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I'm curious to find out where the viewpoint of higher categories may be useful so here is a somewhat vague question (which may or may not have a reasonable answer).

Given a triangulated category, one can consider the set of all possible t-structures on it. Simple examples where one can compute things by hand indicate that this is something complicated but not hopelessly so. See for example the paper http://arxiv.org/abs/0909.0552 by Jon Woolf which describes a three parametric family of t-structures on the constructible bounded derived category of $\mathbf{P}^1(\mathbf{C})$ stratified by a point and its complement. Some of these t-structures are more interesting than others and there is one that is the most interesting of them all since by taking the bounded derived category of its heart one gets back the triangulated category one started with. (For that t-structure the heart is the category of perverse sheaves on $\mathbf{P}^1(\mathbf{C})$.)

On the other hand, the set of t-structures on a triangulated category is interesting since there lurks somewhere the conjectural motivic t-structure whose existence implies Grothendieck's standard conjectures. See the recent paper http://arxiv.org/abs/1006.1116 by Beilinson.

On the triangulated categories page of the n-category lab website it says "Therefore, all the structure and properties of a triangulated category is best understood as a 1-categorical shadow of the corresponding properties of stable (infinity,1)-categories". See http://ncatlab.org/nlab/show/triangulated+category. Note that this is quite a strong statement, since it is referring to all, and not just some, properties and structure of a triangulated category.

So I'd like to ask: is there a higher categorical analog of a t-structure? More generally, how does the higher categorical viewpoint help one understand the set of all (or maybe all "nice" in an appropriate sense) t-structures on a given trangulated category, provided it is the homotopy category of a stable $(\infty,1)$ category?

upd: as Mike points out in the comments, the answer to the first question is yes and it is given by proposition 6.15 of Lurie's Stable Infinity Categories. The second, more "philosophical" question remains.

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I'm far from an expert, but I thought that stable infinity categories were the higher categorical analog of t-structures. My understanding is that it's actually a strength of the higher categorical theory that being stable is a property rather than an extra structure (as in the case of a t-structure on a triangulated category). You're probably already aware of arxiv.org/abs/math/0608228. –  Mike Skirvin Jul 4 '10 at 18:51
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Mike -- thanks! The first question seems to be answered by proposition 6.15 there. But even for stable $(\infty,1)$ categories a t-structure is still an extra structure, not property. –  algori Jul 4 '10 at 19:21
    
I guess you are implicitly assuming your triangulated categories are small? It is certainly not the case otherwise that there is necessarily a set of t-structures and things do seem a little hopeless in that case. –  Greg Stevenson Jul 4 '10 at 21:14
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Greg -- yes, to be completely precise one should fix a Grothendieck universe and require that set of objects belong to it; then so will the set of t-structures. But somehow it seems to me that these set theoretical subtleties are not very important here, which is why I'm willing to sweep them under the carpet. –  algori Jul 4 '10 at 21:52
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algori: I think I misread the point of your original question: actually you want to formulate things in a purely infinity-categorical way? If so, of course it's reasonable, as you wrote, to define a t-structure as a certain kind of localization as in DAG I Prop. 6.15. I think it's worth keeping in mind, though, that this structure is completely encoded at the homotopy level. Admittedly, I have no idea whether all the structure of the collection of all t-structures still lives there. –  Thomas Nevins Jul 6 '10 at 20:56
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1 Answer 1

So I'd like to ask: is there a higher categorical analog of a t-structure?

As Mike Skirvin pointed out in a comment, higher categorical analog of t-structures have been introduced by Lurie. A more up-to-date reference might be Higher Algebra ($\S$ 1.2.1).

More generally, how does the higher categorical viewpoint help one understand the set of all (or maybe all "nice" in an appropriate sense) t-structures on a given trangulated category, provided it is the homotopy category of a stable (∞,1) category?

I guess the answer can be found at the same place. There, Lurie says that "there is a bijective correspondence between $t$-localizations of $\mathcal C$ (a stable $\infty$-category) and $t$-structures on the triangulated category $h\mathcal C$.

The higher categorical point-of-view also seems to be useful to understand the yoga of derived functors in a more conceptual way. In Section 1.3 of the same reference (Higher Algebra) it is explained that if $\mathcal A$ is an abelian category with enough injectives, then its derived $\infty$-category $\mathcal D^-(\mathcal A)$ is stable, admits a $t$-structure, has homotopy category the standard derived category, and satisfies the following universal property: there is a canonical equivalence of abelian categories $\mathcal A\to \mathcal D^-(\mathcal A)^\heartsuit$, and if $\mathcal C$ is a stable $\infty$-category with a left-complete $t$-structure then any right exact functor $\mathcal A\to\mathcal C^\heartsuit$ extends (in an essentially unique way) to an exact functor $\mathcal D^-(\mathcal A)\to \mathcal C$.

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