Triangulated Category - T-structures

T-structures

Verdier introduced triangulated categories in order to place derived categories in a category-theoretic context: for every abelian category A there exists a triangulated category D(A), containing A as a full subcategory (the "0-complexes" concentrated in cohomological degree 0), and in which we can construct derived functors. Unfortunately, different abelian categories can give rise to equivalent derived categories, so that it is impossible to reconstruct A from the triangulated category D(A).

A partial solution to this problem, is to impose a t-structure on the triangulated category D. Different t-structures on D will give rise to different abelian categories inside it. This notion was presented in (Beilinson, Bernstein & Deligne 1982).

The prototype is the t-structure on the derived category D of an abelian category A. For each n there are natural full subcategories and consisting of complexes whose cohomology is "bounded below" or "bounded above" n, respectively. Since for any complex X, we have, these are related to each other:

These subcategories also have the following properties:

• ,
• Every object Y can be embedded in a distinguished triangle with,

A t-structure on a triangulated category consists of full subcategories and satisfying the conditions above. In Faisceaux pervers a triangulated category equipped with a t-structure is called a t-category.

The core or heart (the original French word is "coeur") of a t-structure is the category . It is an abelian category, whereas a triangulated category is additive but almost never abelian. The core of a t-structure on the derived category of A can be thought of as a sort of twisted version of A, which sometimes has better properties. For example, the category of perverse sheaves is the core of a certain (quite complicated) t-structure on the derived category of the category of sheaves. Over a space with singularities, the category of perverse sheaves is similar to the category of sheaves but behaves better.

A basic example of a t-structure is the "natural" one on the derived category D of some abelian category, where are the full subcategories of complexes whose cohomologies vanish in degrees less than or greater than 0. This t-structure has the following features:

• The truncation functors, or in fact for any n, which are obtained by translating the argument of the original two functors. Abstractly, these are the left adjoint and right adjoint, respectively, to the inclusion functors of in D. In addition, the truncation functors fit into a triangle, and this is in fact the unique triangle satisfying the third axiom above:

• The cohomology functor, or in fact, which is obtained by translating its argument: . Its relationship to the truncation functors is that they are defined so that for any complex A, for and is unchanged for ; likewise for ; in particular, is not independent of them, but in fact . Furthermore, the cohomology is a cohomological functor: for any triangle we obtain a long exact sequence

These properties carry over without change to any t-structure, in that if D is a t-category, then there exist truncation functors into its core, from which we obtain a cohomology functor taking values in the core, and the above properties are satisfied for both.