Timeline for Adjunctions with respect to profunctors
Current License: CC BY-SA 4.0
11 events
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| Jan 25 at 23:46 | comment | added | varkor | @DavidWhite: thanks for the suggestion. I've added a couple of links. My motivation for this question was primarily abstract: it seems a natural concept to consider (and indeed, Kan did), and I was wondering if it had been studied anywhere. Kan does not give any convincing examples. | |
| Jan 25 at 23:44 | history | edited | varkor | CC BY-SA 4.0 |
added 193 characters in body
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| Jan 25 at 21:56 | comment | added | David White | Personally, I think this question could have benefitted from linking to the definition of "profunctor" and a bit more motivation. For example, it would have been nice to mention if there was a concrete application of this notion in Kan's paper. Just my two cents. | |
| Apr 27, 2022 at 10:19 | comment | added | varkor | I've discovered the same question was asked here. | |
| Oct 31, 2021 at 0:48 | comment | added | Mike Shulman | You might need to move to the proarrow equipment $\rm Cat \hookrightarrow Prof$, but it seems unlikely in this case since there are no other functors in sight. | |
| Oct 31, 2021 at 0:48 | comment | added | Mike Shulman | Well, it probably depends on what sort of properties you want to prove. But I would expect that most things you might want to prove about such a thing, and that are actually true, would follow from this expression and the fact that representable (resp. corepresentable) profunctors are left (resp. right) adjoints in Prof. | |
| Oct 29, 2021 at 21:08 | comment | added | varkor | @MikeShulman: indeed. The question then becomes, I suppose, "Can properties of such diagrams be deduced from existing theory (e.g. the theory of adjunctions in a 2-category), without having to reprove various results about adjunctions at this greater level of generality?". | |
| Oct 29, 2021 at 19:33 | comment | added | Mike Shulman | Of course, this can also be written as just a commutative square in Prof, where one of the profunctors is representable and one other is corepresentable. | |
| Oct 29, 2021 at 10:03 | answer | added | Ivan Di Liberti | timeline score: 2 | |
| Oct 29, 2021 at 7:39 | comment | added | fosco | I have studied for a little while the case $P=Q$ (because the step I made was generalising $\hom\mapsto P$); if I'm correct, every adjunction $F \dashv G$ for which $F \dashv_P G$ -meaning that $P(F-,-)\cong P(-,G-)$- induces a "hom-relative" adjunction from the category of elements/collage of $P$ to itself; probably for generic $P,Q$ you get an adjunction between to different collages? | |
| Oct 28, 2021 at 18:55 | history | asked | varkor | CC BY-SA 4.0 |