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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeAug 10th 2014
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   Author: Dmitri Pavlov
   Format: MarkdownItexIt appears that a left Bousfield localization of a [[cartesian model category]] is again cartesian. This can be deduced fairly easily from Lemma 7.7.3 in Simpson's book, which proves that weak equivalences in a cartesian model category are closed under products. Is there an explicit reference for this statement anywhere in the literature?

   It appears that a left Bousfield localization of a cartesian model category is again cartesian. This can be deduced fairly easily from Lemma 7.7.3 in Simpson’s book, which proves that weak equivalences in a cartesian model category are closed under products.

   Is there an explicit reference for this statement anywhere in the literature?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeAug 10th 2014
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   Author: Zhen Lin
   Format: MarkdownItexThat can't be right. Dugger's theorem says that every combinatorial model category is Quillen–equivalent to a left Bousfield localisation of (the Heller model structure on) a simplicial presheaf category, and those are always cartesian. But being a cartesian model category implies that the homotopy category is cartesian closed, and this is invariant under Quillen equivalences; it should go without saying that there are combinatorial model categories whose homotopy category is _not_ cartesian closed.

   That can’t be right. Dugger’s theorem says that every combinatorial model category is Quillen–equivalent to a left Bousfield localisation of (the Heller model structure on) a simplicial presheaf category, and those are always cartesian. But being a cartesian model category implies that the homotopy category is cartesian closed, and this is invariant under Quillen equivalences; it should go without saying that there are combinatorial model categories whose homotopy category is not cartesian closed.

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeAug 10th 2014
   • (edited Aug 10th 2014)
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   Author: Dmitri Pavlov
   Format: MarkdownItexIndeed. Charles Rezk in Section 2.21 of his paper “A cartesian presentation of weak n-categories” explicitly points out the need for a cartesian saturation to ensure that the left Bousfield localization is cartesian.

   Indeed. Charles Rezk in Section 2.21 of his paper “A cartesian presentation of weak n-categories” explicitly points out the need for a cartesian saturation to ensure that the left Bousfield localization is cartesian.