T-structure

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. A t-structure on ${\displaystyle {\mathcal {D}}}$ consists of two subcategories ${\displaystyle ({\mathcal {D}}^{\leq 0},{\mathcal {D}}^{\geq 0})}$ of a triangulated category 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.^[1]