Re: theorem about ends

[Ross Street <ross.street@mq.edu.au>]

Dear Paul

On 07/02/2011, at 11:25 AM, Paul Levy wrote:

> Does the following result (which I learnt from Rasmus Mogelberg)
> appear in the literature somewhere?
>
> Given categories C and D, a functor P : C^op x D --> Set and an
> adjunction F -| U : D --> C
>
> the end over c in C of P(c,Fc) is (isomorphic to) the end over d in D
> of P(Ud,d).

This is the sort of thing that would be used in the course of things
without explicit enunciation as a Lemma or Proposition. It involves
two applications of the end version of Yoneda (take V = Set for the
ordinary case):

end_c P(c,Fc)  =~ end_{c,d} V(D(Fc,d),P(c,d)) =~  end_{c,d}
V(C(c,Ud),P(c,d)) =~ end_d P(Ud,d).

It is quite pretty, I agree.

Ross


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]