In the branch of
mathematics called
homological algebra, a ''t''-structure is a way to axiomatize the properties of an
abelian subcategory of a
derived category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ...
. A ''t''-structure on
consists of two subcategories
of a
triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy cat ...
or stable
infinity category which abstract the idea of complexes whose cohomology vanishes in positive, respectively negative, degrees. There can be many distinct ''t''-structures on the same category, and the interplay between these structures has implications for algebra and geometry. The notion of a ''t''-structure arose in the work of Beilinson, Bernstein, Deligne, and Gabber on
perverse sheaves The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was intro ...
.
Definition
Fix a triangulated category
with translation functor