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    • CommentRowNumber1.
    • CommentAuthorFinnLawler
    • CommentTimeSep 9th 2011

    Everyone knows that you can define (co)ends as certain (co)equalizers, as described at end. A while ago I added the section Connecting the two definitions, which derives the equalizer description from the weighted-limit description by observing that the weight hom C\hom_C for an end is presented as a certain split coequalizer in [C op×C,V][C^{op} \times C, V]. Now I’m puzzled, though, because this would seem to imply that ends are split equalizers, and dually, split coequalizers being absolute colimits. But it’s easy to cook up an example where this seemingly can’t happen, say with a pair of categories and functors F,GF,G such that Nat(F,G)= c[Fc,Gc]Nat(F,G) = \int_c [F c, G c] is empty but c[Fc,Gc]\prod_c [F c, G c] isn’t, so there can’t be any morphism from the latter to the former to make the whole thing a split equalizer.

    I must have made a mistake somewhere, and knowing me it’ll be a stupidly obvious one, but can anyone point it out?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeSep 9th 2011

    Are you sure you mean “split” and not reflexive?

    • CommentRowNumber3.
    • CommentAuthorFinnLawler
    • CommentTimeSep 9th 2011

    OK, so not one but two stupidly obvious mistakes:

    1. The monad on [obC×obC,V][ob C \times ob C, V] for which hom C\hom_C is an algebra is not the one I said it was.
    2. The coequalizer presenting hom C\hom_C is indeed reflexive but not split (the splitting I had in mind is not natural). It’s still a coequalizer, though, so the basic idea stands.

    I’ll fix end accordingly. Thanks.