Re: theorem about ends

[Richard Garner <richard.garner@mq.edu.au>]

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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