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Stream: learning: questions

Topic: enrichment in a 2-category

Fawzi Hreiki (Nov 28 2020 at 17:01):

Equivalences, adjunctions, monads, (Kan) extensions, and (Kan) lifts are all fundamental concepts in (1-)category theory which internalise very straightforwardly to arbitrary 2-categories. With a little bit more work, the Yoneda embedding and profunctors internalise (kind of) as well via Yoneda structures and proarrow equipments. I'm curious if there is an analogous notion of enrichment internal to a 2-category - i.e. given objects $A$ and $B$ in a 2-category $\mathscr{K}$, what would it mean for $A$ to be $B$-enriched (or to be equipped with a $B$-enrichment)?

Fawzi Hreiki (Nov 28 2020 at 17:03):

I imagine this will require some sort of 'representable Yoneda structure' (perhaps as in https://arxiv.org/abs/math/0606393)

Matteo Capucci (he/him) (Nov 28 2020 at 22:53):

See Section 2 here https://arxiv.org/abs/1801.01386.
The idea is that a $V$-enriched category $C$ is equivalent to a closed action of $V$ on $C_0$ (here 'closed' means 'parametrized left adjoint'). This condition is something expressible in any monoidal bicategory, so it's natural to consider it a definition in that general case

Matteo Capucci (he/him) (Nov 28 2020 at 22:54):

I find it cool since it shows enrichment has nothing to do with categories, in a sense. It's an orthogonal structure.

Reid Barton (Nov 28 2020 at 23:01):

This equivalence is only for tensored (and/or cotensored) enriched categories though, right? That's the version I know and it seems to be what is explained there as well

Reid Barton (Nov 28 2020 at 23:02):

I always wondered whether you could get exactly enriched categories as some kind of pro- (as in profunctor) modules

Reid Barton (Nov 28 2020 at 23:05):

For example, $V$-valued presheaves on a $V$-enriched category $C$ is a closed $V$-module, but I don't know how to describe it entirely in the "module" language.

Alexander Campbell (Nov 28 2020 at 23:06):

Reid Barton said:

I always wondered whether you could get exactly enriched categories as some kind of pro- (as in profunctor) modules

I do this (in a slightly more general context) in Section 4 of my paper on Skew-enriched categories.

Matteo Capucci (he/him) (Nov 28 2020 at 23:12):

Reid Barton said:

This equivalence is only for tensored (and/or cotensored) enriched categories though, right? That's the version I know and it seems to be what is explained there as well

Indeed, I should have said for a closed monoidal category $V$, which amounts to having co/tensors. In Vasilakopoulou's paper Theorem 2.4 only shows that every such closed action produces an enrichment. You need co/tensors to have an equivalence.
That said, it shouldn't be a problem to say 'closed pseudomonoid' in any monoidal bicategory :thinking:

Reid Barton (Nov 28 2020 at 23:25):

Alexander Campbell said:

Reid Barton said:

I always wondered whether you could get exactly enriched categories as some kind of pro- (as in profunctor) modules

I do this (in a slightly more general context) in Section 4 of my paper on Skew-enriched categories.

Thanks! Can I understand the non-skew case first, or does this only work in the skew setting?

Alexander Campbell (Nov 28 2020 at 23:47):

Reid Barton said:

Alexander Campbell said:

Reid Barton said:

I always wondered whether you could get exactly enriched categories as some kind of pro- (as in profunctor) modules

I do this (in a slightly more general context) in Section 4 of my paper on Skew-enriched categories.

Thanks! Can I understand the non-skew case first, or does this only work in the skew setting?

Some skewness is essential. So for a monoidal category $\mathcal{V}$, there is an equivalence between $\mathcal{V}$-categories and the skew $\mathcal{V}$-proactegories which satisfy the representability condition of Prop 4.12(b) and whose unit constraint is invertible. But in general the skew $\mathcal{V}$-proactegory corresponding to a $\mathcal{V}$-category will not have invertible associativity constraint.