nForum - Discussion Feed (Ore localization and model structures)2024-03-28T15:24:01+00:00https://nforum.ncatlab.org/
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Urs comments on "Ore localization and model structures" (74092)https://nforum.ncatlab.org/discussion/9308/?Focus=74092#Comment_740922018-11-25T15:33:27+00:002024-03-28T15:24:01+00:00Urshttps://nforum.ncatlab.org/account/4/
Welcome Amnon! Thanks for forwarding your question here.
Welcome Amnon! Thanks for forwarding your question here.
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amyekut comments on "Ore localization and model structures" (74091)https://nforum.ncatlab.org/discussion/9308/?Focus=74091#Comment_740912018-11-25T14:49:44+00:002024-03-28T15:24:01+00:00amyekuthttps://nforum.ncatlab.org/account/420/
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 ...
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.