Definition 17 (0N8F) · ScalingStacks

Definition 17. A braided monoidal $2$ -functor consists of

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a monoidal 2-functor $(\mathcal{F},\xi,\alpha)$
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a modification

$\mathcal{F}_{R}:\xi\circ\mathcal{F}(R)\Rightarrow R^{\prime}\circ\xi,$

such that the following two diagrams commute, expressing the fact that $\mathcal{F}$ respects the modifications $\tilde{R}_{(-|-,-)}$ and $\tilde{R}_{(-,-|-)}$ up to $\xi$ .

\begin{array}[]{llll}1.\hat{=}\>R^{\prime}_{(\mathcal{F}(X),\xi)}&2.\hat{=}\>\mathcal{F}_{R}&3.\hat{=}\>\tilde{R^{\prime}}_{(\mathcal{F}(X)|\mathcal{F}(Y),\mathcal{F}(Z))}&4.\hat{=}\>\mathcal{F}(\tilde{R}_{(X|Y,Z)})\\ 5.\hat{=}\>\mathcal{F}_{R}\otimes\mathcal{F}(Z)&6.\hat{=}\>\mathcal{F}(Y)\otimes\mathcal{F}_{R}&7.\hat{=}\>\alpha_{Y,X,Z}&8.\hat{=}\>\xi_{R,Z}\\ 9.\hat{=}\>\xi_{Y,R}&10.\hat{=}\>\alpha_{Y,Z,X}&11.\hat{=}\>\alpha_{X,Y,Z}\end{array}

\begin{array}[]{llll}1.\hat{=}\>R^{\prime}_{(\xi,\mathcal{F}(Z))}&2.\hat{=}\>\mathcal{F}_{R}&3.\hat{=}\>\tilde{R^{\prime}}_{(\mathcal{F}(A),\mathcal{F}(B)|\mathcal{F}(Z))}&4.\hat{=}\>\mathcal{F}(\tilde{R}_{(X,Y|Z)})\\ 5.\hat{=}\>\mathcal{F}(X)\otimes\mathcal{F}_{R}&6.\hat{=}\>\mathcal{F}_{R}\otimes\mathcal{F}(Y)&7.\hat{=}\>\alpha_{X,Z,Y}&8.\hat{=}\>\xi_{X,R}\\ 9.\hat{=}\>\xi_{R,Y}&10.\hat{=}\>\alpha_{Z,X,Y}&11.\hat{=}\>\alpha_{X,Y,Z}\end{array}