Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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.