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Stream: learning: questions

Topic: Reference for transporting Kan extension across 2-adjunction

Graham Manuell (Sep 19 2023 at 16:20):

I am looking for something to cite for the result that 'taking adjuncts preserves Kan extensions'. The situation is well-described by this question on mathoverflow, which also asks for a reference, but unfortunately got no answer. Does anyone here have any suggestions?

Kevin Arlin (Sep 19 2023 at 18:08):

Since pointwise left Kan extensions are certain weighted colimits in the $\mathbf{Cat}$-enriched context, left adjoint 2-functors preserve them by the fact that enriched left adjoints preserve all weighted colimits (see Kelly's Basic Concepts, paragraph 3.2). I'm not immediately sure whether we can make a similar argument for naive Kan extensions (but who cares about those, right? :smile: )

Graham Manuell (Sep 19 2023 at 18:35):

Thanks. The restriction to pointwise Kan extensions might be fine, but I think what I want is slightly different to preservation by a left adjoint, since it also involves composing with the (co)unit.

Kengo Hirata (Sep 20 2023 at 14:07):

Can I ask what you mean by pointwise left Kan extensions are certain weighted colimits in the Cat-enriched context?
Cat is a (co)complete Cat-enriched category but not all pairs of 1-cells in Cat F: A->B and G: A->C admit Lan_F G when C is not cocomplete.

Kevin Arlin (Sep 20 2023 at 17:14):

Hmm, maybe I'm having a level shift problem here and this is nonsense...What I mean is that $\mathrm{Lan}_F G(b)$ is the colimit of $G$ weighted by the profunctor $B(F(-),b),$ or globally that $\mathrm{Lan}_F G$ is the colimit of $G$ weighted by the profunctor $B(F(-),-):A\to B.$

Kevin Arlin (Sep 20 2023 at 17:15):

Probably I need to be in the proarrow equipment of categories, functors, and profunctors rather than the 2-category of categories to make the earlier claim make sense.