Lemma 7 (0N83) · ScalingStacks

0N83

Lemma 7. A braided monoidal 2-category $(\mathcal{C},\otimes,1,R,\tilde{R}_{(-|-,-)},\tilde{R}_{(-,-|-)})$ consists of the following data:

(1)

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

$(\bullet\otimes\bullet)$ For any two objects $A,B\in\mathcal{C}$ an equivalence $R_{A,B}\colon A\otimes B\to B\otimes A$

(3)

$({\to}\otimes\bullet)$ For any 1-morphism $f:A\to A^{\prime}$ and any object $B\in\mathcal{C}$ a 2-isomorphism

(4)

$(\bullet\otimes{\to})$ For any object $A\in\mathcal{C}$ and any 1-morphism $g\colon B\to B^{\prime}$ a 2-isomorphism

(5)

$((\bullet\otimes\bullet)\otimes\bullet)$ For any objects $A,B,C\in\mathcal{C}$ a 2-iso

(6)

$(\bullet\otimes(\bullet\otimes\bullet))$ For any objects $A,B,C\in\mathcal{C}$ a 2-isomorphism

Moreover, these data must satisfy the following conditions:

$({\to}\otimes{\to})$ For any 1-morphisms $f\colon A\to A^{\prime}$ and $g\colon B\to B^{\prime}$ the following cube commutes:

1.\>=\>\otimes_{f,g}\qquad 2.\>=\>\otimes_{g,f}\qquad 3.\>=\>R_{A,g}\qquad 4.\>=\>R_{A^{\prime},g}\qquad 5.\>=\>R_{f,B^{\prime}}\qquad 6.\>=\>R_{f,B}

$(\bullet\otimes{\Downarrow})$ For any object $A\in\mathcal{C}$ , any 1-morphisms $f,f^{\prime}\colon B\to B^{\prime}$ , and any 2-morphism $\beta\colon f\Rightarrow f^{\prime}$ , the following prism commutes:

$({\Downarrow}\otimes\bullet)$ A similar prism, left to the reader.

$(\to{\to}\otimes\bullet)$ For any pair of 1-morphisms $A\stackrel{{\scriptstyle f}}{{\to}}A^{\prime}\stackrel{{\scriptstyle f^{\prime}}}{{\to}}A^{\prime\prime}$ and any object $B\in\mathcal{C}$ , the 2-isomorphism $R_{ff^{\prime},B}$ coincides with the pasting

$(\bullet\otimes{\to}\to)$ A similar pasting law, left to the reader.

$((\bullet\otimes\bullet)\otimes{\to})$ For any objects $A,B,C\in\mathcal{C}$ and any 1-morphism $f\colon C\to C^{\prime}$ , the following triangular prism commutes:

1.\>=\>A\otimes R_{B,f}\qquad 2.\>=\>R_{A,f}\otimes B\qquad 3.\>=\>\tilde{R}_{(A,B|C)}\qquad 4.\>=\>\tilde{R}_{(A,B|C^{\prime})}\qquad 5.\>=\>R_{AB,f}

$({\to}\otimes(\bullet\otimes\bullet))$ A similar prism, left to the reader.

$(({\to}\otimes\bullet)\otimes\bullet)$ For any objects $A,B,C\in\mathcal{C}$ and any 1-morphism $f\colon A\to A^{\prime}$ , the following triangular prism commutes:

1.\>=\>\otimes_{(f,R_{B,C})}\qquad 2.\>=\>R_{f,C}\otimes B\qquad 3.\>=\>\tilde{R}_{(A,B|C)}\qquad 4.\>=\>\tilde{R}_{(A^{\prime},B|C)}\qquad 5.\>=\>R_{f\otimes B,C}

$((\bullet\otimes{\to})\otimes\bullet)$ , $(\bullet\otimes({\to}\otimes\bullet))$ and $(\bullet\otimes(\bullet\otimes{\to}))$ Similar prisms, left to the reader.

$((\bullet\otimes\bullet\otimes\bullet)\otimes\bullet)$ , $(\bullet\otimes(\bullet\otimes\bullet\otimes\bullet))$ , $((\bullet\otimes\bullet)\otimes(\bullet\otimes\bullet))$ As in Definition 6.

$S^{+}=S^{-}$ As in Definition 6.

0N84

Proof. The 1-equivalences $R_{A,B}$ and 2-isomorphisms $R_{f,B}$ and $R_{A,g}$ comprise the pseudonatural equivalence $R\colon\otimes\to\otimes^{\rm op}$ , and conditions $({\to}\otimes{\to})$ , $({\bullet}\otimes{\Downarrow})$ , $({\Downarrow}\otimes{\bullet})$ , $(\to{\to}\otimes{\bullet})$ and $({\bullet}\otimes{\to}\to)$ state that it is indeed a pseudonatural transformation. The 2-morphisms $\tilde{R}_{(A|B,C)}$ and $\tilde{R}_{(A,B|C)}$ comprise the invertible modifications $\tilde{R}_{(-|-,-)}$ and $\tilde{R}_{(-,-|-)}$ , and the commuting triangular prisms state that these are indeed modifications, expressing naturality in each argument. The remaining 4 conditions come from Definition 6. ∎