Tstructure

Tstructure, usually written t-structure, is a fundamental concept in category theory and homological algebra. It provides a way to organize a triangulated category into interacting “nonpositive” and “nonnegative” parts, yielding a natural abelian category called the heart. A t-structure on a triangulated category D consists of two full subcategories D^{≤0} and D^{≥0} that satisfy three basic properties: Hom_D(X,Y) = 0 for all X in D^{≤0} and Y in D^{≥1}; D^{≤0} is closed under taking shifts and extensions, and D^{≥0} is closed under shifts in the opposite direction; and for every object X in D there exists a distinguished triangle A → X → B → A[1] with A in D^{≤0} and B in D^{≥1}. The intersection D^{≤0} ∩ D^{≥0} is called the heart and is an abelian category, enabling the definition of cohomology objects H^i(X).

Standard examples include the canonical t-structure on the bounded derived category D^b(A) of an abelian category A, where the heart recovers A itself; and the perverse t-structure on D^b(X) for a topological space or algebraic variety X, whose heart is the category of perverse sheaves. T-structures are compatible with functors and are central to modern homological algebra, derived geometry, and representation theory. They also underpin the construction of stability conditions in the sense of Bridgeland.

Note that Tstructure without the hyphen is uncommon in mathematical literature; when encountered, it often refers to the same concept in contexts where a hyphen is omitted or to domain-specific terminology. See also t-structure, derived category, heart of a t-structure, perverse sheaf.