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Stream: learning: questions

Topic: Subcategories of free objects

Oscar Cunningham (Jun 07 2020 at 14:37):

Given an adjunction, the Kleisli category is a full subcategory of the Eilenberg-Moore category. If I have a category with a full subcategory is there some result that tells me when it arises from some adjunction?

Matteo Capucci (he/him) (Jun 07 2020 at 15:59):

Off the top of my head the only result I can think of is general AFT applied to the inclusion functor of the subcat. This translates to properties of the subcat and the big cat, that you might check.
Also (co)reflectiveness might be something to look for, but that already presupposes you know the inclusion functor has a left/right adjoint.

sarahzrf (Jun 07 2020 at 21:58):

do you mean "arises from" as in "is the Kleisli subcategory of the EM category", or in a more general sense?

Oscar Cunningham (Jun 08 2020 at 08:15):

sarahzrf said:

do you mean "arises from" as in "is the Kleisli subcategory of the EM category"

Yes.

Oscar Cunningham (Jun 08 2020 at 08:17):

If I'm looking at some problem involving a category and a full subcategory, is there a way that I can tell whether they arise as the Kleisli and Eilenberg-Moore categories of some adjunction?

Morgan Rogers (he/him) (Jun 08 2020 at 10:07):

I can come up with a nice necessary condition. Supposing the existence of the adjunction, for each object $Y$ of the EM category, we have $\mathrm{Hom}(FU(Y),Y) \cong \mathrm{Hom}(U(Y),U(Y))$, and moreover every morphism from the image of $F$ to $Y$ must factor through the counit $FU(Y) \to Y$, so the Kleisli category is an initial subcategory of the EM category.

Oscar Cunningham (Jun 08 2020 at 10:12):

What's an 'initial subcategory'? I would have thought the initial object in the category of subcategories, but that's only the empty category.

Morgan Rogers (he/him) (Jun 08 2020 at 10:27):

The history of the terminology means that that's a common confusion. It's the dual of being final.

Oscar Cunningham (Jun 08 2020 at 10:28):

Thanks, I remember now.

Morgan Rogers (he/him) (Jun 08 2020 at 10:28):

This means, for example, that the inclusion of the terminal object into the walking arrow category cannot be expressed as a Kleisli subcategory of an EM category

Oscar Cunningham (Jun 08 2020 at 10:33):

Okay great, that's a useful condition.

Morgan Rogers (he/him) (Jun 08 2020 at 10:35):

Although from the proof I stated it's clear that you can strengthen "initial" (I'm not sure what the the terminology would be, or if it exists), since the comma categories aren't just connected, they have a terminal object... wait now I'm confused, since that sounds like enough to force the comparison functor to have an adjoint...

Soichiro Fujii (Jun 08 2020 at 12:22):

Other necessary conditions may be derived from the fact that every object in the Eilenberg—Moore category is expressible as a reflexive coequaliser of objects in the Kleisli category, and that the Kleisli category is a dense full subcategory of the Eilenberg—Moore category; see this page in nLab.

Oscar Cunningham (Jun 08 2020 at 12:26):

That's good too, thank you.