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Stream: theory: category theory

Topic: accessibility of induced functors

Daniel Teixeira (Nov 06 2023 at 22:44):

If $V$ is accessible/locally presentable then it is well known that the category of categories enriched over $V$ is accessible/locally presentable. Is this assignment functorial?

More precisely: if $F:V\to W$ is an accessible lax monoidal functor between accessible/locally presentable categories, is the induced functor $F_*:V\text{Cat}\to W\text{Cat}$ accessible?

This is true for instance if $F$ is has a lax adjoint $G$, in which case the adjunction lifts to an adjunction between the categories of enriched categories (and adjunctions are accessible).

Kevin Arlin (Nov 06 2023 at 23:22):

What’s making you nervous about this? If $F$ is $\lambda$-accessible then isn’t $F_*$ as well just because $\lambda$-filtered colimits are computed the same on either side on objects and as $\lambda$-filtered colimits in $V$ or $W$ on homs?

Daniel Teixeira (Nov 06 2023 at 23:59):

well, I'm generally afraid of colimits in VCat, but with your encouragement I'll try to get my hands dirty tomorrow morning

John Baez (Nov 07 2023 at 08:34):

Kevin Arlin said:

What’s making you nervous about this? If $F$ is $$\lambda-$$accessible then isn’t $F_*$ as well just because $$\lambda$$-filtered colimits are computed the same on either side on objects and as $\lambda$-filtered colimits in $V$ or $W$ on homs?

An annoying feature of this Zulip is that double dollars don't work unless they have spaces or dashes around them. E.g.

$\lambda$-accessible

works but

$$\lambda-$$accessible

does not. In the first case the dash is to the right of the closing double dollar sign.

Kevin Arlin (Nov 07 2023 at 17:27):

Ah, thanks, John, I assumed that was just some of the bugginess people have been complaining about lately.

Kevin Arlin (Nov 07 2023 at 17:28):

Daniel, that's a very healthy attitude in general but I'd argue the whole reason accessible categories and functors are so important is that filtered colimits are not scary; more precisely, in any accessible category they're calculated as in some presheaf category, and so they're basically just filtered colimits of sets, and so they're basically (in a rougher sense) just unions.