection of $\{(x,1)\ |\ x\in X\}$ and $\{(y,2)\ |\ y\in Y\}$ is empty, hence the name ‘disjoint’ union.

For the disjoint union, we take the empty set $\emptyset$ as the monoidal unit and set

for $f:X\to Y$ and $f^{\prime}:X^{\prime}\to Y^{\prime}$. The natural isomorphisms of the symmetric monoidal structure are

One again easily verifies that diagrams (20), (21) and (9) all commute. Showing that bifunctoriality holds is again somewhat more tedious.

Note that the inverse to $\lambda_{V}$ is

On the other hand, the monoidal unit for the direct sum is the $0$-dimensional vector space. Hence this monoidal structure only admits a single ‘scalar’. The following subsection discusses scalars in more detail.

A dagger monoidal category ${\bf C}$ is a monoidal category which comes with an identity-on-objects contravariant involutive functor

satisfying eq.(17), and for which all unit and associativity natural isomorphisms are unitary. A dagger symmetric monoidal category ${\bf C}$ is both a dagger monoidal category and a symmetric monoidal category, in which the symmetry natural isomorphism is also unitary.

The category ${\bf FdHilb}$ admits two dagger symmetric monoidal structures, respectively provided by the tensor product and by the direct sum. In both cases, the adjoint of Example 21 is the dagger functor.

As we will see in great detail in Sections 3.2 and 4.4, the category ${\bf Rel}$ which has sets as objects and relations as morphisms also admits two symmetric monoidal structures, just like ${\bf Set}$: these are again the Cartesian product and the disjoint union. Moreover, ${\bf Rel}$ is dagger symmetric monoidal relative to both monoidal structures with the relational converse as the dagger functor. This is a first very important difference between ${\bf Rel}$ and ${\bf Set}$, since the latter does not admit a dagger functor for either of the monoidal structures we identified on it.

The category ${\bf 2Cob}$ has 1-dimensional closed manifolds as objects, and 2-dimensional cobordisms between these as morphisms, it is dagger symmetric monoidal with the disjoint union of manifolds as its monoidal product and with the reversal of cobordisms as the dagger. This category will be discussed in great detail in Section 3.3.

Of course, in ${\bf FdHilb}$ the tensor product $\otimes$ and the direct sum $\oplus$ are very different monoidal structures as exemplified by the particular role each of these plays within quantum theory. In particular, as pointed out by Schrödinger in the 1930’s Schrodinger, the tensor product description of compound quantum systems is what makes quantum physics so different from classical physics. We will refer to monoidal structures which are somewhat like $\otimes$ in ${\bf FdHilb}$ as quantum-like, and to those that are rather like $\oplus$ in ${\bf FdHilb}$ as classical-like. As we will see below, the quantum-like tensors allow for correlations between subsystems, so the joint state can in general not be decomposed into states of the individual subsystems. In contrast, the classical-like tensors can only describe ‘separated’ systems, that is, the state of a joint system can always be faithfully represented by states of the individual subsystems.

The tensors considered in this paper have the following nature:

While $\times$ behaves ‘classical-like’ in ${\bf Set}$, it behaves ‘quantum-like’ in ${\bf Rel}$, and this despite the fact that ${\bf Rel}$ contains ${\bf Set}$ as a subcategory with the same objects as ${\bf Rel}$, and which inherits its monoidal structures from ${\bf Rel}$.

There is a remarkable parallel between the role that the pair $(\oplus,\otimes)$ plays for ${\bf FdHilb}$ and the role that the pair $(+,\times)$ plays for ${\bf Rel}$.

In ${\bf nCob}$ the direct sum even becomes ‘quantum-like’ — a point which has been strongly emphasized for a while by John Baez B2004.

All of this clearly indicates that being either quantum-like and classical-like is something that involves not just the objects, but also the tensor and the morphism structure.

Sections 3 and 4 provide a detailed discussion of these two very distinct kinds of monoidal structures, which will shed more light on the above table.

5 Scalar valuation and multiples

The following is a fascinating fact discovered by Kelly and Laplaza in KellyLaplaza: even for “non-symmetric” monoidal categories, the scalar monoid is always commutative. The proof is given by the following commutative diagram:

Equality of the two outer paths both going from the left-lower-corner to the right-upper-corner boils down to equality between:

the outer left/upper path which consists of $t\circ s$, and the composite of isomorphism ${\rm I}\simeq{\rm I}\otimes{\rm I}$ with its inverse, so nothing but $1_{\rm I}$, giving all together $t\circ s$, and

the outer lower/right path, giving all together $s\circ t$.

Their equality relies on bifunctoriality (cf. middle two rectangles) and naturality of the left- and right-unit isomorphisms (cf. the four squares).

Diagrammatically commutativity is subsumed by the fact that scalars do not have wires, and hence can ‘move freely around in the picture’:

Thus, the scalar structure on $({\bf Set},\times,\{*\})$ is trivial. On the other hand, in ${\bf Rel}$ there are two relations of type $\{*\}\to\{*\}$, the identity and the empty relation, so

So scalars and scalar multiples are more closely related to the ‘multiplicative’ tensor product structure than to the ‘additive’ direct sum structure. We also have

In general, it is the quantum-like monoidal structures which admit non-trivial scalar structure. This might come as a surprise to the reader, given that for vector spaces one typically associates these scalars with linear combinations of vectors, which are very much ‘additive’ in spirit.

The right half of commutative diagram (28) states that {diagram} We generalize this by defining scalar multiples of a morphism $A\rTo{f}B$ as {diagram} These scalars satisfy the usual properties, namely

Diagrammatically these properties are again implicit and require ‘artificial’ brackets to be made explicit, for example, eq.(29) is hidden as:

Of course, we could still prove these properties with commutative diagrams. For eq.(29) the left-hand-side and the right-hand-side are respectively the top and the bottom path of the following diagram:

where we use the fact that $t\circ s=\lambda^{-1}_{\rm I}\circ(s\otimes t)\circ\rho_{\rm I}$. The diamond on the left commutes by naturality of $\rho_{\rm I}$. The top triangle commutes because both paths are equal to $1_{{\rm I}\otimes B}$ as $\lambda_{\rm I}=\rho_{\rm I}$. The bottom triangle commutes by eq.(15). Finally, the right diamond commutes by naturality of $\lambda_{\rm I}$.

Quantum-like tensors

So what makes $\otimes$ so different from $\oplus$ in the category ${\bf FdHilb}$, what makes $\times$ so different in the categories ${\bf Rel}$ and ${\bf Set}$, and what makes $\times$ so similar in the category ${\bf Rel}$ to $\otimes$ in the category ${\bf FdHilb}$?

A compact (closed) category ${\bf C}$ is a symmetric monoidal category in which every object $A\in|{\bf C}|$ comes with

which are such that the following two diagrams commute:

In the case that ${\bf C}$ is strict the above diagrams simplify to

Definition 13 can also be expressed diagrammatically, provided we introduce some new graphical elements:

As before $A$ will be represented by an upward arrow:

On the other hand, we depict $A^{*}$, the dual object to $A$, either by an upward arrow labelled by $A^{*}$, or by a downward arrow labelled $A$:

The unit $\eta_{A}$ and counit $\epsilon_{A}$ are respectively depicted as

Commutation of the two diagrams now boils down to:

When expressed diagrammatically, these equational constraints admit the simple interpretation of ‘yanking a wire’. While at first sight compactness of a category as stated in Definition 13 seems to be a somewhat ad hoc notion, this graphical interpretation establishes it as a very canonical one which extends the graphical calculus for symmetric monoidal categories with cup- and cap-shaped wires. As the following lemma shows, the equational constraints imply that we are allowed to ‘slide’ morphisms also along these cups and caps.

Given a morphism $f:A\rTo B$ define its transpose to be

Diagrammatically, when depicting the morphism $f$ as

Anticipating what will follow, we abbreviate this notation for $f^{*}$ to

that is, we can ‘slide’ morphisms along cup- and cap-shaped wires.

The proof of the first equality simply is

The proof for the second equality proceeds analogously.

where $\{e_{i}\}_{i=1}^{n}$ is a basis of $V$ and $f_{j}\in V^{*}$ is the linear functional such that $f_{j}(e_{i})=\delta_{i,j}$ for all $1\leq i,j\leq n$. Finally, we take the counit to be

We leave it to the reader to verify commutation of diagrams 31 and 32. Two important points need to be made here:

The linear maps $\eta_{V}$ and $\epsilon_{V}$ do not depend on the choice of the basis $\{e_{i}\}_{i=1}^{n}$. It suffices to verify that there is a canonical isomorphism

which does not depend on the choice of basis. The unit $\eta_{V}$ is the image of $1_{V}$ under this isomorphism and since $1_{V}$ is independent of the choice of basis it follows that $\eta_{V}$ does not depend on any choice of basis. The argument for $\epsilon_{V}$ proceeds analogously.

make diagrams (31) and (32) commute. Indeed, graphically we have:

The category ${\bf Rel}$ of sets and relations is also compact relative to the Cartesian product as we shall see in detail in Section 3.2.

The category $\mathbf{QuantOpp}$ is compact. We can pick Bell-states as the units and the corresponding Bell-effects as counits. As shown in AC2004; C2005c, compactness is exactly what enables modeling protocols such as quantum teleportation:

where the trapezoid is assumed to be unitary and hence, its adjoint coincides with its inverse. The classical information flow is (implicitly) encoded in the fact that the same trapezoid appears in the left-hand-side picture both at Alice’s and Bob’s side.

Given a morphism $f:A\rTo B$ in a compact category, its name

Following AC2004 we can show that for $f:A\rTo B$ and $g:B\rTo C$

always holds. The graphical proof is again trivial:

In contrast a (non-strict) symbolic proof goes as follows:

Both paths on the outside are equal to $g\circ f$. We want to show that the pentagon labelled ‘Result’ commutes. To do this we will ‘unfold’ arrows using equations which hold in compact categories in order to pass from the composite $g\circ f$ at the left/bottom/right to $\lambda^{-1}_{C}\circ(\llcorner f\lrcorner\otimes 1_{C})\circ(1_{A}\otimes\ulcorner g\urcorner)\circ\rho_{A}$. This will transform the tautology $g\circ f=g\circ f$ into commutation of the pentagon labelled ‘Result’. For instance, we use compactness to go from the identity arrow at the bottom of the diagram to the composite $\lambda_{B}^{-1}\circ(\epsilon_{B}\otimes 1_{B})\circ(1_{B}\otimes\eta_{B})\circ\rho_{B}$. The outer left and right trapezoids express naturality of $\rho$ and $\lambda$. The remaining triangles/diamonds express bifunctoriality and the definitions of name/coname.

A dagger compact category ${\bf C}$ is both a compact category and a dagger symmetric monoidal category, such that for all $A\in|{\bf C}|$, $\epsilon_{A}=\eta_{A}^{\dagger}\circ\sigma_{A,A^{*}}$.

The category ${\bf FdHilb}$ is dagger compact.

2 The category of relations

We now turn our attention to the category ${\bf Rel}$ of sets and relations, a category which we briefly encountered in previous sections. Perhaps surprisingly, ${\bf Rel}$ possesses more ‘quantum features’ than the category ${\bf Set}$ of sets and functions. In particular, just like ${\bf FdHilb}$ it is a dagger compact category.

A relation $R:X\rightarrow Y$ between two sets $X$ and $Y$ is a subset of the set of all their ordered pairs, that is, $R\subseteq X\times Y$. Thus, given an element $(x,y)\in R$, we say that $x\in X$ relates to $y\in Y$, which we denote as $xRy$. The set

is also referred to as the graph of the relation.

The monoidal category ${\bf Rel}$ is defined as follows:

The morphisms are all relations $R:X\rightarrow Y$.

For $R_{1}:X\rightarrow Y$ and $R_{2}:Y\rightarrow Z$ the composite $R_{2}\circ R_{1}\subseteq X\times Z$ is

Composition is easily seen to be associative. For $X\in|{\bf Rel}|$ we have

The monoidal product of two sets is their Cartesian product, the unit for the monoidal structure is any singleton, and for two relations $R_{1}:X_{1}\rightarrow Y_{1}$ and $R_{2}:X_{2}\to Y_{2}$ the monoidal product $R_{1}\times R_{2}\subseteq X_{1}\times X_{2}\to Y_{1}\times Y_{2}$ is

We mentioned before that ${\bf Set}$ was contained in ${\bf Rel}$ as a ‘sub-monoidal category’. In ${\bf Rel}$, the left- and right-unit natural isomorphisms respectively are

and the associativity natural isomorphism is

These relations are all single-valued, so they are also functions, and they are the same functions as the natural isomorphisms for the Cartesian product in ${\bf Set}$. Let us verify the coherence conditions for them:

Unfolding the definition of relational composition we obtain

which by the definition of $\alpha$ simplifies to

The bottom path yields the same result, hence making the pentagon commute. For the remaining diagrams we leave the details to the reader.

As $\times$ is symmetric in ${\bf Set}$ we also expect ${\bf Rel}$ to be symmetric monoidal. For any $X$ and $Y\in|{\bf Rel}|$, the natural isomorphism

commute since both paths of the left triangle are equal to

while the paths of the right triangle are equal to

So ${\bf Rel}$ is indeed a symmetric monoidal category as expected. ${\bf Rel}$ shares many common characteristics with ${\bf FdHilb}$, one of them being a $\dagger$-compact structure. Firstly, ${\bf Rel}$ is compact closed with self-dual objects that is, $X^{*}=X$ for any $X\in|{\bf Rel}|$. Moreover, for any $X\in|{\bf Rel}|$ let

such that there exists an $(x^{\prime\prime\prime\prime},*)\in X\times\{*\}$ with

By definition of $\rho$ and $1_{X}$, and of the product of relations, this entails that $x,x^{\prime\prime\prime\prime}$ and $x^{\prime}$ are all equal. Moreover, by definition of $\eta_{X}$, and of the product of relations, we have that $x^{\prime\prime}$ and $x^{\prime\prime\prime}$ are also equal. Thus,

such that there exists an $((x^{\prime\prime},x^{\prime\prime\prime}),x^{\prime\prime\prime\prime})\in(X\times X)\times X$ with

By the computation in (b) we have that $x=x^{\prime\prime}$ and $x^{\prime\prime\prime}=x^{\prime\prime\prime\prime}$. By definition of $\epsilon_{X}$, $1_{X}$ and the product of relations we have $x^{\prime\prime}=x^{\prime\prime\prime}$ and $x^{\prime\prime\prime\prime}=x^{\prime}$. All this together yields $x=x^{\prime\prime}=x^{\prime\prime\prime}=x^{\prime\prime\prime\prime}=x^{\prime}$ and hence

(d) Post-composing the previous composite with the natural isomorphism $\lambda^{-1}_{X}$ yields a morphism of type $X\rightarrow X$, namely

which is the identity relation as required.

Commutation of the dual diagram is done analogously. From this, we conclude that ${\bf Rel}$ is compact closed. The obvious candidate for the dagger

is the relational converse. For relation $R:X\rightarrow Y$ its converse $R^{\cup}:Y\rightarrow X$ is

We define the contravariant identity-on-objects involutive functor

Note that the adjoint and the transpose coincide, that is,

which the reader may easily check. Finally, we verify that ${\bf Rel}$ is dagger compact:

The category ${\bf Rel}$ is dagger monoidal:

(i) From the definition of the monoidal product of two relations

(ii) The fact that $\alpha^{\dagger}=\alpha^{-1}$, $\lambda^{\dagger}=\lambda^{-1}$, $\rho^{\dagger}=\rho^{-1}$ and $\sigma^{\dagger}=\sigma^{-1}$ is trivial as the inverse of all these morphisms is the relational converse.

and hence $\sigma\circ\epsilon^{\dagger}_{X}=\epsilon^{\dagger}_{X}=\eta_{X}$.

So ${\bf Rel}$ is indeed a dagger compact category.

3 The category of 2D cobordisms

The category ${\bf 2Cob}$ can be informally described as a category whose morphisms, so-called cobordisms, describe the ‘topological evolution’ of manifolds of dimension $2-1=1$ through time. For instance, consider some snapshots of two circles which merge into a single circle, with time going upwards:

Passing to the continuum, the same process can be described by the cobordism

Thus, we take a cobordism to be a (compact) 2-dimensional manifold whose boundary is partitioned in two. We take these closed one-dimensional manifolds to be the domain and the codomain of the cobordism. Since we are only interested in the topology of the manifolds, each (co)domain consists of a finite number of closed strings.

The category ${\bf 2Cob}$ is defined as follows:

For each object $n$, the identity $1_{n}:n\rightarrow n$ which is given by $n$ parallel cylinders:

Composition is given by “gluing” manifolds together, e.g.

where the cobordism $M^{\prime}:1\rightarrow 2$ is glued to $M:2\rightarrow 1$ along the object $1$.

The disjoint union of manifolds provides this category with a monoidal structure. For example, if $M:1\rightarrow 0$ and $M^{\prime}:2\rightarrow 1$ are cobordisms, then the cobordism $M+M^{\prime}:1+2\rightarrow 0+1$ depicts as:

The empty manifold 0 is the identity for the disjoint union.

The twist cobordism provides symmetry. For example, the twist

of the compact structure on $1$ are the cobordisms

We recover the equations of compactness as

which hold since all cobordisms involved are homeomorphically equivalent. The generalisation of the units to arbitrary $n$ is again obvious:

These together with corresponding counits are easily seen to always satisfy the equations of compactness.

The dagger consists in ‘flipping’ the cobordisms, e.g. if $M:2\rightarrow 1$ is

Clearly the dagger is compatible with the disjoint union which makes ${\bf 2Cob}$ a dagger monoidal category. It is also dagger compact since $\sigma_{1,1}\circ\epsilon_{1}^{\dagger}$ is

which is again easily seen to be true for arbitrary $n$.

Obviously, we have been very informal here. For a more elaborated discussion and technical details we refer the reader to B2004; BaezDolan; Kock; Turaev. The key thing to remember is that there are important ‘concrete’ categories in which the morphisms are nothing like maps from the domain to the codomain.

Note also that we can conceive — again somewhat informally — the diagrammatic calculus of the previous sections as the result of contracting the diameter of the strings in ${\bf 2Cob}$ to zero. These categories of cobordisms play a key role in topological quantum field theory (TQFT). We discuss this topic in Section 5.5.

Classical-like tensors

The tensors to which we referred as classical-like are not compact. Instead they do come with some other structure which, in all non-trivial cases, turns out to be incompatible with compactness AbrClone. In fact, this incompatibility is the abstract incarnation of the No-Cloning theorem which plays a key role in quantum information Dieks; WZ.

Consider the category ${\bf Set}$ with the Cartesian product as the monoidal tensor, as defined in Example 28. Given sets $A_{1},A_{2}\in|{\bf Set}|$, their Cartesian product $A_{1}\times A_{2}$ consists of all pairs $(x_{1},x_{2})$ with $x_{1}\in A_{1}$ and $x_{2}\in A_{2}$. The fact that Cartesian products consist of pairs is witnessed by the projection maps

which identify the respective components, together with the fact that, in turn, we can pair $x_{1}=\pi_{1}(x_{1},x_{2})\in A_{1}$ and $x_{2}=\pi_{2}(x_{1},x_{2})\in A_{2}$ back together into $(x_{1},x_{2})\in A_{1}\times A_{2}$, merely by putting brackets around them. We would like to express this fact purely in category-theoretic terms. But both the projections and the pairing operation are expressed in terms of their action on elements, while categorical structure only recognises hom-sets, and not the internal structure of the underlying objects. Therefore, we consider the action of projections on hom-sets, namely

which we can combine into a single operation ‘decompose’

so $dec_{C}^{A_{1},A_{2}}$ and $rec_{C}^{A_{1},A_{2}}$ are now effectively each others inverses. In the light of Example 4, setting $C:=\{*\}$, we obtain

which corresponds to projecting and pairing elements exactly as in the discussion at the beginning of this section. All of this extends in abstract generality.

A product of $A_{1}$ and $A_{2}\in|{\bf C}|$ is a triple which consists of another object $A_{1}\times A_{2}\in|{\bf C}|$ together with two morphisms

and which is such that for all $C\in|{\bf C}|$ the mapping

admits an inverse $\langle-,-\rangle_{C,A_{1},A_{2}}$.