ScalingStacks

Braided monoidal categories are especially interesting because they give efficient procedures for constructing tangle invariants and 3-dimensional topological quantum field theories (TQFTs). Braided monoidal 2-categories appear to have analogous applications to 2-tangle invariants and 4-dimensional TQFTs. As the TQFT applications are more intimately related to the center construction, we begin with these. To see the patterns involved, it is helpful to consider first the rather trivial case of 2-dimensional TQFTs.

A 2-dimensional TQFT is a particular sort of symmetric monoidal functor $\mathcal{F}\colon 2\mathsf{Cob}\to\mathsf{Vect}$. Here the category $2\mathsf{Cob}$ has compact oriented 1-manifolds as objects and compact oriented cobordisms between them as morphisms, and it has a monoidal structure given by disjoint union. Similarly, the category $\mathsf{Vect}$ of finite-dimensional vector spaces and linear maps has a monoidal structure given by the usual tensor product. In both cases these categories have a natural symmetric structure, as described in HDA and the references therein. The sphere with 3 open discs removed, or ‘trinion’, can be thought of as a morphism in $2\mathsf{Cob}$:

and it gives rise to a product on the vector space $\mathcal{F}(S^{1})$:

One can easily check that this product is associative and commutative. Similarly, the closed disc can be thought of as a morphism

which gives rise to a unit for the product on $\mathcal{F}(S^{1})$:

Thus any 2-dimensional TQFT assigns to the circle a commutative algebra.

The true significance of this fact takes a bit of work to unearth. First, we can define a ‘commutative monoid object’ in any symmetric monoidal category to be an object $A$ equipped with a product and unit

satisfying analogs of the axioms for a commutative monoid. In particular, $\mathcal{F}(S^{1})$ is a commutative monoid object in $\mathsf{Vect}$, that is, a commutative algebra. However, this is really just a corollary of the fact that $S^{1}$ is a commutative monoid object in $2\mathsf{Cob}$, since a symmetric monoidal functor takes commutative monoid objects to commutative monoid objects. The real question is therefore, why is $S^{1}$ a commutative monoid object in $2\mathsf{Cob}$?

We shall not address this question directly. Instead, note that whenever $A$ is a commutative monoid object in a symmetric monoidal category, ${\rm hom}(1,A)$ is a commutative monoid. Thus ${\rm hom}(\emptyset,S^{1})$ is a commutative monoid. Conversely, understanding this commutative monoid should help us understand why $S^{1}$ is a commutative monoid object. Moreover, by following the patterns in Figure 1, we can learn something about the role of braided monoidal categories for 3-dimensional TQFTs, and braided monoidal 2-categories in 4-dimensional TQFTs.

An element of ${\rm hom}(\emptyset,S^{1})$ is an equivalence class of compact oriented 2-manifolds $M$ whose boundary has been identified with $S^{1}$. Alternatively, by fitting the circle inside a square in a standard way, we can think of $M$ as a 2-manifold with corners whose boundary is a square. Then, given $x,y\in{\rm hom}(\emptyset,S^{1})$ we can define a ‘vertical’ product $xy$ and a ‘horizontal’ product $x\otimes y$ as shown in Figure 2.

The appearance of the Eckmann–Hilton argument here suggests that we really have a 2-category with one object and one 1-morphism on our hands. Now, the ‘extended TQFT hypothesis’ in HDA suggests that the best way to understand $n$-dimensional TQFTs is in terms of a weak $n$-category $\mathcal{C}_{n,\infty}$ whose objects are 0-manifolds, whose morphisms are equivalence classes of 1-manifolds with boundary, whose 2-morphisms are equivalence classes of 2-manifolds with corners, and so on, each $(j+1)$-morphism being a kind of cobordism between $j$-morphisms. (Of course these manifolds should be compact and oriented; in general they should also be ‘framed’, but here we neglect this subtlety.) Making this hypothesis precise would require a general definition of weak $n$-categories, and also some careful differential topology. Even in its current vague form, though, it sheds some light on the situation at hand. $\mathcal{C}_{n,\infty}$ should have a distinguished object $\ast$, the positively oriented point. The 1-morphism $1_{\ast}$ should then correspond to the closed unit interval. When $n=2$, ${\rm hom}(1_{\ast},1_{\ast})$ should then be the set of all cobordisms from the interval to itself. These are just equivalence classes of 2-manifolds with corners whose boundary is the square! Thus ${\rm hom}(1_{\ast},1_{\ast})$ is isomorphic to ${\rm hom}(\emptyset,S^{1})$, but now the commutative monoid structure has a purely algebraic explanation: there is a 2-category having one object $\ast$, one 1-morphism $1_{\ast}$, and the set ${\rm hom}(1_{\ast},1_{\ast})$ as its 2-morphisms. Understand the isomorphism between ${\rm hom}(\emptyset,S^{1})$ and ${\rm hom}(1_{\ast},1_{\ast})$ in purely algebraic terms remains an interesting challenge; the solution will probably involve the theory of duality in $n$-categories.

Similarly, in the study of 3-dimensional TQFTs we expect to have a 3-category $\mathcal{C}_{3,\infty}$, and sitting inside this there should be a 3-category with one object $\ast$, one 1-morphism $1_{\ast}$, and the category ${\rm hom}(1_{\ast},1_{\ast})$ as its 2-morphisms and 3-morphisms. This category should thus be a braided monoidal category whose objects are 2-manifolds with corners having a square as boundary, and whose morphisms are cobordisms between these. Likewise, in the 4-dimensional case ${\rm hom}(1_{\ast},1_{\ast})$ would be a braided monoidal 2-category, and so on.

In fact, results along these lines already appear in the literature in the cases of dimensions 3 and 4, but in terms of ${\rm hom}(\emptyset,S^{1})$ rather than ${\rm hom}(1_{\ast},1_{\ast})$. This is less natural algebraically, but simpler topologically, because the theory of cobordisms between manifolds with corners is not well developed. So far, the clearest description of ${\rm hom}(\emptyset,S^{1})$ as a braided monoidal category in dimension 3 and a braided monoidal 2-category in dimension 4 has been given by Crane and Yetter [8]. There are many interesting projects left to do, however. For example, in dimension 3 it should be possible to use existing results of Kerler [24] and others to obtain a presentation of ${\rm hom}(\emptyset,S^{1})$ as a braided monoidal category, and to compare the answer to what one would predict using the extended TQFT hypothesis. This presentation should explain the already known conditions required to construct 3-dimensional TQFTs, such as Chern–Simons theory, which associate a braided monoidal category to the circle [6, 29]. In dimension 4 one still needs to carefully check whether ${\rm hom}(\emptyset,S^{1})$ meets our definition of a braided monoidal 2-category, and then if possible obtain a presentation of it. This may allow the construction of 4-dimensional TQFTs from braided monoidal 2-categories meeting certain conditions. If so, our center construction may serve as a source of 4-dimensional TQFTs.