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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.
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?
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!)
@Fosco: Yes, 0704.4003 is the right reference.
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.
@domenico_fiorenza: I would be very interested in reading such a followup.
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