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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Paul, This is (a special case of) Lemma 2.1 in G.M. Kelly & Stephen Lack, Finite-product-preserving functors, Kan extensions, and strongly finitary 2-monads. Applied Categorical Structures 1:85-94, 1993. Steve. On 07/02/2011, at 11:25 AM, Paul Levy 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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Paul Blain Levy asks:
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).
For well over 40 years I've always thought of that as the Beck/Lawvere vision of what an adjunction *is* :-) . Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Paul, The following answer is essentially the same as Richard's, Ross's, and possibly Steve's (whose paper with Max I haven't looked up). But it provides a slightly different way of looking at things, which might be helpful if you haven't seen it already. Any adjunction F -| U: D --> C gives rise to a (D, C)-bimodule Q, that is, a functor Q: C^op x D --> Set. (Read "profunctor" or "distributor" if you prefer.) It is defined by Q(c, d) = D(Fc, d) = C(c, Ud). Your initial data gives such a (D, C)-bimodule Q, plus another (D, C)-bimodule P. Given such, we can ask: What is Hom(Q, P)? Calculating it one way, it's Hom(Q, P) = \int_{c, d} [Q(c, d), P(c, d)] = \int_{c, d} [D(Fc, d), P(c, d)] = \int_c P(c, Fc) by Yoneda. This used the definition of Q(c, d) as D(Fc, d). But we could also use the definition of Q(c, d) as C(c, Ud), and then a very similar calculation shows that Hom(Q, P) = \int_d P(Ud, d). So both your sets are canonically isomorphic to Hom(Q, P). Best wishes, Tom
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).
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Fred E.J. Linton -
Paul Levy -
Richard Garner -
Steve Lack -
Tom Leinster