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Stream: theory: category theory

Topic: Functors between presentable categories

John Baez (Apr 30 2020 at 03:20):

Someone told @Joe Moeller that in the 2-category Pres with:

• locally presentable categories as objects
• cocontinuous functors as morphisms
• natural transformations as 2-morphisms

the morphisms with right adjoints are precisely the cocontinuous functors whose right adjoints preserve filtered colimits!

Does someone know a reference for this?

Reid Barton (Apr 30 2020 at 03:46):

I don't think this is correct. It should be the cocontinuous functors whose right adjoints preserve all colimits.

Reid Barton (Apr 30 2020 at 03:47):

Because whatever the right adjoint G of a morphism F of Pres is, the functor G will also be right adjoint to F in Cat. Then G must preserve colimits, by definition of the morphisms of Pres.

Reid Barton (Apr 30 2020 at 03:49):

And conversely, as soon as the right adjoint (in the ordinary sense) of F preserves colimits, then it is a morphism of Pres and therefore the right adjoint in Pres also, because the 2-morphisms of Cat and Pres are the same. Right?

sarahzrf (Apr 30 2020 at 03:49):

sounds plausible

Joe Moeller (Apr 30 2020 at 05:03):

Turns out I misunderstood what the person was telling me. They actually said take left adjoints in [presentable categories, preserves filtered colimits, natural transformations], which now that I realize that's what they meant, is completely reasonable.

John Baez (Apr 30 2020 at 06:05):

Yes, that makes perfect sense. Thanks for straightening me out.

sarahzrf (Apr 30 2020 at 06:07):

pictured: straightening out image.png

John Baez (Apr 30 2020 at 06:07):

It seems that the class of functors I was interested in - cocontinuous functors whose right adjoints preserve filtered colimits - actually exist and are interesting. But this particular attempted characterization of them was completely wrong. Thanks!

Mike Shulman (Apr 30 2020 at 13:28):

Do you actually mean "filtered colimits" as in "$\omega$-filtered colimits", but between categories that are only locally presentable and not locally finitely presentable?

Mike Shulman (Apr 30 2020 at 13:29):

That seems an odd mismatch. Although every right adjoint between locally presentable categories preserves $\kappa$-filtered colimits for some $\kappa$ (i.e. is accessible), so that more natural version wouldn't be any extra condition at all.

Simon Burton (Apr 30 2020 at 13:55):

Btw, if anyone knows a reference for locally presentable categories apart from the book by Adamek and Rosicky please let me know... i'm wondering if there is a paper out there somewhere that has a lot of examples of working with locally presentable categories.

Reuben Stern (they/them) (May 01 2020 at 18:13):

Here's a nice statement that you might be familiar with: let $\text{Pr}^{L,\mathsf{st}}$ be the $\infty$-category of $\omega$-presentable stable $\infty$-categories and left-adjoint functors. A functor $F: \mathcal{C} \to \mathcal{D}$ in $\text{Pr}^{L,\mathsf{st}}$ preserves $\omega$-compact objects if and only if its right adjoint preserves $\omega$-filtered colimits. In this case, $F$ restricts to a functor $F^\omega: \mathcal{C}^\omega \to \mathcal{D}^\omega$ between idempotent-complete stable $\infty$-categories, and in fact this is an equivalence of categories: there is a fully faithful functor $\text{Ind}: \mathsf{St}^{\mathsf{idem}} \to \text{Pr}^{L,\mathsf{st}}$ from the $\infty$-category of small, stable, idempotent-complete $\infty$-categories, whose essential image is the wide subcategory $\text{Pr}^{L,\mathsf{st}}_\omega \subseteq \text{Pr}^{L,\mathsf{st}}$ on those left-adjoints whose right-adjoints preserve $\omega$-filtered colimits.

Reid Barton (May 01 2020 at 18:26):

This also works without "stable" (without other minor adjustments perhaps).

Reuben Stern (they/them) (May 01 2020 at 18:30):

Great; I only knew of it in the stable context!

John Baez (May 02 2020 at 02:26):

Zounds, no I'm not familiar with that, and I don't have any sense for what "idempotent-complete" stable $\infty$-categories are like.

Reid Barton (May 02 2020 at 02:28):

In the 1-categorical (and therefore non-stable) context it is the equivalence between small, finitely cocomplete categories and locally finitely presentable categories, given by Ind(-) in one direction and taking the compact objects in the other direction.

Reid Barton (May 02 2020 at 02:29):

aka Gabriel-Ulmer duality, if you replace the finitely cocomplete categories by their opposites.

John Baez (May 02 2020 at 02:30):

Thanks! (Btw, I'm wishing Reuben would edit his post and add a dollar sign to make the TeX beautiful.)

John Baez (May 02 2020 at 02:30):

I'll need to think about this stuff, it might help me with a paper I'm writing.

Reid Barton (May 02 2020 at 02:31):

In this setting the splitting of an idempotent can be expressed a finite colimit, so you don't need the idempotent-complete condition, but "morally" it should be there (if you generalize to accessible categories, for example)

Reid Barton (May 02 2020 at 02:33):

And just to be clear the morphisms on the locally finitely presentable category side are still these ones which preserve compact objects (equivalently, whose right adjoint preserves filtered colimits)--otherwise taking the compact objects wouldn't be functorial.

John Baez (May 02 2020 at 02:33):

Mike Shulman said:

Do you actually mean "filtered colimits" as in "$\omega$-filtered colimits", but between categories that are only locally presentable and not locally finitely presentable?

I really meant it, but I'm probably doing something not quite right (above and beyond the fact that my question was sort of "stupid", in the sense that as soon as Reid Barton answered it I realized I should have been able to figure it out myself).

Reuben Stern (they/them) (May 02 2020 at 20:23):

Fixed the TeX in the original post; sorry about that. One thing I'm coming to understand is that the homotopy theory of stable $\infty$-categories (i.e., the $(\infty, 2)$-category $\mathsf{St}$) is much less well-behaved than that of idempotent-complete stable $\infty$-categories (i.e., the $(\infty, 2)$-category $\mathsf{St}^{\mathsf{idem}}$). This isn't new (though it is new-to-me), but one example is that in $\mathsf{St}^{\mathsf{idem}}$, the cofiber (idempotent-completion of the Verdier quotient) of a fully-faithful functor $\mathcal{A} \hookrightarrow \mathcal{B} \to \mathsf{cofib}(\mathcal{A} \hookrightarrow \mathcal{B})$ is also a fiber sequence; this is not the case in $\mathsf{St}$. But now I'm rambling...

Ben MacAdam (May 08 2020 at 16:02):

Reuben Stern said:

Fixed the TeX in the original post; sorry about that. One thing I'm coming to understand is that the homotopy theory of stable $\infty$-categories (i.e., the $(\infty, 2)$-category $\mathsf{St}$) is much less well-behaved than that of idempotent-complete stable $\infty$-categories (i.e., the $(\infty, 2)$-category $\mathsf{St}^{\mathsf{idem}}$). This isn't new (though it is new-to-me), but one example is that in $\mathsf{St}^{\mathsf{idem}}$, the cofiber (idempotent-completion of the Verdier quotient) of a fully-faithful functor $\mathcal{A} \hookrightarrow \mathcal{B} \to \mathsf{cofib}(\mathcal{A} \hookrightarrow \mathcal{B})$ is also a fiber sequence; this is not the case in $\mathsf{St}$. But now I'm rambling...

Are you familiar with the work that has been done on limit doctrines? Idempotent completeness fits into that framework quite nicely.
Non-enriched: http://www.tac.mta.ca/tac/volumes/10/20/10-20.pdf
Enriched: https://arxiv.org/pdf/0810.2578.pdf
I don't know if it's been worked out at that level of generality for quasicategories, but I know Joyal has looked at sifted colimits in quasicategories.