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.

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.

Tangles can be regarded as certain equivalence classes of 1-manifolds with boundary embedded in $[0,1]^{3}$, possibly equipped with extra structure such as an orientation or framing. Tangles are important because they make clear the relation between knot theory and braided monoidal categories. For example, framed oriented tangles form the ‘free balanced braided monoidal category on one object’ [16, 19, 31, 33], and this fact permits the construction of knot invariants from the categories of representations of quantum groups and other quasitriangular Hopf algebras [28].

The ‘tangle hypothesis’ of HDA suggests that this is part of a more general relationship between ‘$k$-tangles in $(n+k)$ dimensions’ and $k$-tuply monoidal $n$-categories. A $k$-tangle in $(n+k)$ dimensions is something like an isotopy equivalence class of $k$-manifolds with corners embedded in $[0,1]^{n+k}$. The tangle hypothesis proposes that these may be described algebraically using a specific $k$-tuply monoidal $n$-category $\mathcal{C}_{n,k}$, which has the cobordism $n$-category $\mathcal{C}_{n,\infty}$ as a limiting case.

A very interesting example is the case of 2-tangles in 4 dimensions: $n=2$, $k=2$. Topologists have already studied these 2-tangles, and the work of Carter and Saito [5] strongly suggests that they form a braided monoidal 2-category. In fact, Fischer [13] claims to have already shown this. His work is unfortunately rather unclear, but Kharlamov and Turaev [25] have begun to redo it more carefully. It should also be re-evaluated in the light of our definition of braided monoidal 2-category. One would eventually like to construct 2-tangle invariants from certain braided monoidal 2-categories, such as the category of representations of quasitriangular Hopf 2-algebras. Our center construction is a small step in this direction.

We cannot conclude this introduction without a word or two about the Zamolodchikov tetrahedron equation. In a braided monoidal category, the braiding automatically satisfies the Yang–Baxter equation. In other words, given objects $A,B,C$, the following diagram commutes:

In a braided monoidal 2-category, the Yang–Baxter equation holds only up to a 2-isomorphism. Topologically, this 2-isomorphism corresponds to a 2-tangle which intersected with $\{0\}\times[0,1]^{3}\subset[0,1]^{4}$ looks like the left side of Figure 4, and which intersected with $\{1\}\times[0,1]^{3}$ looks like the right side of Figure 4.

In Kapranov and Voevodsky’s theory [21] there are in fact two distinct such 2-isomorphisms, $S^{\pm}_{A,B,C}$, corresponding to two distinct proofs of the Yang–Baxter equation in a braided monoidal category. However, they give the same 2-tangle. There is also a deep relationship between $n$-category theory and homotopy theory, described in HDA and the references therein, and using this, Breen [4] has deduced that the condition $S^{+}=S^{-}$ should hold.

These facts constitute topological evidence that in the correct definition of a braided monoidal category, there should be an extra coherence law asserting that $S^{+}=S^{-}$. We also find algebraic evidence for this, as follows. It follows heuristically from our rough definition of center that a $k$-tuply monoidal $n$-category should embed canonically in its center when $\mathcal{C}$ happens to be already $(k+1)$-tuply monoidal. More precisely, if $\mathcal{C}$ is a $(k+1)$-tuply monoidal $n$-category and $\mathcal{C}_{0}$ is the underlying $k$-tuply monoidal $n$-category, there should be a faithful $(k+1)$-tuply monoidal $n$-functor from $\mathcal{C}$ to $\mathcal{Z}(\mathcal{C}_{0})$. For example, the center of a set $S$ works out to be the monoid ${\rm End}(S)$, but when $S$ happens already to be a monoid, there is a natural embedding $S\hookrightarrow{\rm End}(S)$ given by the left action of $S$ on itself. Similarly, a monoid equals its center when it is commutative, and a monoidal category naturally embeds in its center when it is braided [19, 20]. The third main goal of this paper is to show that a monoidal 2-category $\mathcal{C}$ embeds into $\mathcal{Z}(\mathcal{C})$ when $\mathcal{C}$ happens to be braided. However, for any monoidal 2-category $\mathcal{C}$ it turns out that $S^{+}=S^{-}$ in $\mathcal{Z}(\mathcal{C})$. Thus we can only achieve our goal if our definition of braided monoidal 2-category includes a coherence law saying that $S^{+}=S^{-}$. (It is worth noting that all our results except Theorem 18 hold without this extra coherence law.)

Finally, if $S=S^{+}=S^{-}$, Kapranov and Voevodky’s work [21] implies that the 2-morphisms $S_{A,B,C}$ satisfy an equation of their own, the Zamolodchikov tetrahedron equation. This is the higher-dimensional analogue of the Yang–Baxter equation, and it plays an important role in the theory of 2-tangles. Pictures of the Zamolodchikov tetrahedron equation in terms of 2-tangles can be found in the work of Carter and Saito [5]. Kapranov and Voevodsky, who do not assume $S^{+}=S^{-}$, write down 8 different versions of the Zamolodchikov equation and claim that these all follow from their definition of a braided monoidal 2-category. In our framework there is only one Zamolodchikov equation.