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1. • CommentRowNumber1.
   • CommentAuthoramyekut
   • CommentTimeNov 25th 2018
   • PermaLink
   Author: amyekut
   Format: TextThe question is this: Suppose C is a category, with a given multiplicatively closed set of morphisms S ⊆ C. The role of the denominator conditions on S is rather similar to the role of a Quillen model structure on C, for which S is the set of weak equivalences. However, the precise relationship between these concepts is not clear to me. This question is included in the book Derived Categories 3rd prepubllication version: https://arxiv.org/abs/1610.09640 v3 In more detail: in Example 6.2.29 in the book I discuss the derived category of commutative DG rings. The main innovation is that there is a congruence on the category of comm DG rings by the quasi-homotopy relation. The passage from the corresponding homotopy category to the derived category is a right Ore localization. (There is a similar story for NC DG rings, but another homotopy is used to formulate quasi-homotopies.) The question above is Remark 6.2.30 there. This issue is also touched upon in my paper The Squaring Operation for Commutative DG Rings https://www.math.bgu.ac.il/~amyekut/publications/squaring-DG/squaring-DG.html and in the lecture notes The Derived Category of Sheaves of Commutative DG Rings https://www.math.bgu.ac.il/~amyekut/lectures/shvs-dgrings/abstract.html If participants of the forum have some ideas on this matter, I would like to hear them, and maybe also mention them in my book. Amnon Yekutieli

   The question is this:
   
   Suppose C is a category, with a given multiplicatively closed set of morphisms S ⊆ C. The role of the denominator conditions on S is rather similar to the role of a Quillen model structure on C, for which S is the set of weak equivalences. However, the precise relationship between these concepts is not clear to me.
   
   This question is included in the book
   Derived Categories
   3rd prepubllication version: https://arxiv.org/abs/1610.09640 v3
   
   In more detail: in Example 6.2.29 in the book I discuss the derived category of commutative DG rings. The main innovation is that there is a congruence on the category of comm DG rings by the quasi-homotopy relation. The passage from the corresponding homotopy category to the derived category is a right Ore localization. (There is a similar story for NC DG rings, but another homotopy is used to formulate quasi-homotopies.) The question above is Remark 6.2.30 there.
   
   This issue is also touched upon in my paper
   The Squaring Operation for Commutative DG Rings
   https://www.math.bgu.ac.il/~amyekut/publications/squaring-DG/squaring-DG.html
   and in the lecture notes
   The Derived Category of Sheaves of Commutative DG Rings
   https://www.math.bgu.ac.il/~amyekut/lectures/shvs-dgrings/abstract.html
   
   If participants of the forum have some ideas on this matter, I would like to hear them, and maybe also mention them in my book.
   
   Amnon Yekutieli
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2018
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   Author: Urs
   Format: MarkdownItexWelcome Amnon! Thanks for forwarding your question here.

   Welcome Amnon! Thanks for forwarding your question here.