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Let F,F’: D -> SSet be filtered diagrams. Suppose F->F’ is an objectwise weak equivalence (and hence a weak equivalence in either the projective or injective model structure on the diagram category). Is it true that the induced map colim F -> colim F’ is a weak equivalence? If so, why? If not, why not?
Alright, it’s one of the axioms of a combinatorial model category, but how can we show it for SSet?
I found a proof of a generalization in Cisinski for presheaf categories and localizers on them. It’s pretty easy to show that the class of weak equivalences on SSet is in fact a localizer (since Cisinski gives a nice way to generate them using the cylinder/generator pair (donnée homotopique elementaire) , so that gives a proof, but I’m wondering if there’s any quicker way to show it for SSet.
It’s not one of the axioms, it’s a theorem that in a combinatorial model category, κ-filtered colimits preserve objectwise weak equivalences for sufficiently large κ (i.e. the subcategory is accessible and accessibly embedded). I don’t know how the proof goes, but presumably if the combinatorial model category is sufficiently finitary, probably including sSet, then ordinary filtered colimits ( = ω-filtered ones) are good enough to preserve weak equivalences.
One way you could prove it for sSet is to use a combinatorial characterization of simplicial homotopy groups, which is finitary and thus preserved by filtered colimits.
Ah, it’s one of the axioms for a perfect class of weak equivalences.
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