On the Classification of Topological Field Theories

Unless otherwise specified, we will use the word manifold to refer to a compact smooth manifold $M$, possibly with boundary (or with corners). If $M$ is a manifold, we will denote its boundary by $\bd M$. We will say that $M$ is closed if the boundary $\bd M$ is empty. For a brief description of how the ideas of this paper generalize to manifolds which are not smooth, we refer the reader to Remark 2.4.30.

Throughout this paper, we will make informal use of the language of higher category theory. We will always use the term $n$-category to refer to what is sometimes called a weak $n$-category: that is, a collection of objects $\{X,Y,Z,\ldots\}$ together with an $(n-1)$-category $\OHom(X,Y)$ for every pair of objects $X$ and $Y$, which are equipped with a notion of composition which is associative up to coherent isomorphism. We refer the reader to 1.3 for an informal discussion and 2.1 for the outline of a more precise definition.

If $\calC$ is a category (or a higher category) equipped with an associative and unital tensor product $\otimes$, we will let ${\bf 1}$ denote the unit object of $\calC$.

Let $V$ be a finite-dimensional real vector space. By an inner product on $V$ we will mean a symmetric bilinear form $b:V\times V\rightarrow\R$ which is positive-definite (so that $b(v,v)>0$ for $v\neq 0$). More generally, if $X$ is a topological space and $\zeta$ is a real vector bundle on $X$, then by an inner product on $\zeta$ we will mean an inner product on each fiber $\zeta_{x}$, which depends continuously on the point $x\in X$.

Disclaimer

Our objective in this paper is to give an informal account of some ideas relating to the classification of topological field theories. In many instances, we have not attempted to give precise definitions, let alone careful proofs. A more detailed account of the ideas and methods described in this paper will appear elsewhere.

Acknowledgements

I would like to thank David Ben-Zvi, Kevin Costello, Chris Douglas, Dan Freed, Dennis Gaitsgory, Soren Galatius, Andre Henriques, Kiyoshi Igusa, David Nadler, Chris Schommer-Pries, Stefan Stolz, Peter Teichner, Constantin Teleman, and Ulrike Tillmann for helpful conversations about the subject matter of this paper. I would also like to acknowledge a great intellectual debt to the work of John Baez and James Dolan (who proposed the cobordism and tangle hypotheses in and thereby brought into focus the ideas we discuss here) and the work of Kevin Costello (whose beautiful paper inspired the present work). Special thanks are due to Mike Hopkins: he has been a collaborator through various stages of the project described here, though he has declined to be named as a coauthor. Finally, I would like to thank the American Institute of Mathematics for supporting me during the time this paper was written.

Topological Field Theories and Higher Categories

The starting point for this paper is Atiyah’s definition of a topological field theory, which we will review in §1.1. This notion is fairly concrete, and it is not difficult to explicitly classify topological field theories of dimensions $\leq 2$. In §1.2, we will discuss some of the difficulties that we encounter when attempting to generalize this classification to the higher-dimensional setting. To address these difficulties, we will introduce the notion of an extended topological field theory. We will then formulate a version of the Baez-Dolan cobordism hypothesis (Theorem 1.2.16), which provides an elegant classification of extended topological field theories.

The notion of an extended topological field theory and the cobordism hypothesis itself are most naturally expressed using the language of higher category theory, which we will review informally in §1.3. This language can also be used to introduce a more refined version of topological field theory which takes into account the homotopy types of diffeomorphism groups of manifolds; we will discuss this definition in §1.4, and formulate an appropriate generalization of the cobordism hyothesis (Theorem 1.4.9).

In this section, we will review the notion of a topological field theory as axiomatized by Atiyah; for details, we refer the reader to .

Let $n$ be a positive integer. We define a category ${\text{\bf Cob}}(n)$ as follows:

An object of ${\text{\bf Cob}}(n)$ is a closed oriented $(n-1)$-manifold $M$.

Given a pair of objects $M,N\in{\text{\bf Cob}}(n)$, a morphism from $M$ to $N$ in ${\text{\bf Cob}}(n)$ is a bordism from $M$ to $N$: that is, an oriented $n$-dimensional manifold $B$ equipped with an orientation-preserving diffeomorphism $\bd B\simeq\overline{M}\coprod N$. Here $\overline{M}$ denotes the manifold $M$ equipped with the opposite orientation. We regard two bordisms $B$ and $B^{\prime}$ as defining the same morphism in ${\text{\bf Cob}}(n)$ if there is an orientation-preserving diffeomorphism $B\simeq B^{\prime}$ which extends the evident diffeomorphism $\bd B\simeq\overline{M}\coprod N\simeq\bd B^{\prime}$ between their boundaries.

For any object $M\in{\text{\bf Cob}}(n)$, the identity map $\id_{M}$ is represented by the product bordism $B=M\times$.

Composition of morphisms in ${\text{\bf Cob}}(n)$ is given by gluing bordisms together. More precisely, suppose we are given a triple of objects $M,M^{\prime},M^{\prime\prime}\in{\text{\bf Cob}}(n)$, and a pair of bordisms $B:M\rightarrow M^{\prime}$, $B^{\prime}:M^{\prime}\rightarrow M^{\prime\prime}$, the composition $B^{\prime}\circ B$ is defined to be the morphism represented by the manifold $B\coprod_{M^{\prime}}B^{\prime}$.

The composition law for bordisms described in Definition 1.1.1 is potentially ambiguous, because we did not explain how to endow the manifold $B\coprod_{M^{\prime}}B^{\prime}$ with a smooth structure. To do so, we need to make some auxiliary choices (for example, the choice of a smooth collar around $M^{\prime}$ inside of $B$ and $B^{\prime}$), which ultimately turn out to be irrelevant (different choices of collar lead to different smooth structures on $B\coprod_{M^{\prime}}B^{\prime}$, but the resulting bordisms are nevertheless diffeomorphic). We will not press this technical point any further here; later, we will introduce more elaborate versions of Definition 1.1.1 in which the issue does not arise.

Recall that a symmetric monoidal category is a category $\calC$ equipped with a functor $\otimes:\calC\times\calC\rightarrow\calC$ and a unit object ${\bf 1}_{\calC}\in\calC$, together with isomorphisms

which express the idea that $\otimes$ is a commutative and associative product on $\calC$ (with unit by ${\bf 1}_{\calC}$). These isomorphisms should be required to satisfy a list of coherence conditions which we do not recall here; see for a complete definition.

For each $n>0$, the category ${\text{\bf Cob}}(n)$ can be endowed with the structure of a symmetric monoidal category, where the tensor product operation $\otimes:{\text{\bf Cob}}(n)\times{\text{\bf Cob}}(n)\rightarrow{\text{\bf Cob}}(n)$ is given by the disjoint union of manifolds. The unit object of ${\text{\bf Cob}}(n)$ is the empty set (regarded as a manifold of dimension $(n-1)$).

Let $k$ be a field. Then the category $\Vect(k)$ of vector spaces over $k$ can be regarded as a symmetric monoidal category with respect to the usual tensor product functor $\otimes:\Vect(k)\times\Vect(k)\rightarrow\Vect(k)$. The unit object of $\Vect(k)$ is the vector space $k$ itself.

Given a pair of symmetric monoidal categories $\calC$ and $\calD$, a symmetric monoidal functor from $\calC$ to $\calD$ is a functor $F:\calC\rightarrow\calD$ together with a collection of isomorphisms

These isomorphisms are required to be compatible with the commutativity and associativity constraints on the tensor products in $\calC$ and $\calD$; we refer the reader again to for a more complete discussion.

Let $k$ be a field. A topological field theory of dimension $n$ is a symmetric monoidal functor $Z:{\text{\bf Cob}}(n)\rightarrow\Vect(k)$.

Unwinding Definition 1.1.5, we see that a topological field theory $Z$ of dimension $n$ is given by the following data:

For every oriented closed manifold $M$ of dimension $(n-1)$, a vector space $Z(M)$.

For every oriented bordism $B$ from an $(n-1)$-manifold $M$ to another $(n-1)$-manifold $N$, a linear map of vector spaces $Z(B):Z(M)\rightarrow Z(N)$.

Moreover, these data are required to satisfy a number of natural coherence properties which we will not make explicit.

Let $M$ be a closed oriented manifold of dimension $n$. Then we can regard $M$ as a bordism from the empty $(n-1)$-manifold to itself. In this way, $M$ determines a morphism $\emptyset\rightarrow\emptyset$ in the category ${\text{\bf Cob}}(n)$. If $Z$ is a topological field theory of dimension $n$, then $M$ determines a map $Z(M):Z(\emptyset)\rightarrow Z(\emptyset)$. Since $Z$ is a tensor functor, it preserves unit objects: that is, $Z(\emptyset)$ is canonically isomorphic to the ground field $k$. Consequently, we can think of $Z(M)$ as an element of the endomorphism ring $\Hom_{\Vect(k)}(k,k)$: that is, as an element of $k$. In other words, the functor $Z$ assigns a number to every closed oriented manifold of dimension $n$.

Let $B$ be an oriented $n$-manifold with boundary $\bd B$. Then $B$ can usually be interpreted as a morphism in ${\text{\bf Cob}}(n)$ in many different ways: one for every decomposition of the boundary $\bd B$ as a disjoint union of two components.

For example, let us suppose that $M$ is a closed oriented manifold of dimension $(n-1)$, and let $\overline{M}$ denote the same manifold with the opposite orientation. The product manifold $M\times$ has boundary $\overline{M}\coprod M$. It therefore determines a bordism from $M$ to itself: when so regarded, it represents the identity map $\id_{M}$ in the category ${\text{\bf Cob}}(n)$. However, there are several other ways to view $M\times$ as a morphism in ${\text{\bf Cob}}(n)$, corresponding to other decompositions of the boundary $\bd(M\times)$. For example:

We can regard $M\times$ as a bordism from $\overline{M}$ to itself; it then represents the identity map $\id_{\overline{M}}$ in the category ${\text{\bf Cob}}(n)$.

We can regard $M\times$ as a bordism from $\overline{M}\coprod M$ to the empty set. In this case, $M\times$ represents a morphism $\overline{M}\coprod M\rightarrow\emptyset$ in ${\text{\bf Cob}}(n)$, which we will denote by $\ev_{M}$ and refer to as the evaluation map for $M$.

We can regard $M\times$ as a bordism from the empty set to $M\coprod\overline{M}$. It then represents a morphism $\emptyset\rightarrow M\coprod\overline{M}$ in the category ${\text{\bf Cob}}(n)$, which we will denote by $\coev_{M}$ and refer to as the coevaluation map for $M$.

Suppose now that $Z$ is a topological field theory of dimension $n$, and let $M$ be a closed oriented $(n-1)$-manifold. Applying the functor $Z$ to the evaluation map $\ev_{M}$, we obtain a map of vector spaces

In other words, there is a canonical bilinear pairing of $Z(M)$ with $Z(\overline{M})$.

Let $Z$ be a topological field theory of dimension $n$. Then for every closed $(n-1)$-manifold $M$, the vector space $Z(M)$ is finite dimensional, and the pairing $Z(\overline{M})\otimes Z(M)\rightarrow k$ is perfect: that is, it induces an isomorphism $\alpha$ from $Z(\overline{M})$ to the dual space of $Z(M)$.

The proof is completely formal: we can use the coevaluation map of $M$ to explicitly construct an inverse to $\alpha$. More precisely, let $Z(M)^{\vee}$ denote the dual space to $Z(M)$. Applying $Z$ to the coevaluation map $\coev_{M}$, we obtain a map

Tensoring this map with $Z(M)^{\vee}$ and composing with the natural pairing of $Z(M)^{\vee}$ with $Z(M)$, we get a map

By judiciously applying the axioms for a topological field theory, one can deduce that $\beta$ is an inverse to $\alpha$: this proves that $\alpha$ is an isomorphism. Because every element in the tensor product $Z(M)\otimes Z(\overline{M})$ belongs to $Z(M)\otimes V$ for some finite dimensional subspace $V$ of $Z(\overline{M})$, the image of the map $\beta$ is necessarily finite dimensional; since $\beta$ is an isomorphism, we conclude that $Z(M)^{\vee}$ is finite dimensional (so that $Z(M)$ is also finite dimensional).

In low dimensions, it is possible to describe topological field theories very explicitly.

Let $Z$ be a $1$-dimensional topological field theory. Then $Z$ assigns a vector space $Z(M)$ to every closed oriented $0$-manifold $M$. A zero dimensional manifold $M$ is simply a finite set of points. An orientation of $M$ determines a decomposition $M=M_{+}\coprod M_{-}$ of $M$ into “positively oriented” and “negatively oriented” points. In particular, there are two oriented manifolds which consist of only a single point, up to orientation-preserving diffeomorphism. Let us denote these manifolds by $P$ and $Q$. Applying the functor $Z$, we obtain vector spaces $Z(P)$ and $Z(Q)$. However, these vector spaces are related to one another: according to Proposition 1.1.8, we can write $Z(P)=V$ and $Z(Q)=V^{\vee}$, for some finite-dimensional vector space $V$.

Once we have specified $V$, the remainder of the field theory is uniquely determined (up to isomorphism). For example, the value of $Z$ on any oriented $0$-manifold $M$ is canonically isomorphic to the tensor product

Of course, this does not yet determine $Z$: we must also specify the behavior of $Z$ on $1$-manifolds $B$ with boundary. However, since $Z$ is a symmetric monoidal functor, it suffices to specify $Z(B)$ when $B$ is connected. In this case, the $1$-manifold $B$ is diffeomorphic either to a closed interval $$or to a circle$S^{1}$. There are five cases to consider, depending on how we decompose$\bd B$ into “incoming” and “outgoing” pieces:

Suppose that $B=$, regarded as a bordism from $P$ to itself. Then $Z(B)$ coincides with the identity map $\id:V\rightarrow V$.

Suppose that $B=$, regarded as a bordism from $Q$ to itself. Then $Z(B)$ coincides with the identity map $\id:V^{\vee}\rightarrow V^{\vee}$.

Suppose that $B=$, regarded as a bordism from $P\coprod Q$ to the empty set. Then $Z(B)$ is a linear map from $V\otimes V^{\vee}$ into the ground field $k$: namely, the evaluation map $(v,\lambda)\mapsto\lambda(v)$.

Suppose that $B=$, regarded as a bordism from the empty set to $P\coprod Q$. Then $Z(B)$ is a linear map from $k$ to $Z(P\coprod Q)\simeq V\otimes V^{\vee}$. Under the canonical isomorphism $V\otimes V^{\vee}\simeq\End(V)$, this linear map is given by $x\mapsto x\id_{V}$.

Suppose that $B=S^{1}$, regarded as a bordism from the empty set to itself. Then $Z(B)$ is a linear map from $k$ to itself, which we can identify with an element of $k$ (Remark 1.1.6). To compute this element, it is convenient to decompose the circle $S^{1}\simeq\{z\in\mathbf{C}:|z|=1\}$ into two intervals

It follows that $Z(S^{1})$ is given as the composition of the maps

These maps were described by $(c)$ and $(d)$ above. Under the identification of $Z(\pm 1)$ with $V\otimes V^{\vee}\simeq\End(V)$, the map $Z(S^{1}_{-}):k\rightarrow\End(V)$ is given by $x\mapsto x\id_{V}$, while $Z(S^{1}_{+}):\End(V)\rightarrow k$ is given by $A\mapsto\tr(A)$. Consequently, $Z(S^{1})$ is given by the trace of the identity map from $V$ to itself: in other words, the dimension of $V$.

Example 1.1.9 illustrates some central themes which will reappear (in a more sophisticated form) later in this paper. A priori, the specification of a topological field theory $Z$ involves a large quantity of data: one must give a vector space $Z(M)$ for every closed oriented manifold $M$ of the appropriate dimension, together with a number of linear maps satisfying various conditions. However, the field theory $Z$ is often determined by only a tiny fragment of this data, given by evaluating $Z$ on a small class of manifolds (in Example 1.1.9, the entire field theory $Z$ is determined by the single vector space $V=Z(P)$). Nevertheless, it can be interesting to consider the values of $Z$ on arbitrary manifolds. In Example 1.1.9, the value $Z(S^{1})$ recovers the dimension of the vector space $V$. This is the most important numerical invariant of $V$, and is in some sense the only invariant: any two vector spaces with the same (integer) dimension are isomorphic to one another.

Let $Z$ be a $2$-dimensional topological field theory. Then $Z$ assigns a vector space $Z(M)$ to every closed, oriented $1$-manifold $M$. In particular, $Z$ determines a vector space $A=Z(S^{1})$. Since $Z$ is a symmetric monoidal functor, the values of $Z$ on objects are determined by $A$: every closed $1$-manifold $M$ is a disjoint union of $n$ circles for some $n\geq 0$, so that $Z(M)\simeq A^{\otimes n}$.

Evaluating the field theory $Z$ on bordisms between $1$-manifolds, we obtain some algebraic structure on the vector space $A$. For example, let $B$ denote a pair of pants, regarded as a bordism from two copies of $S^{1}$ to a third copy of $S^{1}$. Then $Z(B)$ determines a linear map

which we will denote by $m$. We can view $m$ as endowing $A$ with a bilinear multiplication. It follows easily from the definition that this multiplication is commutative and associative: for example, the commutativity results from the observation that there is a diffeomorphism of $B$ which permutes the two “incoming” boundary circles and restricts to the identity on the third.

There is also a unit for the multiplication on $A$: namely, the image of $1\in k$ under the linear map

where we regard the disk $D^{2}=\{z\in\mathbf{C}:|z|\leq 1\}$ as a bordism from the empty set to the boundary circle $S^{1}=\bd D^{2}=\{z\in\mathbf{C}:|z|=1\}$. We can also interpret $D^{2}$ as a bordism from $S^{1}$ to the empty set, in which case it determines a linear map $\tr:A\rightarrow k$. The composition

is the linear map associated to the cylinder $S^{1}\times$, and therefore determines a perfect pairing of $A$ with itself (Proposition 1.1.8). (Note that the $1$-sphere $S^{1}$ admits an orientation-reversing diffeomorphism, so that $S^{1}\simeq\overline{S^{1}}$.)

It is convenient to summarize the analysis up to this point by introducing a definition.

Let $k$ be a field. A commutative Frobenius algebra over $k$ is a finite dimensional commutative $k$-algebra $A$, together with a linear map $\tr:A\rightarrow k$ such that the the bilinear form $(a,b)\mapsto\tr(ab)$ is nondegenerate.

The above analysis shows that if $Z$ is a $2$-dimensional topological field theory, then the vector space $A=Z(S^{1})$ is naturally endowed with the structure of a commutative Frobenius algebra over $k$. In fact, the converse is true as well: given a commutative Frobenius algebra $A$, one can construct a $2$-dimensional topological field theory $Z$ such that $A=Z(S^{1})$ and the multiplication and trace on $A$ are given by evaluating $Z$ on a pair of pants and a disk, respectively. Moreover, $Z$ is determined up to unique isomorphism: in other words, the category of $2$-dimensional topological field theories is equivalent to the category of commutative Frobenius algebras.

2 Extending Down: Lower Dimensional Manifolds

In §1.1, we analyzed the structure of an $n$-dimensional topological field theory $Z$ for $n=1$ and $n=2$. In both cases, we accomplished this by emphasizing the value of the field theory $Z$ on closed manifolds of dimension $n-1$. However, it is possible to proceed differently: according to Remark 1.1.6, for every closed oriented manifold $M$ of dimension $n$ we can identify the value $Z(M)$ with an element of the ground field $k$. In other words, a topological field theory gives rise to a diffeomorphism invariant for closed manifolds of dimension $n$. Suppose we take the point of view that these diffeomorphism invariants are the main objects of interest. Of course, a topological field theory provides more data: we can evaluate $Z$ not only on closed manifolds of dimension $n$, but also on manifolds with boundary and manifolds of dimension $(n-1)$. This data can be viewed as supplying a set of rules which allow us to compute the invariant $Z(M)$ associated to a closed manifold $M$ by breaking $M$ up into pieces.

Let $A$ be a commutative Frobenius algebra over a field $k$. According to Example 1.1.11, the algebra $A$ determines a $2$-dimensional topological field theory $Z$. In particular, we can evaluate $Z$ on closed oriented $2$-manifolds $M$. Such manifolds are classified (up to orientation-preserving diffeomorphism) by a single invariant $g$, the genus, which ranges over the nonnegative integers. Consequently, for each $g\geq 0$, we can evaluate $Z$ on a closed surface $\Sigma_{g}$ of genus $g$, to obtain an element $Z(\Sigma_{g})\in k$. Let us compute the value of $Z(\Sigma_{g})$ for small values of $g$.

Suppose that $g=0$. In this case, $\Sigma_{g}$ is diffeomorphic to a $2$-sphere $S^{2}$, which we can view as obtained by gluing together hemispheres $S^{2}_{+}$ and $S^{2}_{-}$ along the equator $S^{2}_{+}\cap S^{2}_{-}\simeq S^{1}$. Consequently, $Z(S^{2})$ is obtained by composing the linear maps

Note that $Z(S^{1})$ coincides with the Frobenius algebra $A$, $Z(S^{2}_{-}):k\rightarrow A$ corresponds to the inclusion of the identity element of $A$, and $Z(S^{2}_{+}):A\rightarrow k$ is the trace map $\tr$. It follows that the invariant $Z(\Sigma_{g})$ is given by $\tr(1)\in k$.

Suppose that $g=1$. In this case, $\Sigma_{g}$ is diffeomorphic to a torus $S^{1}\times S^{1}$, which we can decompose into cylinders $S^{1}_{+}\times S^{1}$ and $S^{1}_{-}\times S^{1}$ meeting in the pair of circles

It follows that $Z(\Sigma_{g})$ is given by composing the linear maps

As in Example 1.1.9, we can identify $Z(\{\pm 1\}\times S^{1})$ with the tensor product $Z(S^{1})\otimes Z(\overline{S}^{1})\simeq A\otimes A^{\vee}=\End(A)$ (of course, $A$ is isomorphic to its dual, since the trace pairing $(a,b)\mapsto\tr(ab)$ is a nondegenerate bilinear form on $A$, but we will not use this observation). In terms of this identification, the map $Z(S^{1}_{-}):k\rightarrow\End(A)$ corresponds to the inclusion of the identity element $\id_{A}$, while $Z(S^{1}_{+}):\End(A)\rightarrow k$ is given by the trace (on the matrix ring $\End(A)$, which is unrelated to the trace on $A$). It follows that $Z(\Sigma_{g})$ is equal to the trace of $\id_{A}$: in other words, the dimension of the Frobenius algebra $A$.

It is possible to continue this analysis, and to compute all of the invariants $\{Z(\Sigma_{g})\}_{g\geq 0}$ in terms of the structure constants for the multiplication and trace on $A$; we leave the details to the interested reader.

We can attempt to use the reasoning of Example 1.2.1 in any dimension. Suppose that $Z$ is a topological field theory of dimension $n$. For every oriented $n$-manifold $M$, we can regard $M$ as a bordism from the empty set to $\bd M$, so that $Z(M):Z(\emptyset)\rightarrow Z(\bd M)$ can be regarded as an element of the vector space $Z(\bd M)$. The requirement that $Z$ be a functor can be translated as follows: suppose that we are given a closed $(n-1)$-dimensional submanifold $N\subseteq M$ which partitions $M$ into two pieces $M_{0}$ and $M_{1}$. Then $Z(M)$ is the image of

under the map $Z(\bd M)\otimes Z(N)\otimes Z(\overline{N})\rightarrow Z(\bd M)$ induced by the perfect pairing $Z(N)\otimes Z(\overline{N})\rightarrow k$ of Proposition 1.1.8. In other words, Definition 1.1.5 provides a rule for computing the invariant $Z(M)$ in terms of any decomposition $M=M_{0}\coprod_{N}M_{1}$ along a closed submanifold $N$ of codimension $1$. We might now ask: is it possible to break $M$ up into “simple” pieces by means of the above procedure? To put the question another way, is it possible to specify a short list of “simple” $n$-manifolds with boundary $\{M_{\alpha}\}$, such that any $n$-manifold can be assembled by gluing together manifolds appearing in the list $\{M_{\alpha}\}$ along components of their boundaries? When $n=2$, this question has an affirmative answer: every oriented surface $\Sigma$ can be obtained by gluing together disks, cylinders, and pairs of pants.

Unfortunately, the above method becomes increasingly inadequate as the dimension $n$ grows. If $M$ is a manifold of large dimension, then it is generally not possible to simplify $M$ very much by cutting along closed submanifolds of codimension $1$ (and these submanifolds are themselves very complicated objects when $n\gg 0$). What we would really like to do is to chop $M$ up into very small pieces, say, by choosing a triangulation of $M$. We might then hope to somehow recover the invariant $Z(M)$ in terms of the combinatorics of the triangulation. A triangulation of an $n$-manifold $M$ allows us to write $M$ as a union $\bigcup_{\alpha}\Delta^{n}_{\alpha}$ of finitely many $n$-simplices, which we can regard as a very simple type of $n$-manifold with boundary. In other words, $M$ can be obtained by gluing together a collection of simplices. However, the nature of the gluing is somewhat more complicated in this case: in general, we must allow ourselves to glue along submanifolds which are not closed, but which themselves have boundary. Definition 1.1.5 makes no provision for this sort of generalized gluing, which requires us to contemplate not only manifolds of dimension $n$ and $n-1$, but also manifolds of lower dimension. For this reason, various authors have proposed refinements of Definition 1.1.5, such as the following:

Let $k$ be a field. A topological field theory $Z$ gives rise to the following data:

For every closed oriented $n$-manifold $M$, an element $Z(M)\in k$.

For every closed oriented $(n-1)$-manifold $M$, a $k$-vector space $Z(M)$. When $M$ is empty, the vector space $Z(M)$ is canonically isomorphic to $k$.

For every oriented $n$-manifold $M$, an element $Z(M)$ of the vector space $Z(\bd M)$. In the special case where $M$ is closed, this should coincide with the element specified by $(a)$ under the isomorphism $Z(\bd M)=Z(\emptyset)\simeq k$ of $(b)$.

A $2$-extended topological field theory consists of data $(a)$ through $(c)$ as above, together with the following:

For every closed oriented $(n-2)$-manifold $M$, a $k$-linear category $Z(M)$. That is, $Z(M)$ is a category such that for every pair of objects $x,y\in Z(M)$, the set of morphisms $\Hom_{Z(M)}(x,y)$ has the structure of a $k$-vector space, and composition of morphisms is given by bilinear maps. Furthermore, when $M$ is empty, the category $Z(M)$ should be (canonically equivalent to) the category $\Vect(k)$ of vector spaces over $k$.

For every oriented $(n-1)$-manifold $M$, an object $Z(M)$ of the $k$-linear category $Z(\bd M)$. In the special case where $M$ is closed, $Z(M)$ should coincide with the vector space specified by $(b)$ under the equivalence $Z(\bd M)=Z(\emptyset)\simeq\Vect(k)$ of $(d)$.

Definition 1.2.2 is very much incomplete: it specifies a large number of invariants, but does not say very much about how they should be related to one another. For example, if $Z$ is an ordinary topological field theory and $B$ is a bordism from an oriented $(n-1)$-manifold $M$ to another oriented $(n-1)$-manifold $N$, then $Z$ associates a linear map $Z(B):Z(M)\rightarrow Z(N)$; clause $(c)$ of Definition 1.2.2 describes only the case where $M$ is assumed to be empty. Similarly, we should demand that if $B$ is a bordism from an oriented $(n-2)$-manifold $M$ to another oriented $(n-2)$-manifold $N$, then $B$ determines a functor $Z(B):Z(M)\rightarrow Z(N)$ (the data described in $(e)$ is reduces to the special case where $M$ is empty, so that $Z(M)\simeq\Vect(k)$, and we evaluate the functor on the vector space $k\in\Vect(k)$). Of course, this is only the tip of the iceberg: we should also demand coherence conditions which describe the behavior of the invariant $Z(B)$ when $B$ is obtained by gluing bordisms together, or as a disjoint union of bordisms, or varies by a bordism between $(n-1)$-manifolds with boundary (to properly formulate the relevant structure, we need to contemplate $n$-manifolds with corners). In the non-extended case, the language of category theory allowed us to summarize all of this data in a very succinct way: a topological field theory is simply a symmetric monoidal functor from the category ${\text{\bf Cob}}(n)$ to the category $\Vect(k)$. There is an analogous picture in the $2$-extended situation, but it requires us to introduce the language of $2$-categories.

A strict $2$-category is a category enriched over categories. In other words, a strict $2$-category $\calC$ consists of the following data:

A collection of objects, denoted by $X,Y,Z,\ldots$

For every pair of objects $X,Y\in\calC$, a category $\OHom_{\calC}(X,Y)$.

For every object $X\in\calC$, a distinguished object $\id_{X}\in\OHom_{\calC}(X,Y)$.

For every triple of objects $X,Y,Z\in\calC$, a composition functor

The objects $\{\id_{X}\}_{X\in\calC}$ are units with respect to composition: in other words, for every pair of objects $X,Y\in\calC$, the functors

given by composition with $\id_{X}$ are the identity.

Composition is strictly associative: that is, for every quadruple of objects $W,X,Y,Z\in\calC$, the diagram of functors

Let $k$ be a field. There is a strict $2$-category $\Vect_{2}(k)$, which may be described as follows:

The objects of $\Vect_{2}(k)$ are cocomplete $k$-linear categories: that is, $k$-linear categories $\calC$ which are closed under the formation of direct sums and cokernels.

Given a pair of objects $\calC,\calD\in\Vect_{2}(k)$, we define $\OHom_{\Vect_{2}(k)}(\calC,\calD)$ to be the category of cocontinuous $k$-linear functors from $\calC$ to $\calD$: that is, functors $F:\calC\rightarrow\calD$ which preserve cokernels and direct sums, and such that for every pair of objects $x,y\in\calC$, the induced map $\Hom_{\calC}(x,y)\rightarrow\Hom_{\calD}(Fx,Fy)$ is $k$-linear.

Composition and identity morphisms in $\Vect_{2}(k)$ are defined in the obvious way.

For every nonnegative integer $n\geq 2$, we can attempt to define a strict $2$-category ${\text{\bf Cob}}_{2}(n)$ as follows:

The objects of ${\text{\bf Cob}}_{2}(n)$ are closed oriented manifolds of dimension $(n-2)$.

Given a pair of objects $M,N\in{\text{\bf Cob}}_{2}(n)$, we define a category $\calC=\OHom_{{\text{\bf Cob}}_{2}(n)}(M,N)$ as follows. The objects of $\calC$ are bordisms from $M$ to $N$: that is, oriented $(n-1)$-manifolds $B$ equipped with a diffeomorphism $\bd B\simeq\overline{M}\coprod N$. Given a pair of objects $B,B^{\prime}\in\calC$, we let $\Hom_{\calC}(B,B^{\prime})$ denote the collection of all (oriented) diffeomorphism classes of (oriented) bordisms $X$ from $B$ to $B^{\prime}$. Here we require that $X$ reduce to the trivial bordism along the common boundary $\bd B\simeq\overline{M}\coprod N\simeq\bd B^{\prime}$, so that $X$ can be regarded as an $n$-manifold with boundary

Unfortunately, this definition does not (immediately) yield a strict $2$-category, because it is difficult to define a strictly associative composition law

Roughly speaking, given a bordism $B$ from $M$ to $M^{\prime}$ and another bordism $B^{\prime}$ from $M^{\prime}$ to $M^{\prime\prime}$, we would like to define $c(B,B^{\prime})$ to be the bordism obtained by gluing $B$ to $B^{\prime}$ along $M^{\prime}$. We encounter two difficulties:

In order to endow $c(B,B^{\prime})$ with a smooth structure, we need to make some additional choices, such as a smooth collar neighborhood of $M^{\prime}$ in both $B$ and $B^{\prime}$. These choices were irrelevant in Definition 1.1.1, because we were only interested in the bordism $c(B,B^{\prime})$ up to diffeomorphism. However, to fit into the mold of Definition 1.2.3, we need $c(B,B^{\prime})$ to be defined on the nose.

Given a triple of composable bordisms $B:M\rightarrow M^{\prime}$, $B^{\prime}:M^{\prime}\rightarrow M^{\prime\prime}$, and $B^{\prime\prime}:M^{\prime\prime}\rightarrow M^{\prime\prime\prime}$, the associative law of Definition 1.2.3 requires that we have an equality of bordisms

This may be difficult to arrange: what we see in practice is a canonical homeomorphism between the right and left hand sides (which can be promoted to a diffeomorphism, provided that we have correctly dealt with problem $(i)$).

For the purposes of studying $2$-extended topological field theories, it is vitally important that our categorical formalism should incorporate Example 1.2.5. There are two possible means by which we might accomplish this:

Adjust the definition of ${\text{\bf Cob}}_{2}(n)$ so that issues $(i)$ and $(ii)$ do not arise. For example, we can address problem $(i)$ by introducing a more complicated notion of bordism which keeps track of collar neighborhoods of the boundary. Issue $(ii)$ is a bigger nuisance: though it is possible to “rectify” the composition law on ${\text{\bf Cob}}_{2}(n)$ to make it strictly associative, it is somewhat painful and technically inconvenient to do so.

Adjust Definition 1.2.3 so that it incorporates Example 1.2.5 more easily. This can be accomplished by introducing the definition of a (nonstrict) $2$-category (also called a weak $2$-category or a bicategory), where we do not require composition to be associative on the nose but only up to coherent isomorphism.

We will adopt approach $(b)$, and work with the $2$-categories rather than strict $2$-categories. The advantage of this approach is that it more easily accomodates Example 1.2.5 and variations thereof. The disadvantage is that the definition of a $2$-category is more complicated than Definition 1.2.3, because we need to define “up to coherent isomorphism” precisely. We will defer a more precise discussion until §2.1; for the moment, we will simply take for granted that there is a good theory of $2$-categories which incorporates the examples above and use it to give a more complete formulation of Definition 1.2.2:

Let $k$ be a field. An $2$-extended topological field theory of dimension $n$ is a symmetric monoidal functor $Z:{\text{\bf Cob}}_{2}(n)\rightarrow\Vect_{2}(k)$ between $2$-categories.

In order to make sense of Definition 1.2.6, we need to understand ${\text{\bf Cob}}_{2}(n)$ and $\Vect_{2}(k)$ not only as $2$-categories, but as symmetric monoidal $2$-categories. In the case of ${\text{\bf Cob}}_{2}(n)$, this is straightforward: the tensor product operation is simply given by disjoint unions of manifolds, just as for ${\text{\bf Cob}}(n)$. The tensor product on $\Vect_{2}(k)$ is a bit more subtle. To describe it, let us first recall how to define the tensor product of a pair of vector spaces $U$ and $V$ over $k$. Given a third $k$-vector space $W$, we can define the notion of a bilinear map from $U\times V$ into $W$: this is a map $b:U\times V\rightarrow W$ which is linear separately in each variable (in other words, we require that for each $u\in U$ the map $v\mapsto b(u,v)$ is linear, and similarly for each $v\in V$ the map $u\mapsto b(u,v)$ is linear). The tensor product $U\otimes V$ is defined to be the recipient of a universal bilinear map $U\times V\rightarrow U\otimes V$. In other words, $U\otimes V$ is characterized by the following universal property: giving a linear map from $U\otimes V$ into another $k$-vector space $W$ is equivalent to giving a bilinear map $U\times V\rightarrow W$.

We can apply the same reasoning to define a tensor product operation in the setting of (cocomplete) $k$-linear categories. We begin by defining the analogue of the notion of a bilinear map: given a triple of cocomplete $k$-linear categories $\calC$, $\calD$, and $\calE$, we will say that a functor $F:\calC\times\calD\rightarrow\calE$ is $k$-bilinear if for every object $C\in\calC$ the functor $D\mapsto F(C,D)$ is cocontinuous and $k$-linear, and for every object $D\in\calD$ the functor $C\mapsto F(C,D)$ is cocontinuous and $k$-linear. We can then attempt to define a tensor product $\calC\otimes\calD$ by demanding the following universal property: for every cocomplete $k$-linear category $\calE$, there is an equivalence between the category of cocontinuous $k$-linear functors $\calC\otimes\calD\rightarrow\calE$ with the category of $k$-bilinear functors $\calC\times\calD\rightarrow\calE$. Of course, it takes some effort to prove that the tensor product $\calC\otimes\calD$ exists (and a bit more work to show that it is associative); we will not dwell on this point, since the strict $2$-category $\Vect_{2}(k)$ will soon disappear from our discussion of topological field theories.

Definition 1.2.6 should be regarded as a more elaborate version of Definition 1.1.5. To explain this, we note that if $\calC$ is an arbitrary symmetric monoidal $2$-category, then we can extract a symmetric monoidal category $\Omega\calC=\bHom_{\calC}({\bf 1},{\bf 1})$ of morphisms from the unit object to itself in $\calC$. Applying this construction in the situations of Example 1.2.4 and 1.2.5, we obtain equivalences

Consequently, any $2$-extended field theory $Z:{\text{\bf Cob}}_{2}(n)\rightarrow\Vect_{2}(k)$ determines a symmetric monoidal functor $\Omega Z:{\text{\bf Cob}}(n)\rightarrow\Vect(k)$, which we can regard as an $n$-dimensional topological field theory in the sense of Definition 1.1.5.

In general, a $2$-extended topological field theory $Z:{\text{\bf Cob}}_{2}(n)\rightarrow\Vect_{2}(k)$ contains a great deal more information than its underlying topological field theory $\Omega Z$, which can be useful in performing calculations. For example, suppose that we wish to compute $Z(M)=(\Omega Z)(M)$, where $M$ is a closed oriented $n$-manifold. Knowing that $Z(M)$ is the value of a topological field theory $\Omega Z$ on $M$ allows us to compute $Z(M)$ by cutting $M$ along closed submanifolds of codimension $1$. The $2$-extended field theory $Z$ itself gives us more flexibility: we can cut $M$ along $(n-1)$-manifolds with boundary. However, this freedom is still somewhat limited: given a decomposition $M=M_{0}\coprod_{N}M_{1}$, we can try to reconstruct $Z(M)$ in terms of the constituents $Z(M_{0})$, $Z(M_{1})$, and $Z(N)$. In particular, we need to understand $Z(N)$, where $N$ has dimension $(n-1)$. If $n$ is large, we should expect $N$ to be quite complicated. It is therefore natural to try to break $N$ into simpler pieces. Definition 1.2.6 gives us a limited amount of freedom to do so: given a decomposition $N=N_{0}\coprod_{P}N_{1}$, where $P$ is a closed $(n-2)$-manifold, we can compute $Z(N)$ in terms of $Z(N_{0})$, $Z(P)$, and $Z(N_{1})$. However, we cannot generally simplify $N$ very much by cutting along closed submanifolds: it is again necessary to allow cutting along $(n-2)$-manifolds with boundary. For this, we need to consider topological field theories which are even more “extended”. To make sense of these ideas, we need to introduce a bit more terminology.

Let $n$ be a nonnegative integer. We define the notion of a strict $n$-category by induction on $n$:

If $n=0$, then a strict $n$-category is a set.

If $n>0$, then a strict $n$-category is a category enriched over strict $(n-1)$-categories. In other words, a strict $n$-category $\calC$ consists of the following data:

For every pair of objects $X,Y\in\calC$, a strict $(n-1)$-category $\OHom_{\calC}(X,Y)$.

Identity objects $\id_{X}\in\OHom_{\calC}(X,X)$ and composition maps

satisfying the usual unit and associativity conditions.

Let us take a moment to unwind Definition 1.2.9. A strict $n$-category $\calC$ is a mathematical structure in which one has:

For every pair of objects $X,Y\in\calC$, a collection of morphisms from $X$ to $Y$, called $1$-morphisms.

For every pair of objects $X$ and $Y$ and every pair of morphisms $f,g:X\rightarrow Y$, a collection of morphisms from $f$ to $g$, called $2$-morphisms.

For every pair of objects $X$ and $Y$, every pair of morphisms $f,g:X\rightarrow Y$, and every pair of $2$-morphisms $\alpha,\beta:f\rightarrow g$, a collection of morphisms from $\alpha$ to $\beta$, called $3$-morphisms.

Moreover, these morphisms are equipped with various notions of composition, which are strictly associative at every level.

When $n=1$, Definition 1.2.9 recovers the usual notion of category. When $n=2$, it reduces to Definition 1.2.3. For $n>2$, Definition 1.2.9 is poorly behaved. However, there is a related notion of $n$-category (or weak $n$-category), where one requires composition to be associative only up to isomorphism, rather than “on the nose”. Most of the examples of $n$-categories which arise naturally (such as the example we will discuss below) are not equivalent to strict $n$-categories. We will review the theory of $n$-categories in §1.3, and sketch a more precise definition in §2.1.

Suppose given a pair of nonnegative integers $k\leq n$. Then there exists a $k$-category which we will denote by ${\text{\bf Cob}}_{k}(n)$, which can be described informally as follows:

The objects of ${\text{\bf Cob}}_{k}(n)$ are closed oriented $(n-k)$-manifolds

Given a pair of objects $M,N\in{\text{\bf Cob}}_{k}(n)$, a $1$-morphism from $M$ to $N$ is a bordism from $M$ to $N$: that is, a $(n-k+1)$-manifold $B$ equipped with a diffeomorphism $\bd B\simeq\overline{M}\coprod N$.

Given a pair of objects $M,N\in{\text{\bf Cob}}_{k}(n)$ and a pair of bordisms $B,B^{\prime}:M\rightarrow N$, a $2$-morphism from $B$ to $B^{\prime}$ is a bordism from $B$ to $B^{\prime}$, which is required to be trivial along the boundary: in other words, a manifold with boundary

A $k$-morphism in ${\text{\bf Cob}}_{k}(n)$ is an $n$-manifold $X$ with corners, where the structure of $\bd X$ is determined by the source and target of the morphism. Two $n$-manifolds with (specified) corners $X$ and $Y$ determine the same $n$-morphism in ${\text{\bf Cob}}_{k}(n)$ if they differ by an orientation-preserving diffeomorphism, relative to their boundaries.

Composition of morphisms (at all levels) in ${\text{\bf Cob}}_{k}(n)$ is given by gluing of bordisms.

Here we encounter the same issues as in Definition 1.2.5, but they are somewhat more serious. For $n>2$, it is not possible to massage the above definition to produce a strict $n$-category: gluing of bordisms is, at best, associative up to diffeomorphism. Nevertheless, ${\text{\bf Cob}}_{k}(n)$ is a perfectly respectable example of a (nonstrict) $n$-category: we will sketch a more precise definition of it in §2.2.

When $k=1$, the category ${\text{\bf Cob}}_{k}(n)$ described in Example 1.2.11 is just the usual bordism category ${\text{\bf Cob}}(n)$ of Definition 1.1.1. When $k=0$, we can identify ${\text{\bf Cob}}_{k}(n)$ with the set of diffeomorphism classes of closed, oriented $n$-manifolds.

We might now attempt to define an extended topological field theory to be a symmetric monoidal functor from the $n$-category ${\text{\bf Cob}}_{n}(n)$ of Example 1.2.11 into a suitable $n$-categorical generalization of $\Vect(k)$. Of course, there are many possible candidates for such a generalization. We will skirt the issue by adopting the following more general definition:

Let $\calC$ be a symmetric monoidal $n$-category. An extended $\calC$-valued topological field theory of dimension $n$ is a symmetric monoidal functor

At a first glance, Definition 1.2.13 appears much more complicated than its more classical counterpart, Definition 1.1.5. First of all, it is phrased in the language of $n$-categories, which we have not yet introduced. Second, an extended topological field theory supplies a great deal more data than that of an ordinary topological field theory: we can evaluate an extended field theory $Z$ on manifolds (with corners) of arbitrary dimension, rather than simply on closed $(n-1)$-manifolds and $n$-manifolds with boundary. Finally, the values of $Z$ on manifolds of low dimension are typically invariants of a very abstract and higher-categorical nature.