Edit: The claim below is wrong, as explained in the comments, because infinite homotopy products of simplicial sets require their components to be fibrantly replaced first, unlike finite homotopy products.

Simplicial homotopy colimits (i.e., homotopy colimits indexed by the opposite category of simplices) commute with (small) homotopy products in spaces.

This can be easily seen by presenting spaces as simplicial sets, simplicial homotopy colimits as the diagonals of the corresponding bisimplicial sets, homotopy products as products, and observing that the diagonal functor, being induced by a restriction along an inclusion of categories, preserves all (co)limits because (co)limits are computed componentwise in any category of functors. In particular, the diagonal functor commutes with small products, which establishes the desired property.

The same argument extends to all Grothendieck ∞-toposes (e.g., quasicategorical toposes or model toposes), because they can be presented as left Bousfield localizations of the categories of simplicial presheaves on small sites, and homotopy (co)limits are computed componentwise in any category of simplicial presheaves, or a left Bousfield localization thereof.

Of course, the opposite category of simplices is homotopy sifted and homotopy sifted colimits commute with finite homotopy products, so the interesting part is commutation of simplicial homotopy colimits with infinite (but small) homotopy products.

I would like to obtain a model-independent proof of this property. Recall that Grothendieck ∞-toposes can be characterized by the appropriate version of Giraud's axioms, as proposed by Rezk, namely, they are precisely those locally presentable ∞-categories in which homotopy colimits are universal, homotopy coproducts are disjoint, and every groupoid object is effective.

Is it possible to prove that small homotopy products commute with simplicial homotopy colimits directly from Giraud's axioms without resorting to a particular model of spaces?

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    $\begingroup$ The statement that colimits commute with products (in each variable) is a special case of the assumption that colimits are universal (namely, that they are preserved by pullback along the map X -> *), which is one of the axioms on your list. $\endgroup$ Sep 18, 2014 at 11:41
  • $\begingroup$ @JacobLurie: Of course, but how does one extend this property to infinite products? $\endgroup$ Sep 18, 2014 at 11:45
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    $\begingroup$ Ah, I misunderstood you. Your original statement is false, even in the infty-category of spaces. (Also, it wouldn't follow from a general infty-topos from there, because left exact functors only preserve finite products. Also, sifted colimits don't commute with finite products in a general presentable infty-category.) $\endgroup$ Sep 18, 2014 at 11:57
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    $\begingroup$ For a counterexample in spaces, let X_* be the simplicial set with vertices the integers and edges joining each integer n to n+1. Regard X_* as a simplicial object of spaces whose geometric realization is contractible. Then a product of infinitely many copies of |X_| is contractible. But the geometric realization of a product of infinitely many copies of X_ is not connected: for example, there's no path joining the identity map Z -> Z with the constant map joining Z -> {0} -> Z. (Contradicting your "elementary computation".) $\endgroup$ Sep 18, 2014 at 12:17
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    $\begingroup$ @JacobLurie: Many thanks for the counterexample, it certainly disproves the claim. The problem with the elementary arguments is that infinite homotopy products of simplicial sets require their arguments to be fibrantly replaced, unlike finite homotopy products, and the diagonal functor does not preserve fibrancy. $\endgroup$ Sep 18, 2014 at 12:56


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