Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: characterizing presheaf categories

John Baez (Apr 03 2025 at 01:41):

I have a feeling I've asked this before... but I can't find the answer on the nLab so I'll ask it again:

How can we characterize those categories that are (equivalent to categories) of the form $\mathsf{Set}^{\mathsf{C}^{\text{op}}}$ for some category $\mathsf{C}$?

John Baez (Apr 03 2025 at 01:43):

I don't want an answer like "they're the categories that are free cocompletions of categories", because that forces one to find the $\mathsf{C}$.

James Deikun (Apr 03 2025 at 02:03):

You can find the $\mathsf C$ by looking for the objects whose Hom functor has a right adjoint.

Kevin Carlson (Apr 03 2025 at 03:24):

Is "cocomplete categories with a strong generator formed of small-projectives" a satisfactory kind of answer?

Kevin Carlson (Apr 03 2025 at 03:24):

"Small projective" here meaning "the hom functor commutes with all small colimits."

John Baez (Apr 03 2025 at 04:11):

And is a "strong generator" the same as an extremal separator, i.e. a family of objects $x_i$ such that the functors $\text{hom}(x_i, -)$ are jointly faithful and jointly reflect isomorphisms? (I'm just looking this stuff up, I never think about this kind of thing.)

To find these small-projective $x_i$ can we take the representables corresponding to the objects of $\mathsf{C}$?

John Baez (Apr 03 2025 at 04:13):

James has made me much happier about the need to find the objects of $\mathsf{C}$ lurking amid $\mathsf{Set}^{\mathsf{C}^{\text{op}}}$.

Mike Shulman (Apr 03 2025 at 04:55):

In general, the small-projective objects in $\rm Set^{C^{op}}$ are the retracts of representables, so what you can find lurking in $\rm Set^{C^{op}}$ is not quite $C$ itself but its idempotent-completion.

John Baez (Apr 03 2025 at 05:04):

Oh, right - if I hand you the category $\mathsf{Set}^{\mathsf{C}^{\text{op}}}$ you could never expect to recover $\mathsf{C}$ itself, just its Cauchy completion, since that has the same presheaf category.

Nathanael Arkor (Apr 03 2025 at 06:18):

A characterisation of presheaf categories is described on the nLab page for [[atomic categories]].

Kevin Carlson (Apr 03 2025 at 06:40):

Hmm, there it claims Marta Bunge only proved that a regular cocomplete category with a set of small-projective generators (just plain generators, i.e. representables jointly faithful, being sufficient here) is of presheaf type. But there's no real reference. I wonder if there's really an irregular counterexample?

Kevin Carlson (Apr 03 2025 at 06:43):

There are two other characterizations on the page for "Presheaf Category" itself, but one says that presheaf categories are the cocomplete, well-powered, co-well-powered, atomic, regular categories, and cites Bunge again. This is a ridiculously, er, overpowered characterization, so this needs some cleanup. Then there is: A category is equivalent to a presheaf topos if and only if it is locally small, extensive, exact and has a small set of projective (here meaning just that the representables preserve regular epimorphisms, like in homological algebra) and indecomposable (i.e. under coproduct) generators.

Nathanael Arkor (Apr 03 2025 at 06:57):

The attribution on the page was incorrect; I've fixed it. The most general statement is from Centazzo–Rosický–Vitale's A characterization of locally $D$-presentable categories.

John Baez (Apr 03 2025 at 16:24):

Thanks! For some reason I was looking at the page [[presheaf]] and missed all these delights at [[category of presheaves]].

It would be nice if there were some characterizations with fewer adjectives... or else counterexamples showing all these adjectives are needed.

Nathanael Arkor (Apr 03 2025 at 16:27):

There are two adjectives: cocomplete and atomic. That hardly seems like a lot.

John Baez (Apr 03 2025 at 16:39):

That one is nice. It's not on the page [[presheaf category]], so I'll add it there. I was talking about the characterizations on the page [[presheaf category]]:

• cocomplete, well-powered, co-well-powered, atomic, regular

and

• locally small, extensive, exact and has a small set of projective and indecomposable generators.

John Baez (Apr 03 2025 at 16:41):

I guess the first one can simply be retired, because it's "cocomplete and atomic" plus a bunch more adjectives. (It may be so lengthy because Bunge was studying the more general enriched case - I don't know.)

Kevin Carlson (Apr 03 2025 at 16:52):

I think the word "atomic" for enriched categories wasn't something you could have really defined in 1966; somebody just copied a preliminary/old result into the nLab.

John Baez (Apr 03 2025 at 17:52):

I updated the nLab article [[presheaf category]] to add the result characterizing presheaf categories as cocomplete atomic categories. There was still another defect: to understand the definition of [[atomic category]] you needed to click on [[atomic objects]]... but that redirects you to [[tiny object]], and then there's a warning about how different people use 'atomic object' in different senses. So, I tried to improve the page [[atomic category]] so that it explains the relevant concept of 'atomic object'.

Simon Burton (Apr 03 2025 at 18:01):

I'm not sure if i noticed before, that the category of $G$-Sets, for $G$ a finite group, is a presheaf category. Which makes me wonder, what is the Cauchy completion of the one object category $G$ in this case?

Kevin Carlson (Apr 03 2025 at 18:10):

The Cauchy completion of a plain (i.e. Set-enriched) category is its idempotent splitting. A group doesn't have any nontrivial idempotents, because of invertibility, so it's already idempotent complete! That is, categories Morita equivalent to a group must be equivalent to that group. This isn't true for monoids, though. I think I remember a nice description of the idempotent completion of a monoid but I won't try to reproduce it just now in case somebody else wants to try and/or I'm remembering wrong.

John Baez (Apr 03 2025 at 20:24):

@Simon Burton - yes, it's also good to think of the category of representations of a group as a category of Vect-valued presheaves. This viewpoints are good because they encourage you to start replacing the group by a groupoid or category, but also they help you think of new ideas.

Besides looking at presheaves valued in some category other than Set we can look at presheaves in enriched category theory.

Somewhere back, some of us talked about how an algebra is a one-object Vect-enriched category, so we should call a Vect-enriched category $A$ an algebroid. The category of Vect-enriched presheaves on an algebroid is another algebroid! Example: the category of representations of an algebra is an algebroid.

It's also interesting to take the Cauchy completion of an algebra and get an algebroid. For example, I was blathering on recently about how the Cauchy completion of the group algebra $\mathbb{C}[G]$ of a finite group $G$ is the algebroid of all finite-dimensional representations of $G$.

Adittya Chaudhuri (Apr 03 2025 at 20:48):

Little out of context (may be stupid too), but the word "Vect-valued presheaves" just reminds me of chararterising all connections on a vector bundle $\pi \colon E \to M$ (seeing as associated bundles of principle bundles) as the the functor category $[\Pi^{thin}(M), Vect]$ (with certain smoothness compatibilities) (via parallel transport), where $\Pi^{thin}(M)$ is the thin homotopy groupoid of $M$ https://ncatlab.org/nlab/show/thin+homotopy. Although these are covariant functors.

Adittya Chaudhuri (Apr 04 2025 at 04:50):

I now realise "why suddenly" I talked about the above context (Still could be out of context; I am apologising priorly for that):

Given a manifold $M$ and a Lie group $G$, there is a pair of categories:

• $B^{\nabla}G(M)$ (Definition 3.13 of https://arxiv.org/pdf/1509.05000), whose objects are pairs $(\pi \colon E \to M, \omega)$ consisiting of a principal $G$-bundle $\pi \colon E \to M$ equipped with a connection 1-form $\omega$. While morphisms are connection preserving morphisms of principal bundles.
• $Trans_{G}(M)$ (Definition 3.15 of https://arxiv.org/pdf/1509.05000), whose objects are functors $T \colon \Pi^{thin}(M) \to G-Tor$ from the thin homotopy groupoid of $M$ to the category of $G$-torsors such that for each $x \in M$ the restriction $T|_{\pi_1^{thin}(M,x)} \colon \pi_1^{thin}(M,x) \to Aut(T(x))$ is a map of diffeological spaces. While morphisms are arbitrary natural transformations.

Now, the Theorem 4.1 of https://arxiv.org/pdf/1509.05000 shows that the category $B^{\nabla}G(M)$ is equivalent to the category $Trans_{G}(M)$ (via parallel transport). In a way , $Trans_{G}(M)$ is a covariant presheaf category (after incorporating appropriate smoothness) over the groupoid $\Pi^{thin}(M)$, and valued in the category $G$-Tor. On the other hand, $B^{\nabla}G(M)$ is a "differently looking category" but equivalent to the "sort of Preshaef category" $Trans_{G}(M)$.

Previously with different machnieries (smooth Descent data) such a correspondence was studied by Schreiber and Waldorf in https://arxiv.org/pdf/0705.0452 (may be in a bit more complicated/complex way).

In a way, $B^{\nabla}G(M)$ is a category equivalent to a category of the form $G-{\rm{Tor}}^{\Pi^{thin}(M)}$. Of course, the original question was in the context of $\mathsf{Set}^{\mathsf{C}^{\rm{op}}}$.

Morgan Rogers (he/him) (Apr 04 2025 at 14:00):

Kevin Carlson said:

Hmm, there it claims Marta Bunge only proved that a regular cocomplete category with a set of small-projective generators (just plain generators, i.e. representables jointly faithful, being sufficient here) is of presheaf type. But there's no real reference. I wonder if there's really an irregular counterexample?

That's because it's in her thesis, which was written before these things were habitually made available online.

Morgan Rogers (he/him) (Apr 04 2025 at 14:05):

Kevin Carlson said:

I think the word "atomic" for enriched categories wasn't something you could have really defined in 1966; somebody just copied a preliminary/old result into the nLab.

Please do not apply the word atomic in this context. It is in conflict with the very well established term "atomic topos", which refers to a topos with a generating set of atoms in a different (and in my opinion more compelling) sense of having no proper subobjects besides the initial object.

Morgan Rogers (he/him) (Apr 04 2025 at 14:06):

Marta Bunge characterized the generating objects in a presheaf category as projective indecomposable objects, which is a more descriptive name anyhow.

Morgan Rogers (he/him) (Apr 04 2025 at 14:09):

Given that "atomic category" refers to "atomic object" which redirects to "tiny object", I think "tiny-generated category" would be a better 'cute' name for this concept, if one is needed.

John Baez (Apr 04 2025 at 15:33):

You can change the terminology on these pages. I will not myself change the terminology there, simply on the principle of not wanting to mess up something when I'm not familiar with the standard conventions. The pages where these issues come up are mainly [[atomic category]] and [[presheaf category]], though the category [[tiny object]] has a nice place for adding further discussion of terminological ambiguities. The page [[atom]] shows how 'atom' and 'atomic' are used in many ways, perhaps too many ways.

David Corfield (Apr 04 2025 at 15:35):

Remember you can introduce disambiguation pages too, such as [[atom (disambiguation)]].

Kevin Carlson (Apr 04 2025 at 18:50):

Morgan Rogers (he/him) said:

Marta Bunge characterized the generating objects in a presheaf category as projective indecomposable objects, which is a more descriptive name anyhow.

"Projective" doesn't work on its own since it doesn't make obvious which colimits the representable preserves, though I'm happy to disambiguate to "small-projective", "regular-projective", etc. "Indecomposable" is an orthogonal concept, no?

Kevin Carlson (Apr 04 2025 at 18:52):

Generally I don't think there are enough people who use these words often enough relative to the number of past people (including Lawvere and Bunge) who have already used them to have much hope of completely stamping out the use of "atomic" for "small-projective" or "tiny."

Kevin Carlson (Apr 04 2025 at 18:53):

Anyway, there was a mathematical question in there that has been passed over so far, so let me ask it separately.

Kevin Carlson (Apr 04 2025 at 19:04):

Oh, I take it back, Nathanael provided a citation that "cocomplete with a strong generator of tiny objects" is sufficient. I vaguely still wonder whether the generator has to be strong, but only vaguely.