Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
1 to 3 of 3
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 X∈sC, K∈sSet, X⊗K is the simplicial object whose nth component is the coproduct ∐KnXn. One can verify that
These facts imply that with the mapping spaces MapC(X,Y) defined by
MapC(X,Y)n=HomC(X⊗Δn,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:
(F⊗K)(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)(F⊗h(X),G)where h(X) is the presheaf of sets represented by X∈S. 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.
You don’t need colimits or limits. There’s a end formula for the hom-spaces: see here.
Interesting, thanks!
1 to 3 of 3