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Stream: theory: category theory

Topic: plain monoidal categories hurt my head

Tim Campion (Apr 23 2022 at 23:23):

I've been forced lately to do contemplate something I've tried to avoid in the past: plain monoidal categories, with no symmetry or even braiding. (The main example for me is the Gray tensor product on omega-categories.) I'm trying to get the hang of these things. Here is a particular thing I want to prove -- I think I've gotten the right formulation finally, and now there's a diagram chase to do. I am very lazy and like to avoid diagram chases, so I guess I'm wondering: is there some theory to appeal to where somebody else has already (perhaps implicitly) done the requisite diagram chasing for the following:

Claim: Let $(V, \otimes)$ be a monoidal category. Assume that

1. $V$ is half-closed, in that $(-) \otimes X : V \to V$ has a right adjoint $\llbracket X, - \rrbracket$ for every $X \in V$ (I guess that's "right"-closed?).

2. $V$ has finite products.

3. $K$ is a monoid object in $V$ with respect to the cartesian product.

4. $V$ is affine monoidal (i.e. the unit for $\otimes$ is the terminal object $1$).

Then the functor $T : V^{op} \to V$, $A \mapsto \llbracket A, K \rrbracket$, is lax monoidal with respect to the reverse tensor product $\otimes^{rev}$ on $V^{op}$ and the given tensor product $\otimes$ on $V$.

The lax unit constraint $1 \to \llbracket 1, K \rrbracket = K$ is the unit map for the monoid structure on $K$. The lax monoidality constraint $\llbracket A, K \rrbracket \otimes \llbracket B, K \rrbracket \to \llbracket B \otimes A, K \rrbracket$ is adjoint to the map $\llbracket A, K \rrbracket \otimes \llbracket B, K \rrbracket \otimes B \otimes A \to K$ which first uses the "$B$-evaluation map" to get from $\llbracket A, K \rrbracket \otimes \llbracket B, K \rrbracket \otimes B \otimes A$ to $\llbracket A, K \rrbracket \otimes K \otimes A$, then uses the only map you can think of to get from there to $(\llbracket A, K \rrbracket \otimes A) \times K$, then uses "$A$-evaluation" to get to $K \times K$, and finally uses the multiplication on $K$ to get to $K$.

I'm reasonably convinced that this description does yield a lax monoidal functor, because I've checked the unit laws and I've written down a "balanced ternary monoidality constraint" with which both ways of associating should coincide. Before I roll up my sleeves and chase down the diagrams to verify this, how can I save myself the effort?

One possibility might be to consider $(V, \otimes, \times)$ as a duoidal category, but I'm not sure if there's anything there.