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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 25th 2014
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   Author: Dmitri Pavlov
   Format: MarkdownItexThe article [[local model structure on simplicial presheaves]] states that for a site with enough points stalkwise weak equivalences of simplicial presheaves coincide with weak equivalences of simplicial presheaves in the Bousfield localization of componentwise weak equivalences with respect to all hypercovers (i.e., weak equivalences in a local model structure). Is a proof of this statement written up somewhere? (The article cited above gives a reference to Jardine, which claims, but does not prove this statement.) Also, is it possible to formulate an analog of this statement for sites that do not have enough points? (Presumably we would have to talk about sufficiently refined (hyper)covers instead of points.)

   The article local model structure on simplicial presheaves states that for a site with enough points stalkwise weak equivalences of simplicial presheaves coincide with weak equivalences of simplicial presheaves in the Bousfield localization of componentwise weak equivalences with respect to all hypercovers (i.e., weak equivalences in a local model structure).

   Is a proof of this statement written up somewhere? (The article cited above gives a reference to Jardine, which claims, but does not prove this statement.)

   Also, is it possible to formulate an analog of this statement for sites that do not have enough points? (Presumably we would have to talk about sufficiently refined (hyper)covers instead of points.)

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeJun 25th 2014
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   Author: Zhen Lin
   Format: MarkdownItexActually, the business with left Bousfield localisation with respect to hypercovers is due to Dugger, Hollander, and Isaksen. They proved that the local weak equivalences are precisely the morphisms that induce isomorphisms on sheaves of homotopy groups, which is (more or less) Jardine's definition of weak equivalence. This does not require the enough-points hypothesis. However, sheaves of homotopy groups are preserved by inverse image functors, so Jardine's weak equivalences can be detected stalkwise for a topos with enough points.

   Actually, the business with left Bousfield localisation with respect to hypercovers is due to Dugger, Hollander, and Isaksen. They proved that the local weak equivalences are precisely the morphisms that induce isomorphisms on sheaves of homotopy groups, which is (more or less) Jardine’s definition of weak equivalence. This does not require the enough-points hypothesis. However, sheaves of homotopy groups are preserved by inverse image functors, so Jardine’s weak equivalences can be detected stalkwise for a topos with enough points.