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John Onstead (Jan 14 2025 at 12:28):I had a thought the other day I wanted to run by you. First, define a functor in the following way: it sends objects in to the codomain fibration and morphisms in to the usual functors between slice categories. Now, take the nerve of this functor- that is, find the restricted Yoneda functor .
As a nerve, this functor has a realization , setting up an adjunction between these two categories. Now, here's my question: what would happen if you then decided to restrict this adjunction to an equivalence?
The intent with this question is quite obvious: I want to know if, by any chance at all, this gives anything like the usual equivalence . I tried to figure this out myself for a bit but I kept losing track of what everything was supposed to be doing!
John Onstead (Jan 15 2025 at 10:51):I did some work on this today and I'm sure the answer is "yes". This question on MO suggests so. But I also checked for myself using coends, which I found quite simple to do. It's also 4.33 in the handy book on ends and coends.
I think it's interesting just how far reaching the nerve/realization construction is. Even Isbell duality (which we are covering on another thread) can be given in this form. It really seems that if you have an important adjunction in category theory, there's a good chance it might originate from a nerve/realization.
Tomáš Jakl (Jan 15 2025 at 11:51):This is beautiful, thanks!