Ore localization and model structures

Question

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 (in preparation)

Derived Categories, 3rd prepubllication version

In more detail: in Example 6.2.29 in the book I discuss the derived category of commutative DG rings. There is a congruence on the category of comm DG rings by the quasi-homotopy relation. (This is the same idea as the concept of homotopy in categories of fibrant objects, except that in my case the homotopy is presented by a cylinder object, not by a path object.) The passage from the corresponding homotopy category to the derived category is a right Ore localization. The question above is Remark 6.2.30 there.

There is a similar story for NC DG rings. But in the case there is another homotopy is used to formulate quasi-homotopies. (Here the homotopy is presented by a path object; it is also called a Keller homotopy.) Still one gets a right Ore localization.

This issue is also touched upon in my paper

The Squaring Operation for Commutative DG Rings

and in the lecture notes

The Derived Category of Sheaves of Commutative DG Rings

Note that in the geometric setting above there appears to be no Quillen model structure.

If anybody has ideas on this matter, I would like to hear them, and maybe also mention them in my book. One or two references will be appreciated.

Thanks.

Amnon Yekutieli

Answer 1

The full subcategory spanned by the fibrant objects of a model category always satisfies the right Ore condition, following https://ncatlab.org/nlab/show/calculus+of+fractions .

Similarly the subcategory of cofibrant objects satisfies the left Ore condition.

As a corollary, in many many cases of interest (nonnegatively graded complexes of modules with the projective model structure, topological spaces), all objects are fibrant, so these model categories satisfy the right Ore condition automatically. Dually, model categories in which all objects are cofibrant (simplicial sets with the Kan and Joyal model structures) satisfy the left Ore condition.