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Stream: theory: category theory

Topic: κ-small κ-accessible categories

Josselin Poiret (Jan 25 2023 at 12:08):

Are there any references on the category of κ-small κ-accessible categories with κ-accessible functors between them? I need some general fix point theorems that are only available with accessible categories, but they don't have an internal Hom which I need, and I'm hoping that this restriction would be enough to develop a more tame version of all the usual accessibility machinery. (edit: here κ is inaccessible)

Josselin Poiret (Jan 25 2023 at 12:10):

If that helps, my setup can probably be even simpler: take $U$ a Grothendieck universe and $Cat_U$ the categories internal to $Set_U$, then you can define $Cat_U$-accessible categories in $Cat_U$.

Morgan Rogers (he/him) (Jan 25 2023 at 18:56):

I'm curious: are there many of these? As soon as you strengthen to $\kappa$-presentable you get $\kappa$-cocompleteness and so your category is forced to be a preorder, but I haven't thought about how that argument fails without (co)products, which are all it uses.

Mike Shulman (Jan 25 2023 at 19:07):

A small category is accessible if and only if it is idempotent-complete. Specifically, according to Theorem 2.2.2 in Makkai-Paré, it is $\lambda$-accessible for $\lambda = (\#A + \aleph_0)^{++}$. If $\kappa$ is inaccessible, then $\#A < \kappa$ implies $\lambda <\kappa$, so any idempotent-complete $\kappa$-small category is $\kappa$-accessible.

Josselin Poiret (Jan 25 2023 at 19:39):

Thanks! That should help quite a lot! I still have to wrap my head around accessible categories in general, I've never managed to get any intuition about them...