Re: theorem about ends

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

Actually, since the codomain of your functor P is actually Set, the
argument I gave---which would be valid for any suitably complete
codomain---can be rewritten to avoid the use of weighted limits
entirely, by expressing those necessary in this case as hom-sets in a
functor category. It then becomes the single calculation that

End(P(-,F-)) = [C^op x C, Set](Hom_C, P.(1 x F)) = [C^op x D,
Set](Hom_C.(1 x U), P) = [C^op x D, Set](Hom_D.(F^op x 1), P) = [C^op
x D, Set](Hom_D, P.(U^op x 1)) = End(P(U-,-))

which requires no machinery beyond that of transpositions under adjunction.

Richard

On 7 February 2011 15:35, Richard Garner <richard.garner@mq.edu.au> wrote:
> 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
>

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