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1. • CommentRowNumber1.
   • CommentAuthorFosco
   • CommentTimeJan 12th 2015
   • (edited Jan 12th 2015)
   • PermaLink
   Author: Fosco
   Format: MarkdownItexDomenico Fiorenza and I are completing a [paper](https://www.dropbox.com/s/eqbwis0qwbg9rxv/a-brand-new-file.pdf) about hearts of t-structures in stable $\infty$-categories, which shows that in the $\infty$-categorical setting semiorthogonal decompositions on a stable $\infty$-category $\mathcal{C}$ arise decomposing morphisms in the *Postnikov tower* induced by a chain of t-structures, regarded (thanks to our previous [work](http://arxiv.org/abs/1408.7003)) as multiple factorization systems on $\mathcal{C}$. A slightly unexpected result is that t-structures having stable classes, i.e. those $(\mathcal{C}_{\ge 0}, \mathcal{C}_{\lt 0})$ such that both classes are stable $\infty$-subcategories of $\mathcal{C}$, are precisely the *fixed points* for the natural action of $\mathbb{Z}$ on the set of t-structures, given by the shift endofunctor. As always, any comment, suggestion, criticism is welcome.

   Domenico Fiorenza and I are completing a paper about hearts of t-structures in stable $\infty$-categories, which shows that in the $\infty$-categorical setting semiorthogonal decompositions on a stable $\infty$-category $\mathcal{C}$ arise decomposing morphisms in the Postnikov tower induced by a chain of t-structures, regarded (thanks to our previous work) as multiple factorization systems on $\mathcal{C}$.

   A slightly unexpected result is that t-structures having stable classes, i.e. those $(\mathcal{C}_{\ge 0}, \mathcal{C}_{\lt 0})$ such that both classes are stable $\infty$-subcategories of $\mathcal{C}$, are precisely the fixed points for the natural action of $\mathbb{Z}$ on the set of t-structures, given by the shift endofunctor.

   As always, any comment, suggestion, criticism is welcome.

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJan 14th 2015
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   Author: Dmitri Pavlov
   Format: MarkdownItexSpeaking of t-structures, apparently there is a closely related notion of a _weight structure_ on a triangulated category (see the papers by Bondarko), which axiomatizes chain complexes equivalent to those concentrated in nonnegative resp. nonpositive degrees, as opposed to t-structures, which axiomatize chain complexes whose homology is concentrated in nonnegative resp. nonpositive degrees. Are you aware of any treatments of weight structures in the context of stable ∞-categories?

   Speaking of t-structures, apparently there is a closely related notion of a weight structure on a triangulated category (see the papers by Bondarko), which axiomatizes chain complexes equivalent to those concentrated in nonnegative resp. nonpositive degrees, as opposed to t-structures, which axiomatize chain complexes whose homology is concentrated in nonnegative resp. nonpositive degrees.

   Are you aware of any treatments of weight structures in the context of stable ∞-categories?

3. • CommentRowNumber3.
   • CommentAuthorFosco
   • CommentTimeJan 14th 2015
   • (edited Jan 14th 2015)
   • PermaLink
   Author: Fosco
   Format: MarkdownItexCan you provide us a pointer with the definition of weight structure? There are several papers in [Bondarko's arxiv list](http://arxiv.org/find/math/1/au:+Bondarko_M/0/1/0/all/0/1) (NP, just found [this](http://arxiv.org/pdf/0704.4003.pdf).. but any other reference is welcome!)

   Can you provide us a pointer with the definition of weight structure? There are several papers in Bondarko’s arxiv list (NP, just found this.. but any other reference is welcome!)

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJan 14th 2015
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   Author: Dmitri Pavlov
   Format: MarkdownItex@Fosco: Yes, 0704.4003 is the right reference.

   @Fosco: Yes, 0704.4003 is the right reference.

5. • CommentRowNumber5.
   • CommentAuthordomenico_fiorenza
   • CommentTimeJan 19th 2015
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   Author: domenico_fiorenza
   Format: MarkdownItexWe have now uploaded on arXiv. The comparison with eight structures suggested by Dmitri is indeed very interesting but we realized it needed a development going beyond the aims of the present paper, so we postponed it to a (hopefully) forthcoming follow-up.

   We have now uploaded on arXiv. The comparison with eight structures suggested by Dmitri is indeed very interesting but we realized it needed a development going beyond the aims of the present paper, so we postponed it to a (hopefully) forthcoming follow-up.

6. • CommentRowNumber6.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJan 21st 2015
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   Author: Dmitri Pavlov
   Format: MarkdownItex@domenico_fiorenza: I would be very interested in reading such a followup.

   @domenico_fiorenza: I would be very interested in reading such a followup.