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/ ]