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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 27th 2010
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   Author: Harry Gindi
   Format: MarkdownItexLet 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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 27th 2010
   • (edited Jun 27th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexAlright, 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 [[localizer|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) $(\Delta^1,\emptyset)$, so that gives a proof, but I'm wondering if there's any quicker way to show it for SSet.

   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) $(\Delta^1,\emptyset)$, so that gives a proof, but I’m wondering if there’s any quicker way to show it for SSet.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJun 27th 2010
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   Author: Mike Shulman
   Format: MarkdownItexIt'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 $W\subset Arr(C)$ 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.

   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 $W\subset Arr(C)$ 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.

4. • CommentRowNumber4.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 27th 2010
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   Author: Harry Gindi
   Format: MarkdownItexAh, it's one of the axioms for a perfect class of weak equivalences.

   Ah, it’s one of the axioms for a perfect class of weak equivalences.