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Stream: theory: category theory

Topic: Enriched monoidal categories

Paolo Perrone (Oct 12 2021 at 20:58):

Hello! What is the state of the art in "enriched monoidal categories"? There is this paper for example, which treats the case of strict monoidal categories enriched in a braided monoidal category.
That paper says that "enrichment in a symmetric monoidal category is known". Can anyone tell me where? Also, is the non-strict version worked out in that case?

Evan Patterson (Oct 12 2021 at 21:07):

Great question. I came across that paper recently and wondered the same thing myself.

Nathanael Arkor (Oct 12 2021 at 21:15):

When $\mathcal V$ is symmetric, the 2-category $\mathcal V\text{-}\mathbf{CAT}$ is symmetric monoidal (see §1.4 of Kelly, for instance). You could then define a monoidal $\mathcal V$-category to be a pseudomonoid in $\mathcal V\text{-}\mathbf{CAT}$. I suspect something similar holds if $\mathcal V$ is just braided, and that the authors of that paper worked out explicitly what the definition would be in that case.

Mike Shulman (Oct 12 2021 at 21:17):

Yeah, this is kind of folklore. I don't know of a really good reference unfortunately.

John Baez (Oct 13 2021 at 01:40):

There's some even more folkloric folklore that you can define monoidal, braided monoidal, and so on up to (k-1)-tuply monoidal n-categories enriched in a k-tuply monoidal n-category.

So for example you can define categories enriched in a monoidal category, monoidal categories enriched in a braided monoidal category, braided monoidal categories enriched in a symmetric monoidal category, symmetric monoidal categories enriched in a symmetric monoidal category... and here it stabilizes.

John Baez (Oct 13 2021 at 01:42):

The pattern is slightly obscured by the stabilization.

John Baez (Oct 13 2021 at 01:44):

The general case for arbitrary k, n should probably be called "myth" rather than "folkore" since I wouldn't how to prove it or even make it precise.

Mike Shulman (Oct 13 2021 at 05:50):

There are some very Australian papers by people like Day and Street that study various kinds of pseudomonoids in monoidal bicategories, with applications to enriched categories in mind.

Mike Shulman (Oct 13 2021 at 05:53):

"Monoidal bicategories and Hopf algebroids", for instance.