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Stream: learning: questions

Topic: iterated enrichment?

Bruno Gavranović (Mar 06 2021 at 12:52):

Can we do something like "iterated enrichment" where we consider chain of enrichments, and does that form something like a category?

In more detail:
Category being enriched in itself is like saying $C$ is enriched in $V$ and in some sense $C$ is equivalent to $V$. Is there a way we do "iterated enrichment" i.e. where I could say $C$ is enriched in $V$, and $V$ is enriched in $W$ and, say, $C$ is equivalent to $W$?
It seems interesting to think about what place $Set$ and $Cat$ have in it.

Fawzi Hreiki (Mar 06 2021 at 13:01):

Well enrichment is transitive since if $A$ is enriched in $B$ and $B$ is enriched in $C$, we can enrich $A$ in $C$ by $Hom(X, Y) = B(I_B, A(X, Y))$.

Fawzi Hreiki (Mar 06 2021 at 13:01):

But you'll generally lose information since the unit object $I_B$ may not be a separator.

Bruno Gavranović (Mar 06 2021 at 13:03):

Right, I guess what I'm asking is: has somebody written about this somewhere?

Bruno Gavranović (Mar 06 2021 at 13:03):

Even though we lose information as you say, it still seems interesting

Fawzi Hreiki (Mar 06 2021 at 13:04):

I'm not sure about the details but this is reminiscent of the section on enrichment in Lawvere's Perugia notes.

Fawzi Hreiki (Mar 06 2021 at 13:04):

https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1972-perugia-lecture-notes.pdf

Fawzi Hreiki (Mar 06 2021 at 13:05):

Lesson 3, section 4

Bruno Gavranović (Mar 06 2021 at 13:18):

Thanks, I'll have a look

Nathanael Arkor (Mar 06 2021 at 13:36):

This is considered very briefly in Forcey's Vertically Iterated Classical Enrichment.
image.png

Bruno Gavranović (Mar 06 2021 at 14:23):

Huh, I guess just thinking about this iterated enrichment for a minute; it's not really transitive. If $C$ is enriched in $V$ and $V$ is enriched in $W$, that doesn't mean that $C$ is enriched in $W$. So this doesn't form a category

Mike Shulman (Mar 06 2021 at 14:54):

If $V$ is closed symmetric monoidal, then to give a closed symmetric monoidal $V$-enriched category $C$ with copowers is equivalent to giving a closed symmetric monoidal ordinary category $C$ and a symmetric monoidal adjunction $V \rightleftarrows C$. The right adjoint can then be applied homwise to make any $C$-enriched category into a $V$-enriched category. And of course, symmetric monoidal adjunctions can be composed.

Mike Shulman (Mar 06 2021 at 14:54):

Unfortunately this is kind of folklore; I'm not sure of a reference.

Mike Shulman (Mar 06 2021 at 14:55):

BTW, the word "equivalent" in your original question doesn't seem appropriate; I can't think of a nontrivial sense in which a $V$-enriched category is equivalent to $V$.

Mike Shulman (Mar 06 2021 at 15:00):

One place I'm familiar with where this sort of double enrichment arises naturally is in equivariant topology, see e.g. section 6.3 of http://www.math.uchicago.edu/~may/PAPERS/GMR.pdf.

Bruno Gavranović (Mar 07 2021 at 01:51):

Mike Shulman said:

If $V$ is closed symmetric monoidal, then to give a closed symmetric monoidal $V$-enriched category $C$ with copowers is equivalent to giving a closed symmetric monoidal ordinary category $C$ and a symmetric monoidal adjunction $V \rightleftarrows C$. The right adjoint can then be applied homwise to make any $C$-enriched category into a $V$-enriched category. And of course, symmetric monoidal adjunctions can be composed.

So I'm taking this to mean that enrichment "can be composed"?
I guess I was forgetting that all bases of enrichment have to be closed. But I still practically don't see: if i have $\mathcal{C}$ enriched in $\mathcal{V}$ enriched in $\mathcal{W}$, then that means that $\mathcal{C}(c_1, c_2) \in Ob(\mathcal{V})$, but also that somehow must mean that $\mathcal{C}(c_1, c_2) \in Ob(\mathcal{W})$. Which object is it? I'm not sure if I see a way to coherently assigns objects of $\mathcal{W}$ to $\mathcal{V}$.

Mike Shulman (Mar 07 2021 at 02:44):

The way to assign objects of $W$ to objects of $V$ is the right adjoint in the adjunction $W \rightleftarrows V$ that's equivalent to making $V$ a $W$-enriched closed monoidal category with copowers. In terms of the enrichment, that right adjoint sends $x\in V$ to the $W$-enriched hom-object $V(I,x)$, where $I$ is the unit object for the monoidal structure of $V$. This is just like how any $V$-enriched category has an underlying $\rm Set$-enriched category.

Mike Shulman (Mar 07 2021 at 02:45):

(Bases of enrichment don't have to be closed, but they're better-behaved when they are.)

Asad Saeeduddin (Mar 07 2021 at 04:53):

Just to create a link: I think the answer to this question would also answer https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/Enrichment.20relations.20categorically/near/210088708. You might find some of the comments in that thread useful

Ben MacAdam (Mar 08 2021 at 18:58):

Mike Shulman said:

If $V$ is closed symmetric monoidal, then to give a closed symmetric monoidal $V$-enriched category $C$ with copowers is equivalent to giving a closed symmetric monoidal ordinary category $C$ and a symmetric monoidal adjunction $V \rightleftarrows C$. The right adjoint can then be applied homwise to make any $C$-enriched category into a $V$-enriched category. And of course, symmetric monoidal adjunctions can be composed.

Mike Shulman said:

Unfortunately this is kind of folklore; I'm not sure of a reference.

I think this paper by Rory Lucyshyn-Wright is related: http://www.tac.mta.ca/tac/volumes/31/6/31-06.pdf, where he considers the case of a symmetric monoidal closed adjunction.

Bruno Gavranović (Mar 10 2021 at 10:39):

Mike Shulman said:

The way to assign objects of $W$ to objects of $V$ is the right adjoint in the adjunction $W \rightleftarrows V$ that's equivalent to making $V$ a $W$-enriched closed monoidal category with copowers. In terms of the enrichment, that right adjoint sends $x\in V$ to the $W$-enriched hom-object $V(I,x)$, where $I$ is the unit object for the monoidal structure of $V$. This is just like how any $V$-enriched category has an underlying $\rm Set$-enriched category.

Ah, I see - that makes a lot of sense!