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Added:
Suppose $C$ is a category that admits small coproducts.
Given simplicial objects $A,B\in C^{\Delta^{op}}$, their function complex is a simplicial set
$Hom(A,B)$whose set of $n$-simplices is the set of maps
$\Delta^n\otimes A\to B,$where $\otimes$ denotes the copowering of simplicial objects over simplicial sets given by
$(C\otimes D)_n=\coprod_{i\in C_n}D_n.$The original definition of a function complex in the generality stated above is due to Daniel M. Kan:
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Looking over the entry now, I have adjusted the previous notation a little in order to avoid a couple of clashes:
the notation “Hom” for the simplicial hom-complex I have replaced with “$Hom_\Delta$”
the letter “$C$” used to denote both the ambient category as well as a simplicial set tensor factor. I have changed to writing “$\mathcal{C}$” for the former and “$S$” for the latter.
Also expanded the presentation just a little.
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