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Stream: theory: category theory

Topic: yoneda for virtual double categories

Matteo Capucci (he/him) (Jan 11 2024 at 12:03):

Nathanael Arkor said:

(I was thinking about this at some point because I wanted a Yoneda embedding for virtual double categories, but I ran into issues because there didn't seem to be a functor virtual double category construction. But perhaps if one restricts to (virtual) equipments it will work out after all.)

Couldn't you find an embedding from a vdc to its vdc of 'modules in Set'? I'm thinking of a way to pull out Yoneda without necessarily having preseahves around in the way Garner and Shulman do here

Nathanael Arkor (Jan 11 2024 at 20:26):

Matteo Capucci (he/him) said:

Nathanael Arkor said:

(I was thinking about this at some point because I wanted a Yoneda embedding for virtual double categories, but I ran into issues because there didn't seem to be a functor virtual double category construction. But perhaps if one restricts to (virtual) equipments it will work out after all.)

Couldn't you find an embedding from a vdc to its vdc of 'modules in Set'? I'm thinking of a way to pull out Yoneda without necessarily having preseahves around in the way Garner and Shulman do here

I don't think I follow. There is certainly an embedding of a virtual double category $\mathbb X$ in the virtual double category $\mathbb M\mathbf{od}(\mathbb X)$, which is the analogue of the construction Garner and Shulman consider. However, one would expect the Yoneda embedding of a virtual double category to factor through this construction (assuming that $\mathbb M\mathbf{od}(\mathbb X)$ is indeed a free cocompletion of $\mathbb X$).

Matteo Capucci (he/him) (Jan 12 2024 at 09:32):

(I moved messages from #community: our work)

Matteo Capucci (he/him) (Jan 12 2024 at 09:34):

I'm talking about this:
image.png

Matteo Capucci (he/him) (Jan 12 2024 at 09:35):

One should be able to define a right module (valued in Sets) for a virtual double category and representables should be among these

Nathanael Arkor (Jan 12 2024 at 10:06):

Ah, I see what you mean. They're using the terminology "right module" instead of "presheaf" (which is slightly confusing, because the terminology "module" is often used for monads in a VDC). Yes, it's possible there's a notion of presheaf for VDCs that does not reduce to a functor of VDCs. There's also an evident candidate, since one can derive a definition of distributors of VDCs from the paper of Cruttwell and Shulman, so one could consider a distributor with terminal domain as a notion of presheaf. The question is then whether presheaves on a VDC form a VDC.

Matteo Capucci (he/him) (Jan 13 2024 at 07:32):

Exactly!

Nathanael Arkor said:

The question is then whether presheaves on a VDC form a VDC.

Indeed!

Patrick Nicodemus (Jan 13 2024 at 23:07):

I added several papers by Grandis and Pare to the nlab page on double categories, one of which establishes that presheaves on a (pseudo) DC form a VDC.
It might be possible to extend this from pseudo DC's to VDC's.

Patrick Nicodemus (Jan 13 2024 at 23:07):

Robert Paré Yoneda theory for double categories, Theory and Applications of Categories, Vol. 25, 2011, No. 17, pp 436-489.

Nathanael Arkor (Jan 14 2024 at 00:14):

It might be possible to extend this from pseudo DC's to VDC's.

The difference is that there is a notion of functor double category, but not (seemingly) that of a functor virtual double category.

Christian Williams (Jan 17 2024 at 01:37):

A virtual double category is a multi-monad in SpanCat. The construction of pseudomonads in a 3D category generalizes to one of multi-monads, defining in particular a 3D category of VDCs.

Just like double categories, VDCs have both vertical and horizontal profunctors; and in particular this is how functors between VDCs form a VDC, with vertical and horizontal transformations.

Here's a brief outline of the construction of multi-monads in 3D string diagrams.
MultiMnd.pdf

James Deikun (Jan 17 2024 at 02:22):

I recall coming up with a similar definition in the past but there was some kind of problem with the composition of double transformations.

Christian Williams (Jan 17 2024 at 02:46):

(deleted)

Christian Williams (Jan 17 2024 at 06:05):

Well, I'm pretty sure it all works simply. I'll draw each kind of composition and post tomorrow.

I'm fairly certain that if a (co/lax) triple category has vertical and horizontal sums, then multimonads therein form a (co/lax) triple category.

Nathanael Arkor (Jan 17 2024 at 08:53):

Although a definition of virtual double categories as some kind of "multi-monad" is another interesting topic, it is orthogonal to the Yoneda embedding discussed above, so it is probably better to open a separate topic for this discussion.

Christian Williams (Jan 17 2024 at 16:52):

So y'all are talking about horizontal presheaves, right?

Nathanael Arkor (Jan 17 2024 at 17:14):

The direction that corresponds to lax functors into Set, if that's the one you call horizontal?

Christian Williams (Jan 17 2024 at 18:23):

(by Set you mean SpanSet) Yeah, because horizontal morphisms of a double category form categories.

So if $\mathbb{V}$ is a virtual double category, $\mathbb{V}(A,-)$ gives for each $B$ a multi-category rather than a category; then what is its action on vertical and horizontal morphisms?

John Baez (Jan 17 2024 at 19:07):

(The terms loose/tight avoid confusions about the convention concerning which direction is vertical and which is horizontal.)

Nathanael Arkor (Jan 17 2024 at 19:09):

John Baez said:

(The terms loose/tight avoid confusions about the convention concerning which direction is vertical and which is horizontal.)

I agree this terminology is clearer, but in this setting there are three directions, and there has been no proposed analogue for "transveral" morphisms in the tight/loose nomenclature as far as I'm aware.

John Baez (Jan 17 2024 at 19:15):

Flabby. :upside_down:

Morgan Rogers (he/him) (Jan 17 2024 at 19:15):

Slack?

Nathanael Arkor (Jan 17 2024 at 19:19):

Morgan Rogers (he/him) said:

Slack?

This was exactly the term I had been using in my notes :grinning_face_with_smiling_eyes:

Christian Williams (Jan 17 2024 at 19:40):

In an equipment, a "tight" morphism can "loosen up" to form a companion or conjoint. But in three dimensions, the story is not as simple. A double functor induces a v-profunctor and an h-profunctor; but a v-profunctor does not induce an h-profunctor. So it is no longer a case of "increasing looseness".

[In my thesis I give an interpretation of an equipment as a logic, so a tight (v-)morphism is a process (or term), and a loose (h-)morphism is a relation (or judgement). Then for the "triple category" of double categories, a V-morphism is a meta-process, because it contains processes, and an H-morphism is a meta-relation, because it contains relations. For the transversal dimension, we just need a word that reminds you of "transformation"; I chose flow.]

Nathanael Arkor (Jan 17 2024 at 20:03):

Christian Williams said:

In an equipment, a "tight" morphism can "loosen up" to form a companion or conjoint. But in three dimensions, the story is not as simple. A double functor induces a v-profunctor and an h-profunctor; but a v-profunctor does not induce an h-profunctor. So it is no longer a case of "increasing looseness".

Yes, perhaps "slack" is a little misleading, then.

Mike Shulman (Jan 17 2024 at 20:13):

In most of the triple categories that I'm familiar with, one direction is clearly tighter than the other two, so it would make sense to call that one "tight". This tight direction is the one that Grandis and Pare draw transversally.

Mike Shulman (Jan 17 2024 at 20:14):

I don't personally see a connotation in the words "slack" and "loose" that either of them is "tighter" than the other, so from that perspective at least I wouldn't object to using those two for the other two directions.

Mike Shulman (Jan 17 2024 at 20:15):

Are there naturally-occurring triple categories in which the directions are linearly ordered by tightness?

Nathanael Arkor (Jan 17 2024 at 20:27):

Perhaps one could form a triple category whose objects are toposes, whose tight morphisms are essential geometric morphism, whose loose morphisms are geometric morphisms, and whose slack morphisms are finitely continuous functors (e.g. taking the direction of each class of morphisms in the logical direction)?

John Onstead (Jan 29 2024 at 01:30):

In a virtual double category, one can impose the condition that all composites exist. When they do, the virtual double category reduces to a normal (pseudo) double category, and the class of objects along with horizontal morphisms between them becomes a bicategory. What kind of structure do the class of objects and horizontal morphisms between them form more generally in a virtual double category, if not a bicategory?

Mike Shulman (Jan 29 2024 at 01:54):

To get a bicategory you need not only the objects and the horizontal morphisms, but the "globular" 2-cells. If you likewise include all the 2-cells that have identity vertical arrows in an arbitrary vdc, you get a "virtual bicategory".