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    • CommentRowNumber1.
    • CommentAuthoramyekut
    • CommentTimeNov 25th 2018
    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
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2018

    Welcome Amnon! Thanks for forwarding your question here.

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