T-structure

In mathematics, more specifically in homological algebra, a t-structure is an additional piece of structure that can be put on a triangulated category or a stable infinity category that axiomatizes the properties of complexes whose positive or negative cohomology vanishes. The notion was introduced by Beilinson, Bernstein and Deligne. It allows one to construct an abelian category, namely the heart of the t-structure, from a triangulated category.

The derived category D of an abelian category A contains, for each n, the full subcategories ${\displaystyle D^{\geq n}}$ and ${\displaystyle D^{\leq n}}$ consisting of complexes whose cohomology is "bounded below" or "bounded above" n, respectively, i.e., ${\displaystyle H^{m}(X)=0}$ for ${\displaystyle m<n}$ and ${\displaystyle m>n}$, respectively. The subcategories have the following properties:

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