Stable Homotopy theory
Let be a triangulated category. A t-structure on is a pair of strictly full subcategories
for all and the hom object is the zero object: ;
the subcategories are closed under suspension/desuspension: and .
For all objects there is a fiber sequence with and .
Given a t-structure, its heart is the intersection
In stable -categories
In the infinity-categorical setting -structures arise as torsion/torsionfree classes associated to suitable factorization systems on a stable infinity-category .
In a stable setting, the subcategories are closed under de/suspension simply because they are co/reflective and reflective and these operations are co/limits. Co/reflective subcategories of arise from co/reflective factorization systems on ;
A bireflective factorization system on a -category consists of a factorization system where both classes satisfy the two-out-of-three property.
A bireflective factorization system on a stable -category is called normal if the diagram obtained from the reflection and the coreflection (where the category is obtained as under the adjunction described at reflective factorization system and in CHK; see also FL0, §1.1) is exact, meaning that the square in
is a fiber sequence for any object ; see FL0, Def 3.5 and Prop. 3.10 for equivalent conditions for normality.
Remark. CHK established a hierarchy between the three notions of simple, semi-exact and normal factorization system: in the setting of stable -category the three notions turn out to be equivalent: see FL0, Thm 3.11.
Theorem. There is a bijective correspondence between the class of -structures and the class of normal torsion theories on a stable -category , induced by the following correspondence:
- On the one side, given a normal, bireflective factorization system on we define the two classes of a -structure to be the torsion and torsionfree classes associated to the factorization .
- On the other side, given a -structure on we set
Proof. This is FL0, Theorem 3.13
Theorem. There is a natural monotone action of the group of integers on the class (now confused with the class of normal torsion theories on ) given by the suspension functor: goes to .
This correspondence leads to study families of -structures ; more precisely, we are led to study -equivariant multiple factorization systems .
Theorem. Let and correspond each other under the above bijection; then the following conditions are equivalent:
- , i.e. ;
- is a stable -category;
- the class is closed under pullback.
In each of these cases, we say that or is stable.
Proof. This is FL1, Theorem 2.16
This results allows us to recognize -structures with stable classes precisely as those which are fixed in the natural -action on .
Two “extremal” choices of -chains of -structures draw a connection between two apparently separated constructions in the theory of derived categories: Harder-Narashiman filtrations and semiorthogonal decompositions on triangulated categories: we adopt the shorthand to denote the tuple , each of the being a -structure on , and we denote similarly . Then
- In the stable case the tuple is endowed with a (monotone) -action, and the map is equivariant with respect to this action; the absence of nontrivial -actions on forces each to be stable.
- In the orbit case we consider an infinite family of -structures on , obtained as the orbit of a fixed with respect to the natural -action.
The HN-filtration induced by a -structure and the factorization induced by a semiorthogonal decomposition on both are the byproduct of the tower associated to a tuple :
The heart of a stable -category is an abelian category.
(BBD 82, Higher Algebra, remark 188.8.131.52, FL0, Ex. 4.1 and FL1, §3.1)
Application to spectral sequence
If a the heart of a t-structure on a stable (∞,1)-category with sequential limits is an abelian category, then the spectral sequence of a filtered stable homotopy type converges (see there).
Related Lab entries include Bridgeland stability?.
For triangulated categories
S. I. Gelfand, Yuri Manin, Methods of homological algebra, Nauka 1988, Springer 1998, 2003
Donu Arapura, Triangulated categories and -structures (pdf)
Alexander Beilinson, Joseph Bernstein, Pierre Deligne, Faisceaux pervers, Asterisque 100, Volume 1, 1982
- D. Abramovich, A. Polishchuk, Sheaves of t-structures and valuative criteria for stable complexes, J. reine angew. Math. 590 (2006), 89–130
- A. L. Gorodentsev, S. A. Kuleshov, A. N. Rudakov, t-stabilities and t-structures on triangulated categories, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 4, 117–150
- A. Polishchuk, Constant families of t-structures on derived categories of coherent sheaves, Moscow Math. J. 7 (2007), 109–134
- John Collins, Alexander Polishchuk, Gluing stability conditions, arxiv/0902.0323
For stable (∞,1)-categories
For reflective factorization systems and normal torsion theories in stable -categories
- Cassidy and Hébert and Kelly, “Reflective subcategories, localizations, and factorization systems”. J. Austral. Math Soc. (Series A) 38 (1985), 287–329 (pdf)