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: theory: category theory

Topic: Cardinal bounded presheaves

Brendan Murphy (Apr 02 2024 at 18:36):

Let $\kappa$ be an infinite cardinal number. Say that a set is $\kappa$-small if it is of cardinality $< \kappa$. Let $\mathsf{Set}_{< \kappa}$ be the category of $\kappa$-small sets (note this is essentially small as every set of cardinality $< \kappa$ is isomorphic to a set hereditarily of cardinality $< \kappa$). Given a small category $\mathcal{C}$ we can form the category $\operatorname{Psh}_{< \kappa}(\mathcal{C}) = \operatorname{Fun}(\mathcal{C}^{\mathrm{op}}, \mathsf{Set}_{< \kappa})$ of $\kappa$-small presheaves. This is a full subcategory of $\operatorname{Psh}(\mathcal{C}) = \operatorname{Fun}(\mathcal{C}^{\mathrm{op}}, \mathsf{Set})$ and if $\mathcal{C}$ is locally $\kappa$-small the yoneda embedding $\mathcal{C} \to \operatorname{Psh}(\mathcal{C})$ restricts to a fully faithful functor $\mathcal{C} \to \operatorname{Psh}_{< \kappa}(\mathcal{C})$.

Observe that $\mathsf{Set}_{< \kappa} \subseteq \mathsf{Set}$ is closed under finite limits and under $\operatorname{cof}(\kappa)$-small colimits (in particular under finite colimits). It follows that $\mathsf{Set}_{< \kappa}$ is Barr-exact and $\operatorname{cof}(\kappa)$-extensive, thus is a pretopos. If $\kappa$ is a strong limit cardinal then it is furthermore an elementary topos. I think all of this is inherited by $\operatorname{Psh}_{< \kappa}(\mathcal{C}) = \operatorname{Fun}(\mathcal{C}^{\mathrm{op}}, \mathsf{Set}_{< \kappa})$, at least if $\operatorname{cof}(\kappa) > |\mathcal{C}|$ (the cardinality of a small category here means the cardinality of its morphism set). If $\operatorname{cof}(\kappa) > |\mathcal{C}|$ then the total category/category of elements of a $\kappa$-small presheaf has cardinality $< \kappa$ and the category of elements construction gives an equivalence between $\operatorname{Psh}_{< \kappa}(\mathcal{C})$ and the category of discrete fibrations over $\mathcal{C}$ with $\kappa$-small fibers. In particular the formula for a presheaf as a colimit over its category of elements shows $\operatorname{Psh}_{< \kappa}(\mathcal{C})$ is generated by representables under $\kappa$-small colimits.

Is there a universal property of $\mathcal{C} \hookrightarrow \operatorname{Psh}_{< \kappa}(\mathcal{C})$, assuming $\operatorname{cof}(\kappa) > \mathcal{C}$? Perhaps we want to assume $\kappa$ is either a strong limit cardinal or regular (certainly the theory works out nicely when $\kappa$ is both!). While representables generate under $\kappa$-small colimits, the category doesn't actually have all $\kappa$-small colimits (because $\kappa$ isn't assumed regular). It seems like this is sort of related to determining what limits commute with the $\kappa$-small colimits, by the whole theory of generalized ind-completion/generalized accessible & presentable categories in terms of a limit doctrine (I don't understand this as well as I would like to)

Brendan Murphy (Apr 02 2024 at 18:39):

It's possible we need to "remember" the data of the cof(kappa)-small sets to get a universal property somehow, e.g. track a class of admissible natural transformations F -> G consisting of those whose components have cof(kappa)-small fibers (I have no idea if this notion of admissibility is useful or correct)

John Onstead (Apr 02 2024 at 23:18):

This is a really interesting writeup, especially because I've recently started getting into the relationship between set theory and category theory. As such I wanted to share my thoughts, with the disclaimer that I will likely be getting a lot of stuff wrong since I am just learning about it now. But I do find it helpful to share my understanding so that anything I get wrong can be corrected.

First, I learned that if kappa is not a strong limit cardinal, then kappa-Set is not closed (IE, in the sense of a closed cartesian category) since it will not have all power objects. For instance, the set of natural numbers has a strong limit cardinality, and thus, FinSet, given by kappa-Set where kappa = cardinality of natural numbers, is closed and a topos. This makes sense because the power between any two finite sets will also have a finite cardinality, given a^b will always be finite so long as a and b are finite, and so the category is indeed closed. But if you were to add in just the set of natural numbers to your set category, then the category is no longer closed since, for instance, the powerset of the natural numbers 2^N would not be an object in this category.

If we look at the construction from the perspective of enriched category theory, we can see the issue with trying to define a universal property of the general kappa-Yoneda embedding. Usually, when enriching categories, we want a base with a lot of "good properties"- for instance, the best bases of enrichment are closed symmetric monoidal categories, possibly even complete or cocomplete as well. Topoi are very good examples of these, so as long as kappa is a strong limit cardinal, you will have a very nice theory of locally kappa-small categories (the categories enriched over kappa-Set). More precisely, you can equip a category of enriched categories with a "Yoneda structure" when the base is (I believe) closed and complete, and Yoneda structures allow for the machinery of presheaf categories and the Yoneda lemma to be applicable. For instance, you can derive a Yoneda structure from a KZ monad/doctrine, and since those abstract the concept of free cocompletions, it means you can define a presheaf category in the "nice" enriched settings via the usual standard universal property of free cocompletion. You can't do this if the base of enrichment is not closed like in a general kappa-Set, since then you don't get a Yoneda structure that allows you to do this.

There are some ways of getting around this, though. One is to generalize Yoneda structures to augmented virtual double categories/equipments. Any category of enriched categories, regardless of the properties of the base, is also a virtual equipment, which can be extended to an augmented one in a standard way. You can then define a concept of a "Yoneda morphism" within this context for any (augmented) virtual equipment using a universal property. For instance, this paper by Koudenburg gives a universal definition of a yoneda morphism in Section 4/Definition 4.2.

No idea if what I said means anything or helps but I always like sharing what I've recently learned about category theory!

Reid Barton (Apr 03 2024 at 01:51):

If $\kappa$ is an infinite regular cardinal bigger than $|C|$ then the $\kappa$-small set-valued presheaves on $C$ are the cocompletion for $\kappa$-small colimits. (I'm slightly worried about the case $\kappa = \omega$, but it seems to also be okay.) So I guess you're mostly interested in the case where $\kappa$ is not regular?

Reid Barton (Apr 03 2024 at 01:52):

What about $\mathrm{Set}_{< \kappa}$ itself, does it have any interesting universal property when $\kappa$ is not regular?

Brendan Murphy (Apr 03 2024 at 01:55):

Yeah, I guess so. I suppose I wanted to look at the strong limit case, since there we get a topos. But it's possible that this construction is only really reasonable if κ is regular. In that case we do still have lots of exactness properties and some function sets can still be formed

Reid Barton (Apr 03 2024 at 01:55):

@Jonas Frey and I discussed this kind of example a bit elsewhere. For instance, if $\kappa$ is an uncountable strong limit cardinal but $\mathrm{cof}(\kappa) = \omega$ (such as for $\kappa = \beth_\omega$), then $\mathrm{Set}_{< \kappa}$ is an elementary topos with a standard natural number object, but it doesn't have (external) countable coproducts.

Brendan Murphy (Apr 03 2024 at 01:55):

Oh wow, that's weird

Brendan Murphy (Apr 03 2024 at 02:08):

Reid Barton said:

What about $\mathrm{Set}_{< \kappa}$ itself, does it have any interesting universal property when $\kappa$ is not regular?

Hm, this is a good point since it's certainly not the closure of a one point set under $\kappa$-small colimits/coproducts. What it is is the collection of things that can be written as a $\kappa$-small colimit/coproduct of one point sets. So we don't close under an operation but just apply it once. That doesn't seem very well behaved... Actually, what is the closure under $\kappa$-small colimits? Is it $\mathsf{Set}_{<\lambda}$ where $\lambda$ is the least regular cardinal $\geq \kappa$?

Reid Barton (Apr 03 2024 at 02:11):

I never thought about it before, but I guess that must be how it works?

Reid Barton (Apr 03 2024 at 02:13):

I remember going through LPAC back when I didn't understand regular cardinals very well, and trying to figure out exactly where the regularity hypothesis was needed... pretty quickly I decided the answer was "everywhere", and gave up on considering the non-regular case.

Brendan Murphy (Apr 03 2024 at 02:13):

Yeahh, I think I'm coming around to that

Reid Barton (Apr 03 2024 at 02:13):

I don't have my copy at hand at the moment, but I think there is some exercise in the first chapter about what it means for a functor to commute with $\kappa$-filtered colimits, or something like that, when $\kappa$ is not regular.

Brendan Murphy (Apr 03 2024 at 02:14):

I do, I'll take a look

Brendan Murphy (Apr 03 2024 at 02:17):

Ah yeah, the statement about the closure being the least greater regular cardinal is part of that exercise

Mike Shulman (Apr 03 2024 at 02:24):

I'm not sure what exercise you're looking at, but in case it's relevant let me mention that exercise 1.b(2) of LPAC is in error: the two parts of the exercise seem to contradict each other.

Mike Shulman (Apr 03 2024 at 02:26):

When I mentioned this to Jiří Rosický he said there is a correct treatment of presentability for singular cardinals in his paper https://arxiv.org/abs/1708.06782 with Lieberman and Vasey.

Brendan Murphy (Apr 03 2024 at 02:28):

Oh sorry, I looked at 1.b too quickly and completely misread it

Brendan Murphy (Apr 03 2024 at 02:28):

I'm not even sure how I got what I thought it said from the discussion of cofinalty

Mike Shulman (Apr 03 2024 at 02:31):

Another possibly-relevant paper is https://arxiv.org/abs/1902.06777.