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Stream: learning: questions

Topic: local presentability of Cat

Daniel Teixeira (Nov 14 2023 at 20:12):

The category of small categories is finitely locally presentable, for instance the walking morphism is a compact generator. My knowledge about sizes fails me: is this enough to get locally small categories?

The category of locally small categories is $Cat = VCat$ with $V = Set$, so it should be locally finitely presentable since Set is so, thus I just want to sanity check if this is indeed true or if there is some size issue going under my radar.

Kevin Arlin (Nov 14 2023 at 20:22):

No, $Set-Cat$ is the category of small categories. Locally small categories can't possibly be locally presentable because you can't get something with a large class of objects from a small colimit of small objects.

fosco (Nov 14 2023 at 20:22):

Perhaps are you asking whether the category of locally small categories is presentable? It's not even locally small itself...

fosco (Nov 14 2023 at 20:23):

you have to take small functors of locally small categories; then that category is "legitimate" (=which is legalese for "it exists")

Daniel Teixeira (Nov 14 2023 at 20:27):

Kevin Arlin said:

No, $Set-Cat$ is the category of small categories.

why? I'm working from naivety here, the textbook definition of a $V$-category has a collection of objects and an object of morphisms between them, specializing to $V=Set$ seems to allow for a collection of objects and hom-sets = locally small categories

Kevin Arlin said:

Locally small categories can't possibly be locally presentable because you can't get something with a large class of objects from a small colimit of small objects.

this makes sense, thanks . the compact objects must generate C under colimits indexed by sets

Kevin Arlin (Nov 14 2023 at 20:29):

It's fine to say that a locally small category is a category enriched in Set, but you're basically never going to collect all of those into a single category.

Kevin Arlin (Nov 14 2023 at 20:29):

You could also read the first sentence of the paper you linked in case you don't want to take my word for it :)

Daniel Teixeira (Nov 14 2023 at 20:31):

Kevin Arlin said:

It's fine to say that a locally small category is a category enriched in Set, but you're basically never going to collect all of those into a single category.

I see, so basically VCat is always the category of small V-categories = V-categories with a set of objects, because otherwise sizes go out of hand, just like they do for locally presentable categories as pointed out by fosco

Kevin Arlin (Nov 14 2023 at 20:33):

Yeah, exactly. You can, in a somewhat ironic complication, talk usefully about the (non-locally-small) category of locally presentable categories and cocontinuous functors, or its opposite, whose morphisms are the continuous accessible functors, because these functors are determined by a small amount of data.

fosco (Nov 14 2023 at 20:34):

I might be sloppy here, but the problem seems deeper to me than "sizes go out of hand" :smile: sizes do not even apply for the "category" of locally small categories, because given large, but locally small, categories, there can be a proper class of morphisms in $[C,D]$, because a natural transformation is a family $\alpha_C : \prod_c D[Fc,Gc]$, indexed over a proper class

Kevin Arlin (Nov 14 2023 at 20:35):

I'm not clear what "does not even apply" means.

Kevin Arlin (Nov 14 2023 at 20:35):

I would just say the category of locally small categories is a certain subcategory of the category of small $U$-categories where $U$ is the next universe up.

Daniel Teixeira (Nov 14 2023 at 20:35):

by the way, what is a small functor between locally small categories?

fosco (Nov 14 2023 at 20:35):

"The joy of cats" uses the word "illegitimate" for a category whose hom-classes are not sets; they imply that a category which is not legitimate doesn't exist (implicit: in the universe of reference, but who cares what happens out of it anyway?)

Kevin Arlin (Nov 14 2023 at 20:35):

It's true that locally-small-cat is not self-enriched.

Kevin Arlin (Nov 14 2023 at 20:36):

Yeah, that seems like a rather rude name to me!

Kevin Arlin (Nov 14 2023 at 20:36):

A small functor into Set is a small colimit of representables.

Kevin Arlin (Nov 14 2023 at 20:36):

Then small functors in general are defined representably: $F$ is small if $Hom(x,F-)$ is small for every $x.$

Kevin Arlin (Nov 14 2023 at 20:37):

They're relevant mostly because the result that the presheaf category is the free cocompletion of a small category can be extended to say that the category of small set-valued functors is the free cocompletion of any locally small category.

Kevin Arlin (Nov 14 2023 at 20:38):

This is also something nice about locally small categories relative to "locally large" ones, since those don't even have representable functors valued in small sets so it's kind of pointless to talk about small colimits in that context.

fosco (Nov 14 2023 at 20:38):

Kevin Arlin said:

A small functor into Set is a small colimit of representables.

yes. Another problem: given $C$ locally small, all presheaves $[C^o,Set]$ (live one universe up) and do not form the free cocompletion of $C$. Instead, small presheaves do.

fosco (Nov 14 2023 at 20:39):

Kevin Arlin said:

A small functor into Set is a small colimit of representables.

More generally, given locally small categories $C,D$, a functor $F : C\to D$ is small if there exists a small subcategory $j : A\to C$ such that $F$ arises as a left Kan extension along $j$ of a functor $f : A\to D$

Nathanael Arkor (Nov 14 2023 at 22:05):

Kevin Arlin said:

It's fine to say that a locally small category is a category enriched in Set, but you're basically never going to collect all of those into a single category.

You can certainly consider a (huge) category of large V-categories. Often the category of small V-categories is denoted V-Cat, and the category of large V-categories is denoted V-CAT. The precise formulation will depend on your foundations, but there's no fundamental obstruction.

Kevin Arlin (Nov 14 2023 at 23:07):

Yeah, certainly a category of large categories is a thing, since it's just the small categories a universe up. Specifically a category of locally small categories is not something you need so often.