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Alastair Grant-Stuart (Oct 09 2020 at 14:54):We know that e.g. any functor with a left adjoint sends limits to limits; likewise, functors with right adjoints send colimits to colimits.
Are there analogous properties that could tell us that a 2-functor sends extensions to extensions?
Jade Master (Oct 14 2020 at 03:58):What do you mean by extension?
Fawzi Hreiki (Oct 14 2020 at 09:13):I think he just means the generalisation of a Kan extension to a 2-category, i.e. https://ncatlab.org/nlab/show/Kan+extension#In2cat
Morgan Rogers (he/him) (Oct 14 2020 at 09:27):There is a general result that adjoints preserve extensions, I remember this from the Kan Extensions chapter of Emily Riehl's book. That might help?
Fawzi Hreiki (Oct 14 2020 at 10:29):I don't think he means preservation in that sense - that is preservation in the 'external' Kan extension sense. A 2-functor preserves extensions iff every extension in the domain is taken to an extension in the codomain.
Morgan Rogers (he/him) (Oct 14 2020 at 10:33):Ah gotcha. 2-adjunctions are trickier beasts, as are variants of them
Fawzi Hreiki (Oct 14 2020 at 10:36):I don't know much about 2- (or n-) categories but this paper https://arxiv.org/abs/math/0702535v1 by Stephen Lack seems to have some stuff on extensions.
Fawzi Hreiki (Oct 14 2020 at 10:38):But in the notes he mentions that pointwise extensions are computed as (co)ends, which are then just special (co)limits. So, assuming we still have the usual RAPL/LAPC for 2-adjunctions, that might give a partial answer for pointwise extensions.
Alastair Grant-Stuart (Oct 14 2020 at 12:40):@Fawzi Hreiki interpreted my meaning right — sorry for the ambiguity. I was indeed a little worried about using the phrase "preserves extensions" since that means something quite different for 1-functors than the 2-functor behaviour I'm thinking about.
Thanks for the thoughts. I, too, don't know much about 2-categories yet, but it seems that learning about 2-adjunctions and their interactions with (co)limits is my next step.
Private conversation with tslil (they) also pointed out that an extension can be rephrased as something called a weighted (co)limit — I still need to learn what this is. In this case, something like a right-2-adjoints-preserve-weighted-limits/left-2-adjoints-preserve-weighted-colimits result would be what I'm looking for.
Amar Hadzihasanovic (Oct 14 2020 at 14:27):The fact that left/right adjoint 1-morphisms preserve left/right extensions (which specialise to colimits and limits) is true for purely formal reasons in every 2-category. Presumably if you are looking for generalisations you should look for similar statements that are true in the theory of 3-categories (or tricategories?)
Amar Hadzihasanovic (Oct 14 2020 at 14:31):I think it looks like you are considering "extensions in a 2-category" as a kind of higher-dim generalisation of "limits/colimits in a category", but that seems like the wrong intuition in this context, as in the proof of "preservation by adjoints" it is best to think of the latter as... extensions in the 2-category Cat!
Amar Hadzihasanovic (Oct 14 2020 at 14:33):I think it's more helpful to think of "2-functors preserving extensions" as a generalisation of functors preserving specific properties of morphisms in a category, e.g. functors preserving monomorphisms
Amar Hadzihasanovic (Oct 14 2020 at 14:35):By the way, an example of "2-functors preserving right Kan extensions" on which you may find more information is closed functors between symmetric monoidal closed categories, that is, those that preserve the internal homs.
Amar Hadzihasanovic (Oct 14 2020 at 14:36):Since internal homs in a SMCC are right Kan extensions when you look at the monoidal category as a one-object bicategory.