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The homotopy product of an infinite family of simplicial sets can be computed by deriving the product functor sSetW→sSet, for example, by performing the componentwise fibrant replacement using Kan's Ex∞ functor and then taking the ordinary product.

If the simplicial sets are already Kan, then their product is also their homotopy product. There are families of simplicial sets for which this is false, an example is given in the comments to the question Commutation of simplicial homotopy colimits and homotopy products in spaces.

Is it possible to relax the Kan condition so that the ordinary product still computes the homotopy product? Are there any references for such questions?

For example, one potential relaxation that I have in mind is to require that every horn H→X has a filling up to a homotopy, meaning that there is a simplicial homotopy Δ¹×H→X that does not move any of the vertices of H and whose bottom face is the given map H→X and the top face has a filling in the usual sense.

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  • $\begingroup$ Here's a simple observation: let us say that a simplicial set $X$ has property $C_n$ if two vertices are in the same connected component of $X$ if and only if there is a path of length $\le n$ connecting them. So for example Kan complexes have property $C_1$. Then for every positive integer $n$, $\pi_0$ preserves products of families of simplicial sets having property $C_n$. This seems to suggest that it suffices to satisfy the Kan condition up to some finite subdivision... $\endgroup$
    – Zhen Lin
    Commented Oct 2, 2014 at 20:10

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