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Atrium > Mathematics, Physics & Philosophy: Kan Extension

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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 22nd 2010
    • (edited Mar 22nd 2010)

    Would anyone be opposed to giving the definition of pointwise Kan extension as a Kan extension that is preserved by all co-representable functors (i.e. Ran_kT is a pointwise kan extension if m_a(Ran_K(T))\cong Ran_{m_a(K)}m_a(T) where m_a denotes the functor Hom(a,-).

    I looked this up in Mac Lane's Categories Work, but I'm not sure that I interpreted "preserved by Hom(a,-) for all objects a" correctly.

    Mac Lane also makes the point of assuming that the category has small Hom-sets, but I think that distinction is unnecessary unless we're being cardinality-strict for no reason.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMar 23rd 2010

    Of course not. Mac Lane's assumption of local smallness is just because a category that is not locally small has no hom-functors landing in Set, so the definition is meaningless. Not everyone believes that proper classes are just sets of some bigger universe in disguise.

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 23rd 2010

    I don't see the use of assuming otherwise. Is there a good argument for having hom sets that are actually proper classes? I mean, on questions that are independent of set theory, shouldn't we take the answer that allows us to do the most?

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 23rd 2010
    Given Set (not U-Set, but some non-strict category of all sets, say), then the category of presheaves with values in Set has whopping big hom-sets, if I remember rightly
    • CommentRowNumber5.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 23rd 2010
    • (edited Mar 23rd 2010)

    We can systematically excise every occurrence of Set in any mathematical document by replacing Set with U-Set and making the necessary changes to be relative to U.

    This gives us the Yoneda embedding for any category, which is no trivial matter.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 23rd 2010

    But we might not want to assume the axiom of universes...

    • CommentRowNumber7.
    • CommentAuthorHarry Gindi
    • CommentTimeMar 23rd 2010

    To what end?

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 23rd 2010

    centipede mathematics, perhaps. People around here, I've noticed, tend to try and do this sort of thing. You might as well ask, why do without the axiom of associativity? The answer is, of course, why not? If things break, we want to know precisely why and how.

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