Dear Paul, I do not know anywhere that it appears explicitly, but it can be pieced together quite quickly from results about weighted limits in Kelly's book. First, given any adjunction X -| Y : B --> A, any W : A --> Set, and any G : B --> C, we have {WY, G} = {W, GX} (**) in the sense that the one exists if the other does, and the canonical comparison is then an isomorphism. This follows since Lan_X(W) = WY (by (4.28) of Kelly) and {Lan_X(W), G} = {W, GX} (by (4.58) ibid). Since the end of a functor T: K^op x K --> E is by definition ((3.59) ibid) the limit of H weighted by the hom-functor H_K: K^op x K --> Set, we have, in the situation you describe, that End(P(-,F-)) = {H_C, P.(1 x F)} = {H_C.(1 x U), P} = {H_D.(F^op x 1), P} = {H_D, P.(U^op x 1)} = End(P(U-,-)) by applying (**) twice to the adjointnesses 1 x F -| 1 x U and U^op x 1 -| F^op x 1, and using the natural isomorphism H_C.(1 x U) = H_D.(F^op x 1) obtained from the adjointness F -| U. Richard On 7 February 2011 11:25, Paul Levy <pbl@cs.bham.ac.uk> wrote:
Dear all,
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).
Paul
-- Paul Blain Levy School of Computer Science, University of Birmingham +44 (0)121 414 4792 http://www.cs.bham.ac.uk/~pbl
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