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• CommentRowNumber1.
• CommentAuthordomenico_fiorenza
• CommentTimeAug 12th 2014
• (edited Aug 12th 2014)

With Fosco Loregian we are now fine tuning a short note on t-structures and factorization systems in $\infty$-stable categories. In case you’d like to have a preview of it before we post it to the arXiv, any suggestion, comment or criticism is welcome.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeAug 12th 2014
• (edited Aug 12th 2014)

What is the time span of posting ? I mean I am very interested to read this one and comment as I worked in localization theory for a while (and revived that this Spring), but this current week is rather busy to get to a serious new paper, beyond some minutes in public transport…

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeAug 12th 2014
• (edited Aug 12th 2014)

I have just updated torsion theory for the full reference to Belgiannis-Reiten article which to my surprise is not cited in your paper.

• Apostolos Beligiannis, Idun Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 (2007), no. 883, viii+207 pp. pdf

Regarding that the classical theory of torsion theories is developed somewhat beyond the abelian case, I would expect that in $\infty$-categories one could go somewhat beyond the stable case…

1. Hi Zoran, glad you are interested in this! No hurry: we were planning to upload on arXiv around September 1st or so.

2. Thanks for the reference! I now see that’s indeed extremely relevant to what we did, we were not aware of it. Thanks a lot!

3. Ah, no, I was misatken here: the Belgiannis-Reiten work you are referring to is what we are citing as [BR07]. It appears already on page 1 of our note. Maybe I’m confused here and we are pointing to two different articles?

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeAug 12th 2014
• (edited Aug 12th 2014)

Oh, no, I was mistaken. The reference is indeed in your list but not at alphabetic place and having being very well skilled in alphabetic search I dismissed as Belgiannis is not before Borges :) Sorry for confusing you…

• CommentRowNumber8.
• CommentAuthorFosco
• CommentTimeAug 12th 2014
• (edited Aug 12th 2014)

You have no reason to be sorry: on the contrary, I thank you for pointing out the thing. Improving readability implies avoiding alphabetical mistakes. And I plan to re-read Beligiannis-Reiten first chapter and some others works on which our preprint is based in order to refer to them more precisely.

I am struggling with some minor corrections precisely in the bibliography, but at the moment I have no idea how to prevent BibTeX to put the two references in different order.

4. Hi Fosco, not sure the references should be in a different order: a casual reader is not expected to guess from [BR07] what reference is that, and so will search for [BR07] in the bibliography after [Bor]. In this case the problem seems quit striking since [Bor] is a non-math reference. But it could have been Borcherds here :) So in the end I’d trust BibTeX choice: I don’t see a solution which will be optimal from any point of view.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 18th 2014
• (edited Aug 18th 2014)

Sorry for the slow reaction. I had been somewhat absorbed with other things, and still am.

This here just to say what I believe I had already said when you announced this elsewhere, earlier: it seems to me a really nice result. I’ll look into it, but not this week (am at a meeting once more, and am way behind with my required preparation).

Meanwhile, this statement certainly deserves an nLab entry. Maybe somebody likes to create one with a summary of the main statement and a pointer to the preprint?

• CommentRowNumber11.
• CommentAuthorFosco
• CommentTimeAug 18th 2014

I was planning to do it as soon as possible! Maybe in the following days, certainly for when we will upload on arxiv the paper. Thank you a lot for your interest

• CommentRowNumber12.
• CommentAuthorFosco
• CommentTimeAug 23rd 2014
• (edited Aug 24th 2014)

I’m back home, finally. I plan to add a link to the statement to the existing page about t-structures, and then create a new page, linking the existing reflective factorization system and adding informations on normal simple and semi-left/right-exact factorization system, and on how the three notions coincide in the stable $\infty$-categorical setting.

2. (semi?)orthogonal decompositions are “well-behaved” t-structures, where the reflection and coreflection associated to the factorization system are exact functors; there is a restriction of the correspondence we outlined in the paper, that associates to every semiorthogonal decomposition $(\mathbf{D}_{\ge 0}, \mathbf{D}_{\lt 0})$ a stable factorization system. We are working out the details and this stuff will (maybe) appear in a subsequent paper, especially due to the importance semiorthogonal decompositions seem to have in Algebraic Geometry (see for example Kuznetsov-Lunts “Resolution of singularities”).