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 complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration 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 nforum 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.
    • CommentAuthoradeelkh
    • CommentTimeMay 2nd 2014
    • (edited May 3rd 2014)

    Let C be a category admitting colimits and limits. It is well known that the category of simplicial objects sC in a category C has an action by sSet: for XsC, KsSet, XK is the simplicial object whose nth component is the coproduct KnXn. One can verify that

    1. For a fixed K, the functor K:sCsC admits a right adjoint ()K.
    2. For XC, X preserves colimits and XΔ0=X.
    3. For XC and K,LsSet, A(K×L)=(AK)L.

    These facts imply that with the mapping spaces MapC(X,Y) defined by

    MapC(X,Y)n=HomC(XΔn,Y),

    sC becomes an sSet-enriched category. All this is well-documented, for example in the book of Goerss-Jardine.

    More generally, let S be a small category. Let PSh(S,C) denote the category of presheaves on S with values in C. Define an action of PSh(S)=PSh(S,Set) on PSh(S,C) in the same way as above:

    (FK)(X)=K(X)F(X)

    Presumably, the above three facts are still true and one can define a PSh(S)-valued Hom by

    Hom(F,G)(X)=HomPSh(S,C)(Fh(X),G)

    where h(X) is the presheaf of sets represented by XS. I imagine that this makes PSh(S,C) into a PSh(S)-enriched category.

    Are these things written down somewhere (maybe on the nLab)? The arguments written in Goerss-Jardine probably work without modification, but I am wondering if there is a simpler description of the exponential objects XK than what they do.

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeMay 2nd 2014

    You don’t need colimits or limits. There’s a end formula for the hom-spaces: see here.

    • CommentRowNumber3.
    • CommentAuthoradeelkh
    • CommentTimeMay 2nd 2014

    Interesting, thanks!