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Stream: theory: algebraic topology

Topic: Highly strucured ringoids

Brendan Murphy (Mar 10 2024 at 00:55):

Let $\mathcal{V}$ be some nice monoidal model category of spectra, eg symmetric spectra for concreteness. We can then form a model category of enriched $\mathcal{V}$-categories. Does the tensor product of $\mathcal{V}$-categories make this into a monoidal model category? Does anyone know of prior work about this?

Brendan Murphy (Mar 10 2024 at 00:56):

(I'm eventually interested in bousfield localizing wrt the Morita equivalences, but this is the first step!)

Mike Shulman (Mar 10 2024 at 01:47):

I can't say for sure about spectra, but in general I think it is very rare that for a monoidal model category $V$ the ordinary tensor product of $V$-categories makes $V\mathrm{Cat}$ also a monoidal model category. E.g. it already fails for $V=\mathrm{Cat}$ or even $V=\mathrm{Gpd}$. In fact offhand I can't think of any nontrivial $V$ for which I know it is true.

Brendan Murphy (Mar 10 2024 at 01:47):

Darn, oh well. I didn't really have a sense whether this should work

Mike Shulman (Mar 10 2024 at 01:50):

Roughly speaking, the issue is that in the tensor product of $V$-categories you have a strict interchange law $(f\otimes 1)\circ (1\otimes g) = (1\otimes g) \circ (f\otimes 1)$, but to be homotopically well-behaved this should only be an isomorphism. In particular, the tensor product of cofibrant $V$-categories tends to no longer be cofibrant, because this is a "nontrivial equation" that always holds in it. In the case of Cat there is a modified tensor product called the [[Gray tensor product]] for which $V\mathrm{Cat} = 2\mathrm{Cat}$ is a monoidal model category, and there are some generalizations of that to other cases but none quite as well-behaved.

Brendan Murphy (Mar 10 2024 at 02:01):

Oh, I didn't know this interpretation of the Gray tensor product! I'd heard it mentioned and knew of the semistrictification result but never really learned about it

Brendan Murphy (Mar 10 2024 at 02:01):

The "nontrivial equation" breaking cofibrancy makes sense

Brendan Murphy (Mar 10 2024 at 02:04):

It should be fine in my application to work purely with infinity categories, I just like having a model structure to fall back on if I need to calculate something and I'm stuck

John Baez (Mar 10 2024 at 04:58):

Some people think mathematicians are always trying to prove nontrivial equations, and here you guys are trying to avoid them!