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 ]]>

New entries torsion theory and quotient category.

]]>New entry for Samuel compactification of uniform spaces, and some references at uniform space.

]]>Extended the entry Cohn localization now starting with the ring viewpoint. Urs: I hope you will now agree that it is justified to call it a localization of a ring $R\to \Sigma^{-1} R$.

]]>This is a different approach - more restrictive, I believe - than Mike's work at [[michaelshulman:exact completion of a 2-category]], where he talks about anafunctors in a 2-category (weak, by default, for him). There he talks about anafunctors in the 2-category of 2-congruences in a 2-site, and thinking of them more along the lines as in Cat(S). But I'm very interested in the relation between the two, especially if one could be derived from the other.

One spin-off of this is that I would like to provide another model for the localisation of a 2-category. Here J needs to be weakly cofinal in the class W one wants to invert. One point of my anafunctors paper was to show that the localisation of a 2-category of internal categories had a better model that the default one constructed by Pronk, and this theorem should go through, namely K[W^-1] ~ K_ana. Note that this is (2,2)-category localisation, not (2,1)-category localisation. (As an aside, the approach to localisation via bibundles, which is even simpler to describe, wouldn't work here because that assumes one is in a (2,1)-category.)

The one point which is a bit restrictive is that one needs covers to be an [[ff morphism]] in order to define the bicategory K_ana of anafunctors in K. (This reminds me somewhat of talking about S-local maps in a model theoretic setup, at least when the pretopology J is morally like a cover by open balls or affine schemes. But I haven't thought about this too much yet.)

One direction this may go is if the whole game can be phrased in a suitably 2-categorical way, then perhaps similar techniques could be used to talk about localisation of higher categories (say simplicial categories), at least in special cases. For example, defining weak maps between strict higher categories or something. This is complete speculation, and not a short-term goal by any means.

Thoughts? ]]>

Urs, David Roberts and I got into discussion of locally trivial noncommutative bundles in a discussion with a wrong title (see around here), so let us better move it here. There are still some of my latest posts there which Urs and David might have not yet seen.

I decided to update a bit noncommutative principal bundle, so I will start today a bit.

]]>I have started a stub localization of an abelian category. Added a list of related terms at topologizing subcategory.

]]>New entry localization of an enriched category.

]]>I started an important entry differential monad. According to Lunts-Rosenberg MPI 1996-53 pdf differential calculus on schemes and noncommutative schemes can be derived from the yoga of coreflective topologizing subcategories in the abelian category of quasicoherent sheaves on the scheme, like the $\mathbb{T}$-filtration, and $\mathbb{T}$-part, in the case when the topologizing subcategory is the diagonal in the sense of the smallest subcategory of the category of additive endofunctors having right adjoint which contains the identity functor – in that case we say differential filtration and differential part. The regular differential operators are the elements of the differential part of the bimodule of endomorphisms. Similarly, one can define the conormal bundle etc.

]]>This was the query in topologizing subcategory which I summarized shortly:

]]>Mike: Where does the word ’topologizing’ come from?

Zoran Skoda: I am not completely sure anymore, but I think it is from ring theory, where people looked at the localizations at topologizing categories. There exist some topologies on various sets of ideals like Jacobson topology, so it is something of that sort in the language of subcategories instead of the language of filters of ideals. I’ll consult old references like Popescu, maybe I recall better. In any case it is pretty standard and has long history in usage: both classical and modern. No, it is not in Popescu…old related term is in fact talking about topologizing filters of ideals in a ring, so that must be the source…for example, the classical algebra by Faith, vol I, page 520 defines when the set of right ideals is topologizing. I am not good with that notion, but I can make an entry with quotation to be improved later.