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Stream: learning: questions

Topic: "Pure"/"Faithfully Flat" symmetric monoidal (oo-) functors?

Chris Grossack (they/them) (Oct 28 2024 at 17:10):

There's a problem I'm working on where I need a notion of "base change" for module categories over a (symmetric) monoidal category. It feels like this should be a categorification of base change of a module over a (commutative) ring, and so I'm expecting the "pureness" condition to be something like

$F : \mathcal{C} \to \mathcal{D}$, a (strong? lax?) monoidal functor is pure iff for every $\mathcal{C}$-module category $\mathcal{M}$, the functor $F \boxtimes \mathcal{M}$ is (something like "regular mono"... Maybe fully faithful?) iff the base change functor $F \boxtimes - : \mathcal{C}\text{-Mod} \to \mathcal{D}\text{-Mod}$ is effective descent.

I'm happy to develop a lot of this theory of base change for module categories myself if I need to (and I think it even looks fun!), but it seems like the kind of thing that someone might have done before.

Does anyone have any references?

John Baez (Oct 29 2024 at 00:25):

That looks interesting, and I may know what you're up to here (from our previous conversations), but I don't know any results along these lines.

Peter Arndt (Oct 29 2024 at 09:06):

The PhD thesis of John Bourke might contain useful things for you.

Peter Arndt (Oct 29 2024 at 20:48):

Actually here are some more:
Hermida, Descent on 2-Fibrations and Strongly 2-Regular 2-Categories
Street, Categorical and combinatorial aspects of descent theory
Lucatelli Nunes, Pseudo-Kan extensions and Descent Theory

Chris Grossack (they/them) (Oct 30 2024 at 14:35):

Thanks! I'm excited to look at these!