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    • 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 XsCX \in sC, KsSetK \in sSet, XKX \otimes K is the simplicial object whose nth component is the coproduct K nX n\coprod_{K_n} X_n. One can verify that

    1. For a fixed KK, the functor K:sCsC- \otimes K : sC \to sC admits a right adjoint () K(-)^K.
    2. For XCX \in C, XX \otimes - preserves colimits and XΔ 0=XX \otimes \Delta^0 = X.
    3. For XCX \in C and K,LsSetK,L \in sSet, A(K×L)=(AK)LA \otimes (K \times L) = (A \otimes K) \otimes L.

    These facts imply that with the mapping spaces Map C(X,Y)Map_C(X, Y) defined by

    Map C(X,Y) n=Hom C(XΔ n,Y), Map_C(X,Y)_n = Hom_C(X \otimes \Delta^n, 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)PSh(S, C) denote the category of presheaves on S with values in C. Define an action of PSh(S)=PSh(S,Set)PSh(S) = PSh(S, Set) on PSh(S,C)PSh(S, C) in the same way as above:

    (FK)(X)= K(X)F(X) (F \otimes K)(X) = \coprod_{K(X)} F(X)

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

    Hom(F,G)(X)=Hom PSh(S,C)(Fh(X),G) \mathbf{Hom}(F, G)(X) = Hom_{PSh(S,C)}(F \otimes h(X), G)

    where h(X)h(X) is the presheaf of sets represented by XSX \in S. I imagine that this makes PSh(S,C)PSh(S, C) into a PSh(S)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 X KX^K 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!