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Stream: deprecated: id my structure

Topic: Double category of presheaves

Patrick Nicodemus (Jun 28 2023 at 14:59):

Let $C$ be a small double category.
I am currently working out the definition of a double category of presheaves on $C$ where

• the objects are presheaves on the category of objects $C_{obj}$
• the horizontal maps are natural transformations
• a vertical arrow is a presheaf $P$ on the arrow category $C_{arr}$ together with "domain" and "codomain" presheaves $A,B$ on $C_{obj}$ and natural transformations $\sigma : P\to A\circ\operatorname{dom}, \tau : P \to B\circ\operatorname{cod}$
• domain and codomain functors are the obvious ones
• identity is given by left Kan extension along the functor $id : C_{obj}\to C_{arr}$ (??)
• composition resembles profunctor composition

I am wondering if this has been written up and I can find a reference on this somewhere.

Notification Bot (Jun 28 2023 at 14:59):

A message was moved here from #learning: id my structure > Reference requests for polynomial monads by Patrick Nicodemus.

Patrick Nicodemus (Jun 28 2023 at 15:01):

I don't know if this is the right definition for my project yet, i'm still kind of hazy on what the right definition should look like. But I don't think the definition "A presheaf on a double category C is a lax functor from C to the double category of sets and spans" is adequate for my project

Nathanael Arkor (Jun 28 2023 at 16:05):

In light of your last message, Paré's Yoneda Theory for Double Categories may not be what you're looking for, but seems the most relevant reference for presheaves on double categories. The paper Double Fibrations of Cruttwell–Lambert–Pronk–Szyld may also be relevant.

Patrick Nicodemus (Jun 28 2023 at 17:22):

Yeah I'm struggling to figure out if Paré's notion of presheaf is appropriate for what I'm working on. I don't have many 2-cells in my double category so there are not many cocones.

Patrick Nicodemus (Jun 28 2023 at 18:47):

Regarding Pare

Nathanael Arkor said:

In light of your last message, Paré's Yoneda Theory for Double Categories may not be what you're looking for, but seems the most relevant reference for presheaves on double categories. The paper Double Fibrations of Cruttwell–Lambert–Pronk–Szyld may also be relevant.

Is it known whether the "realization" functor in Pare's paper is a double functor?
I.e. he assigns to each presheaf on a double category a double category of elements, and then the horizontal colimit indexed by the category of elements we can think of as the "realization" of the presheaf. Does this play nice with vertical morphisms? I don't see this stated explicitly and there's a lot going on in this paper.

Patrick Nicodemus (Jun 28 2023 at 18:49):

I need an appropriate double-categorical notion of "density" that says something about vertical arrows.

Patrick Nicodemus (Jun 28 2023 at 18:52):

Ignoring the subtle fact that composition of vertical arrows in the presheaf category is not easily defined, i want to know if "realization" sends vertical arrows to vertical arrows

Nathanael Arkor (Jun 28 2023 at 18:59):

I'm not sure I understand precisely what you're asking, but Lambert proves that Paré's double category of elements construction underlies an equivalence of virtual double categories in Discrete Double Fibrations, which indicates to me that it behaves as nicely as one would hope for.

Patrick Nicodemus (Jun 28 2023 at 19:11):

Maybe you can help me, I have a "paradox" I'm trying to resolve. I have given this some thought already and I am still confused.

Say $\mathbb{C}$ is a cocomplete double category and $J$ is a small 1-category. Let $F: J\to \mathbb{C}_{obj}$ be a functor from $J$ into the underlying 1-category of $\mathbb{C}$, and suppose that $F$ is 1-categorically dense, so the nerve-realization adjunction has a counit which is an isomorphism.

Now I can easily upgrade $J$ to a 2-category $\mathbb{J}$ simply by adding formal identity arrows to every object, and $F$ becomes a double functor. I now have double nerve and double realization functors which I believe should extend the 1-categorical nerve and realization functors. But now if $v : c\to c'$ is a vertical arrow in $\mathbb{C}$, then by applying the nerve and realization I should get a vertical arrow $R(N(v)) : c\to c'$ and presumably some kind of comparison 2-cell $\alpha : R(N(v))\to v'$ over $(id_c,id_{c'})$.

This seems strange to me, how can I construct a vertical cell between $c$ and $c'$ out of some kind of colimit of the vertical arrows in $\mathbb{J}$ when there are only trivial (identity) vertical arrows in $\mathbb{J}$?

Patrick Nicodemus (Jun 28 2023 at 19:16):

If $\mathbb{C}$ is a double category where a 2-cell $\alpha$ is an isomorphism as soon as $\operatorname{dom}\alpha$ and $\operatorname{cod}\alpha$ are, for example if the category of vertical arrows is some full subcategory of the category of all arrows, then the comparison map is an isomorphism. That suggests we can somehow recover the double-categorical structure of $\mathbb{C}$ by probing it with a 1-category, which seems like it must be wrong.

Patrick Nicodemus (Jun 28 2023 at 19:49):

Nathanael Arkor said:

I'm not sure I understand precisely what you're asking, but Lambert proves that Paré's double category of elements construction underlies an equivalence of virtual double categories in Discrete Double Fibrations, which indicates to me that it behaves as nicely as one would hope for.

Ok, this is helpful. So Lambert's construction is showing that if you have a module between presheaves you can make it into a vertical arrow between discrete fibrations over $\mathbb{B}$. My question is whether these vertical arrows induce vertical maps between their colimits. Like if $p : \mathbb{E}\to\mathbb{B}$ is a discrete double fibration and $q : \mathbb{F}\to\mathbb{B}$ is a discrete double fibration, and $F : \mathbb{B}\to\mathbb{C}$ is a double functor, if $\alpha : p\to q$ is a vertical arrow in Lambert's sense, does this necessarily induce a vertical arrow $\operatorname{colim} Fp \to \operatorname{colim}Fq$ which we can call $\operatorname{colim}\alpha$

Nathanael Arkor (Jun 30 2023 at 10:51):

I've not yet given much thought to double categorical colimits, so I don't have a good intuition for your question, unfortunately. I imagine coming up with some small examples to see what's going on would be the most helpful thing to do here. I'd be interested to hear about your progress though; I think this is an interesting topic in general.

Patrick Nicodemus (Jul 02 2023 at 19:38):

@Nathanael Arkor In Grandis and Pare's paper on limits and double categories, they observe two pathologies of limits.

1. They are not automatically (vertically) functorial, so that a vertical natural transformation between diagrams does not necessarily induce a vertical map between their limits
2. They are not necessarily unique up to vertical isomorphism.

(2.) can be solved by adding some extra assumptions on the category, their solution to (1.) is essentially a solution by definition, i.e. say that a category has "functorial vertical limits" if it has functorial vertical limits.

Presumably this same problem will come up in the case of weighted (co)limits.