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A simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets).

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How to define the internal hom between presheaves valued in cotensored categories?

Yes, if $\mathcal M$ is reasonable. The object-wise internal hom $(s,t) \mapsto G(s)^{F(t)}$ is contravariant in $s \in \mathcal S$ and covariant in $t$. The object you want is the end of this funct …
Theo Johnson-Freyd's user avatar