Definition 6 (0N82) · ScalingStacks

0N82

Definition 6. A braided monoidal 2-category $(\mathcal{C},\otimes,I,R,\tilde{R}_{(-|-,-)},\tilde{R}_{(-,-|-)})$ consists of:

(1)

A semistrict monoidal 2-category $(\mathcal{C},\otimes,1)$
(2)

A pseudonatural equivalence $R\colon\otimes\Rightarrow\otimes^{\rm op}$

(3)

Two invertible modifications $\tilde{R}_{(-|-,-)}$ and $\tilde{R}_{(-,-|-)}$ , giving for any objects $A,B,C\in\mathcal{C}$ the 2-isomorphisms

These data must satisfy the following conditions. First, for all objects $A,B,C,D\in\mathcal{C}$ the following diagrams commute:

$((\bullet\otimes\bullet\otimes\bullet)\otimes\bullet)$

\begin{array}[]{ll}1.\>\tilde{R}_{(A\otimes B,C|D)}&2.\>=\>\tilde{R}_{(A,B|D)}\otimes C\\ 3.\>=\>A\otimes\tilde{R}_{(B,C|D)}&4.\>=\>\tilde{R}_{(A,B\otimes C|D)}\end{array}

$(\bullet\otimes(\bullet\otimes\bullet\otimes\bullet))$

\begin{array}[]{ll}1.\>=\>\tilde{R}_{(A|B,C\otimes D)}&2.\>=\>B\otimes\tilde{R}_{(A|C,D)}\\ 3.\>=\>\tilde{R}_{(A|B,C)}\otimes D&4.\>=\>\tilde{R}_{(A|B\otimes C,D)}\end{array}

$((\bullet\otimes\bullet)\otimes(\bullet\otimes\bullet))$

\begin{array}[]{lll}1.\>=\>\tilde{R}_{(A,B|C)}\otimes D&2.\>=\>C\otimes\tilde{R}_{(A,B|D)}&3.\>=\>A\otimes\tilde{R}_{(B|C,D)}\\ 4.\>=\>\tilde{R}_{(A|C,D)}\otimes B&5.\>=\>\tilde{R}_{(A,B|C\otimes D)}&6.\>=\>\otimes_{(R_{A,C},R_{B,D})}\\ 7.\>=\>\tilde{R}_{(A\otimes B|C,D)}&\end{array}

Second, for any objects $A,B,C\in\mathcal{C}$ , we define two 2-isomorphisms corresponding to two proofs of the Yang–Baxter hexagon in a braided monoidal category:

We refer to these 2-morphisms as $S^{+}_{A,B,C}$ and $S^{-}_{A,B,C}$ , respectively. We require them to be equal:

$(S^{+}=S^{-})$ :