ScalingStacks

To efficiently define braided monoidal 2-categories it is useful to exploit the fact that $(2\mathsf{Cat},\otimes_{\rm G},\mathcal{I})$ is closed, i.e., enriched over itself [17]. Put more explicitly, what this means is that $2\mathsf{Cat}$ can be regarded as a semistrict 3-category having small 2-categories as objects, 2-functors as morphisms, ‘pseudonatural transformations’ as 2-morphisms, and ‘modifications’ as 3-morphisms [3, 23]. A pseudonatural transformation $T$ between 2-functors $\mathcal{F},\mathcal{G}\colon\mathcal{C}\to\mathcal{D}$ assigns to each object $A\in\mathcal{C}$ a morphism $T_{A}\colon\mathcal{F}(A)\to\mathcal{G}(A)$ which satisfies the definition of a natural transformation only up to a specified isomorphism. Thus, $T$ also assigns to each morphism $f\colon A\to B$ in $\mathcal{C}$ a 2-isomorphism $T_{f}$ as follows:

As explained in the introduction, in an $n$-category the notion of ‘isomorphism’ can be weakened to a recursively defined notion of ‘equivalence’. In the case of $2\mathsf{Cat}$ this gives the following concepts. A modification $\alpha$ from the pseudonatural transformation $S$ to the pseudonatural transformation $T$ is ‘invertible’ if there is a modification $\alpha^{-1}$ from $T$ to $S$ such that $\alpha\alpha^{-1}=1_{S}$ and $\alpha^{-1}\alpha=1_{T}$. A pseudonatural transformation $T$ from $\mathcal{F}$ to $\mathcal{G}$ is a ‘pseudonatural equivalence’ if there is a pseudonatural transformation $\overline{T}\colon\mathcal{G}\to F$ and invertible modifications

There is a similar notion at the level of 2-functors, but we will not need it.

Every semistrict monoidal 2-category has a second, ‘opposite’ tensor product:

There is an analogous opposite tensor product for strict monoidal categories, and a strict braided monoidal category is just a strict monoidal category equipped with a natural isomorphism $R\colon\otimes\Rightarrow\otimes^{\rm op}$, the ‘braiding’, such that the following triangles commute:

The definition of a semistrict braided monoidal 2-category is very similar. However, instead of a strict monoidal category, one starts with a semistrict monoidal 2-category. Instead of the braiding being a natural transformation, it is a pseudonatural equivalence. Instead of the equations above holding ‘on the nose’, they hold up to specified invertible modifications. Finally, these modifications must satisfy 3 new coherence laws discovered by Kapranov and Voevodsky, together with the equation $S^{+}=S^{-}$ discussed in Section 1.3.

In all that follows, in diagrams we sometimes denote the tensor product of objects simply by juxtaposition. We also label some clauses in the definition using the ‘hieroglyphic’ notation invented by Kapranov and Voevodsky.

We can unpack this definition to obtain an explicit list of operations and laws. In this form the definition is essentially due to Kapranov and Voevodsky, though with certain differences, which we list at the end of this section.

Note that by $(\to{\to}\otimes\bullet)$ resp. $(\bullet\otimes{\to}\to)$ and by the invertibility of the respective 2-morphisms, for any objects $A,B\in\mathcal{C}$ we have $R_{A,1_{B}}=1_{R_{A,B}}$ and $R_{1_{A},B}=1_{R_{A,B}}$.

The above lemma makes it clear that our definition of braided monoidal 2-category differs from that of Kapranov and Voevodsky in precisely the following points:

1. (1)

   Invertibility of the braiding. Our definition implies that the 1-morphisms $R_{A,B}$ are equivalences. Kapranov and Voevodsky make no invertibility assumptions on these 1-morphisms. Our definition would agree with theirs on this point, and otherwise stay the same, if we required $R\colon\otimes\to\otimes^{\rm op}$ to be merely a pseudonatural transformation, rather than a pseudonatural equivalence.

2. (2)

   $S^{+}=S^{-}$. As already noted, Kapranov and Voevodsky omit this condition.

3. (3)

   Naturality of $\tilde{R}_{(-|-,-)}$ and $\tilde{R}_{(-,-|-)}$. Our definition implies the commutativity of 6 triangular prisms expressing the naturality in each argument of these modifications. Kapranov and Voevodsky substitute cubes for 4 of these prisms, namely $(\bullet\otimes({\to}\otimes\bullet))$, $(\bullet\otimes(\bullet\otimes{\to}))$, $((\bullet\otimes{\to})\otimes\bullet)$ and $(({\to}\otimes\bullet)\otimes\bullet)$. By the following lemma one can deduce these cubes from the remaining data — but not, it appears, vice versa. In personal communication, Kapranov agreed that all these prisms should hold.

We refer to this cube with the hieroglyph $(\bullet\otimes({\to}\otimes\bullet))^{\prime}$. One can similarly prove the analogous cube corresponding to the hieroglyph $(\bullet\otimes(\bullet\otimes{\to}))^{\prime}$ commutes. Moreover, we can prove the commutativity of cubes corresponding to the hieroglyphs $((\bullet\otimes{\to})\otimes\bullet)^{\prime}$ and $(({\to}\otimes\bullet)\otimes\bullet)^{\prime}$ using $({\Downarrow}\otimes\bullet)$ and $(\to{\to}\otimes\bullet)$.