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    • CommentRowNumber1.
    • CommentAuthorFosco
    • CommentTimeJan 12th 2015
    • (edited Jan 12th 2015)

    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 𝒞\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 𝒞\mathcal{C}.

    A slightly unexpected result is that t-structures having stable classes, i.e. those (𝒞 0,𝒞 <0)(\mathcal{C}_{\ge 0}, \mathcal{C}_{\lt 0}) such that both classes are stable \infty-subcategories of 𝒞\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.

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 14th 2015

    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?

    • CommentRowNumber3.
    • CommentAuthorFosco
    • CommentTimeJan 14th 2015
    • (edited Jan 14th 2015)

    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!)

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 14th 2015

    @Fosco: Yes, 0704.4003 is the right reference.

  1. 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.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 21st 2015

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