# Coherence for symmetric monoidal categories

Let $\mathcal{C}$ be a monoidal category. By Mac Lane's coherence theorem for monoidal categories, there are strong monoidal functors $F : \mathcal{C} \to \mathcal{C}_s$ and $G : \mathcal{C}_s \to \mathcal{C}$ such that $\mathcal{C}_s$ is a strict monoidal category and $(F, G)$ form an equivalence of categories. [CWM, Ch. XI, §2] Informally, this tells us that we may pretend that the associators $\alpha$ and unitors $\lambda, \rho$ are identity arrows rather than mere isomorphisms.

My question concerns the analogue for symmetric monoidal categories: suppose $\mathcal{C}$ has a braiding $$\gamma_{a, b} : a \otimes b \to b \otimes a$$ such that $\gamma_{b, a} \circ \gamma_{a, b} = \textrm{id}$, and suppose it is compatible with the monoidal structure making $\mathcal{C}$ into a symmetric monoidal category.

1. Does $F \gamma$ make $\mathcal{C}_s$ into a symmetric monoidal category?

2. Assuming that (1) is true, do "all" diagrams involving $\gamma$ with atomic subscripts commute in $\mathcal{C}_s$? More precisely, let $\sigma$ be a permutation on $k$ letters, and suppose that $\sigma = \tau_1 \cdots \tau_m = \tilde{\tau}_1 \cdots \tilde{\tau}_n$ for some transpositions $\tau_i$, $\tilde{\tau}_j$ of adjacent pairs; does the equation $\tau_1 \cdots \tau_m = \tilde{\tau}_1 \cdots \tilde{\tau}_n$ still hold true when I interpret $\tau_i$ and $\tilde{\tau}_j$ as the corresponding braiding operation?

For example, $$(1 \, 2 \, 3) = (1 \, 2) (2 \, 3) = (2 \, 3) (1 \, 2) (2 \, 3) (1 \, 2)$$ so I expect that $$(\gamma_{a, c} \otimes \textrm{id}_b) \circ (\textrm{id}_a \otimes \gamma_{b, c}) = (\textrm{id}_a \otimes \gamma_{b, c}) \circ (\gamma_{a, b} \otimes \textrm{id}_c) \circ (\textrm{id}_a \otimes \gamma_{b, c}) \circ (\gamma_{a, b} \otimes \textrm{id}_c)$$ which is indeed correct.

3. Let $a = a_1 \otimes \cdots \otimes a_n$, $b = b_1 \otimes \cdots \otimes b_m$. Let $\sigma$ be the permutation $$(a_1, \ldots, a_n, b_1, \ldots, b_m) \mapsto (b_1, \ldots, b_m, a_1, \ldots, a_n)$$ Assuming (2) holds, let $\gamma_\sigma : a \otimes b \to b \otimes a$ be the morphism corresponding to the permutation $\sigma$. Then, is it true that $\gamma_{b, a} = \gamma_\sigma$?

For example, take $n = 2$, $m = 1$. I expect it to be true that $$\gamma_{a \otimes b, c} = (\gamma_{a, c} \otimes \textrm{id}_b) \circ (\textrm{id}_a \otimes \gamma_{b, c})$$ and this turns out to be an instance of the hexagon axiom, in the special case of a strict monoidal category. But what if $n > 2$ or $m > 1$?

4. Does this imply that "all" diagrams involving $\alpha, \lambda, \rho, \gamma$ commute in $\mathcal{C}$?

I suspect that the answers to all the questions are yes, at least if I understood Mac Lane [CWM, Ch. XI, §3] correctly, but I haven't found an explicit statement of the implied coherence theorem for symmetric monoidal categories. (There is an obvious way of defining a "free symmetric strict monoidal category" generated by another category, and I imagine there is some way of defining a "free symmetric monoidal category"; the coherence theorem ought to state that the two are equivalent via strong braided monoidal functors.)

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1. Yes. By functoriality and strong monoidality, $F\gamma$ must satisfy the symmetry axioms.
3. The fact that you say "the" symmetry map corresponding the given permutation is already a consequence of coherence for SMC's. All symmetry maps corresponding to the given permutation (including $\gamma_{a,b}$) are equal.