ScalingStacks

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.