ScalingStacks

The second main goal of this paper is to give a procedure for constructing a braided monoidal 2-category as the ‘center’ $\mathcal{Z}(\mathcal{C})$ of of a monoidal 2-category $\mathcal{C}$. To appreciate this rather complicated procedure it is necessary to understand the general concept of ‘center’ proposed in HDA. In essence this concept is simple; all the complications arise from the lack of a good general theory of weak $n$-categories.

There is no ‘set of all sets’, but there is a class of all sets. Better still, there is a category $\mathsf{Set}$ having sets as objects and functions between them as morphisms. Similarly, there is a 2-category $\mathsf{Cat}$ having small categories as objects, functors between them as 1-morphisms, and natural transformations between functors as 2-morphisms. In general, we expect there to be a very important $(n+1)$-category $n\mathsf{Cat}$ having as objects all small $n$-categories (i.e., those for which the $j$-morphisms form a set). This has been worked out quite generally in the strict context, but in the weak context only for $n\leq 2$ [2, 17].

In terms of this idea, the ‘center’ of a small $k$-tuply monoidal $n$-category $\mathcal{C}$ is a small $(k+1)$-tuply monoidal $n$-category $\mathcal{Z}(\mathcal{C})$ defined as follows. Recall that $\mathcal{C}$ is really a special sort of $(n+k)$-category, namely one with only one $j$-morphism for $j<k$. Thus $\mathcal{C}$ is an object in $(n+k)\mathsf{Cat}$. Let $1_{1}=1_{\mathcal{C}}$ denote the identity 1-morphism of $\mathcal{C}$ in $(n+k)\mathsf{Cat}$, and recursively define

so that $1_{j}$ is a $j$-morphism. Then there should be a sub-$(n+k)$-category $\mathcal{Z}(\mathcal{C})$ of $(n+k)\mathsf{Cat}$ having $\mathcal{C}$ as its only object, $1_{\mathcal{C}}$ as its only 1-morphism, $1_{1_{\mathcal{C}}}$ as its only 2-morphism, and so on up to $1_{k}$, and then having all $(k+1)$-morphisms from $1_{k}$ to itself as $(k+1)$-morphisms, all $(k+2)$-morphisms between these as $(k+2)$-morphisms, and so on. Since $\mathcal{Z}(\mathcal{C})$ has only one $j$-morphism for $j<k+1$, it follows that $\mathcal{Z}(\mathcal{C})$ is a $(k+1)$-tuply monoidal $n$-category.

As this construction is a bit mind-boggling at first sight, let us illustrate it in the case $n=0$, $k=1$. Thus we begin with a small category $\mathcal{C}$ with only one object $\ast$. The set $\tilde{\mathcal{C}}$ of 1-morphisms of $\mathcal{C}$ can be an arbitrary monoid. Similarly, $\mathcal{Z}(\mathcal{C})$ is a 2-category with only one object and one 1-morphism, and the 2-morphisms of such a 2-category form a commutative monoid. More precisely, $\mathcal{Z}(\mathcal{C})$ is the sub-2-category of $\mathsf{Cat}$ having $\mathcal{C}$ as its only object, $1_{\mathcal{C}}$ as its only 1-morphism, and all natural transformations $T\colon 1_{\mathcal{C}}\to 1_{\mathcal{C}}$ as 2-morphisms. What is such a natural transformation in concrete terms? It must assign to the one object $\ast$ of $\mathcal{C}$ a morphism $T_{\ast}\colon\ast\to\ast$, such that for all $f\colon\ast\to\ast$ the following diagram commutes:

In other words, it is simply an element $T_{\ast}$ of the center of $\tilde{\mathcal{C}}$. Thus the generalized concept of center reduces in this case to the standard notion.

The case $n=1$, $k=1$ is more interesting. The center of a weak monoidal category is a weak braided monoidal category [19, 21, 26]. In particular, if $H$ is a Hopf algebra, the category $\mathsf{Reps}(H)$ of finite-dimensional comodules of $H$ is a weak monoidal category, and the center $\mathcal{Z}(\mathsf{Reps}(H))$ is then the category of representations of a coquasitriangular Hopf algebra $DH$ called the ‘quantum double’ of $H$. (Working with comodules and coquasitriangular Hopf algebras, rather than modules and quasitriangular Hopf algebras, serves as a technical convenience.) The quantum double construction, invented by Drinfeld [11], gives to many interesting coquasitriangular Hopf algebras. In particular, the quantum groups arising from semisimple Lie groups, while not quantum doubles themselves, are straightforward quotients thereof [20]. Thus the center construction can be regarded as an elegant approach to quantum groups, which, as we shall see, makes their appearance in 3-dimensional topology much less mysterious.

The class of theorems known as ‘Tannaka–Krein reconstruction theorems’ [9, 27, 32] further clarifies the relation between the center construction and quantum doubles. Given a Hopf algebra $H$, the category $\mathsf{Reps}(H)$ is a $\mathbb{C}$-linear abelian rigid monoidal category and equipped with a faithful $\mathbb{C}$-linear exact monoidal functor to $\mathsf{Vect}$. Conversely, given any such category $\mathcal{C}$ equipped with such a functor to $\mathsf{Vect}$, $\mathcal{C}$ is equivalent to $\mathsf{Reps}(H)$ for some Hopf algebra $H$ unique up to natural isomorphism. A similar theorem holds for $H$ coquasitriangular and $\mathcal{C}$ braided. Thus we may construct the quantum double of $H$ by first forming $\mathsf{Reps}(H)$, then taking the center $\mathcal{Z}(\mathsf{Reps}(H))$ of this category, and then applying Tannaka–Krein reconstruction to obtain $DH$.

It is natural to hope that other cases of the center construction will give interesting analogs of these results. The most interesting case that can be handled with our present limited understanding of weak $n$-categories is the case $n=2$, $k=1$: if $\mathcal{C}$ is a monoidal 2-category, one expects that $\mathcal{Z}(\mathcal{C})$ will be a braided monoidal 2-category. The difficulty with proving this result is that we lack a general theory of weak 4-categories. Thus we do not know the definition of a weak braided monoidal 2-category, and cannot use the expected result that $3\mathsf{Cat}$ forms a weak 4-category. Instead, we need to start with a semistrict monoidal category $\mathcal{C}$, explicitly describe the objects, morphisms, and 2-morphisms of $\mathcal{Z}(\mathcal{C})$, and then rather laboriously prove that it is indeed a semistrict monoidal 2-category.

In fact, it is natural to conjecture a kind of ‘categorification’ of the whole theory of quantum doubles. For example, one should be able to start with a ‘Hopf category’ as defined by Crane and Frenkel [7] — or, better, a ‘Hopf 2-algebra’ — and form the monoidal 2-category $\mathsf{Reps}(H)$ of its representations on ‘2-vector spaces’ [21, 34]. The monoidal 2-category $\mathsf{Reps}(H)$ should be equipped with a monoidal 2-functor to $2\mathsf{Vect}$ and satisfy various other conditions, and there should be a Tannaka–Krein theorem saying that, conversely, such data determine a Hopf 2-algebra, unique up to equivalence. The center $\mathcal{Z}(\mathsf{Reps}(H))$ should thus be a braided monoidal 2-category, and by Tannaka–Krein reconstruction should determine a Hopf 2-algebra $DH$, the ‘quantum double’ of $H$. Finally, one expects that this quantum double will be ‘quasitriangular’ in the sense defined by Crane and Frenkel [7]. More ambitiously, one might conjecture a similar correspondence between braided monoidal $n$-categories and quasitriangular Hopf $n$-algebras for higher $n$. We shall not attempt to make these conjectures precise and prove them here. However, it is helpful to keep them in mind when considering the applications of braided monoidal 2-categories to topology.