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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 $X \in sC$, $K \in sSet$, $X \otimes K$ is the simplicial object whose nth component is the coproduct $\coprod_{K_n} X_n$. One can verify that
These facts imply that with the mapping spaces $Map_C(X, Y)$ defined by
$Map_C(X,Y)_n = Hom_C(X \otimes \Delta^n, Y),$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 $PSh(S, C)$ 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:
$(F \otimes K)(X) = \coprod_{K(X)} F(X)$Presumably, the above three facts are still true and one can define a $PSh(S)$-valued Hom by
$\mathbf{Hom}(F, G)(X) = Hom_{PSh(S,C)}(F \otimes h(X), G)$where $h(X)$ is the presheaf of sets represented by $X \in S$. 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 $X^K$ than what they do.
You don’t need colimits or limits. There’s a end formula for the hom-spaces: see here.
Interesting, thanks!
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