$\newcommand{\M}{\mathcal{M}}\newcommand{\B}{\mathfrak{B}}\newcommand{\hom}{\operatorname{Hom}}\newcommand{\BM}{\mathsf{BM}}\newcommand{\SBM}{\mathsf{SBM}}\newcommand{\S}{\mathcal{S}}$I refer to the chapter: "Symmetry and braiding in monoidal categories" from CWM.

I've just finished this chapter, but I am unsatisfied with the statements concerning braided coherence.

The so-called "braided coherence theorem":

For any braided monoidal $\M$, $$\hom_{\BM}(\B,\M)\simeq\M$$Via an equivalence that assigns an $F$ in the LHS to the object $F(1)$ in the RHS.

Mac Lane proves this theorem by instead saying:

We know any braided monoidal $\M$ is equivalent to a strict braided monoidal $\S$ via functors which are strong monoidal in both directions. We will show then that: $$\hom_{\SBM}(\B,\S)\cong\S$$

I don't see why this is sufficient. It seems as if Mac Lane is implying: $$\hom_{\BM}(\B,\M)\simeq\hom_{\BM}(\B,\S)\simeq\hom_{\SBM}(\B,\S)$$If this is true, then his proof would indeed be sufficient. While I can believe the first equivalence, as we need only post-compose functors on either side with the equivalences $\M\to\S,\,\S\to\M$, I can't quite believe the second. I can't believe that we can easily promote a strong braided monoidal functor $F:\B\to\S$ to a strict $F'$, as the axiom on a monoidal natural isomorphism would here be (see the definitions below) - since $\mu_2=1$ by strictness - $\theta\otimes\theta=\theta\circ\mu_1$. But the obvious choice to make an equivalence of categories would just put $F'$ as the same functor $F$, but with a different $\mu$. So $\theta$ would be chosen to be the identity transformation (what other choice is there, that I'm missing?) and the axiom for it to be a monoidal transformation would not be satisfied in general.

Question: Why is it sufficient to prove $\hom_{\SBM}(\B,\S)\cong\S$?


I define "braiding" via the nLab definition, since Mac Lane's definitions are unfortunately outdated.

If $\M,\M'$ are braided categories, with associators, left and right unitors, unit and braidings $(\alpha,\lambda,\rho,e,\gamma)$ and $(\alpha',\lambda',\rho',e',\gamma')$ respectively, then we say: $F:\M\to\M'$ is a strong braided functor when there is a natural isomorphism: $\mu:F(-)\otimes F(-)\implies F(-\otimes-)$ and an isomorphism $\epsilon:e'\to F(e)$ which satisfy:

For all $x,y,z\in\M$, the composite: $$F(\alpha)\circ\mu\circ(1\otimes\mu):F(x)\otimes(F(y)\otimes F(z))\to F(x)\otimes F(y\otimes z)\to F(x\otimes (y\otimes z))\to F((x\otimes y)\otimes z)$$

Is equal to: $$\mu\circ(\mu\otimes1)\circ\alpha:F(x)\otimes (F(y)\otimes F(z))\to (F(x)\otimes F(y))\otimes F(z)\to F(x\otimes y)\otimes F(z)\to F((x\otimes y)\otimes z)$$

So that $F$ associates. Moreover we require: $$\lambda'=F(\lambda)\circ\mu\circ(\epsilon\otimes1):e'\otimes F(x)\to F(e)\otimes F(x)\to F(e\otimes x)\to F(x)$$And: $$\rho'=F(\rho)\circ\mu\circ(1\otimes\epsilon):F(x)\otimes e'\to F(x)\otimes F(e)\to F(x\otimes e)\to F(x)$$

We also require: $$\mu\circ\gamma':F(x)\otimes F(y)\to F(y)\otimes F(x)\to F(y\otimes x)$$To equal: $$F(\gamma)\circ\mu:F(x)\otimes F(y)\to F(x\otimes y)\to F(y\otimes x)$$

A natural transformation $\theta:(F,\mu_1,\epsilon_1)\implies(G,\mu_2,\epsilon_2)$ of (braided) monoidal functors is said to be monoidal when: $$\theta\circ\mu_1:F(x)\otimes F(y)\to F(x\otimes y)\to G(x\otimes y)$$Equals: $$\mu_2\circ(\theta\otimes\theta):F(x)\otimes F(y)\to G(x)\otimes G(y)\to G(x\otimes y)$$

Let $\B$ be the braid category. Its object class is $\Bbb N_0$, and the arrow class $\B(n,m)$ is empty if $n\neq m$. When $n=m$, $\B(n,n)$ is the $n$th Artin braid group (if $n=0$, we leave $\B(0,0)=\{1\}$) with the same composition and identities. $\B$ is a strict braided monoidal category, through the product $n\otimes m=n+m$, and: $f\otimes g:n\otimes m\to n'\otimes m'$ is defined to be the braiding which is $f$ on the first $n$ strings and $g$ on the subsequent $m$ strings (lay them side by side). The unit is $0$, and the braiding on $\B$, $\gamma:n\otimes m\to m\otimes n$ assigns to every braid the 'swizzled' (wording my own) braid where the first $n$ strings are braided to the final $n$ strings in $m+n$, and the final $m$ strings in $n+m$ are braided to the first $m$ strings.

Now, we denote for any braided monoidal $\M$ the category of strong braided monoidal functors $\B\to\M$ as: $\hom_{\BM}(\B,\M)$, and if $\S$ is a strict braided monoidal category then $\hom_{\SBM}(\B,\S)$ denotes the category of strict braided monoidal functors; in both, the arrows are the monoidal natural transformations.

  • $\begingroup$ Cross-posted on MO because I feel my question might be too obscure - as some of my previous questions seemingly have been - to receive an answer (in timely fashion) on MSE. $\endgroup$
    – FShrike
    Oct 20, 2022 at 17:03


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