Let be a triangulated category. A t-structure on is a pair of strictly full subcategories
such that
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 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:
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:
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
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 1.2.1.12, FL0, Ex. 4.1 and FL1, §3.1)
If 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).
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
For reflective factorization systems and normal torsion theories in stable -categories
Domenico Fiorenza and Fosco Loregian, “-structures are normal torsion theories” (arxiv).
Domenico Fiorenza and Fosco Loregian, “Hearts and Postnikov towers in stable -categories” (in preparation).
Last revised on March 25, 2021 at 08:30:04. See the history of this page for a list of all contributions to it.