A t-structure (pronounced "tee-structure") is a fundamental concept in [[higher category theory]], particularly in the study of [[triangulated categories]] and [[derived categories]]. It provides a way to "decompose" a triangulated category into a "heart" which is an [[abelian category]], and two "tails" representing objects concentrated in "negative" and "positive" degrees relative to the t-structure. This allows for the application of homological algebra techniques to objects in triangulated categories, which are often non-abelian.
Formal Definition
A t-structure on a [[triangulated category]] $\mathcal{D}$ is a pair of full subcategories $(\mathcal{D}^{\le 0}, \mathcal{D}^{\ge 0})$ satisfying the following axioms:-
Shift Invariance:
- The subcategory $\mathcal{D}^{\le 0}$ is closed under shifts to the left: if $X \in \mathcal{D}^{\le 0}$, then $X[-1] \in \mathcal{D}^{\le 0}$.
- The subcategory $\mathcal{D}^{\ge 0}$ is closed under shifts to the right: if $X \in \mathcal{D}^{\ge 0}$, then $X[1] \in \mathcal{D}^{\ge 0}$.
- Both $\mathcal{D}^{\le 0}$ and $\mathcal{D}^{\ge 0}$ are closed under direct summands.
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Orthogonality: For any object $X \in \mathcal{D}^{\le 0}$ and any object $Y \in \mathcal{D}^{\ge 1}$ (where $\mathcal{D}^{\ge 1} = \mathcal{D}^{\ge 0}[1]$), the set of morphisms between them is trivial: $\text{Hom}_{\mathcal{D}}(X, Y) = 0$.
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Existence of Truncation Triangles: For every object $X \in \mathcal{D}$, there exists a [[distinguished triangle]] $A \to X \to B \to A[1]$ such that $A \in \mathcal{D}^{\le 0}$ and $B \in \mathcal{D}^{\ge 1}$. This triangle is unique up to isomorphism and is called the truncation triangle for $X$. The objects $A$ and $B$ are often denoted $\tau_{\le 0}X$ and $\tau_{\ge 1}X$ respectively, representing the "left truncation" and "right truncation" of $X$.
From these axioms, one can define sequences of shifted subcategories $\mathcal{D}^{\le n} = \mathcal{D}^{\le 0}[-n]$ and $\mathcal{D}^{\ge n} = \mathcal{D}^{\ge 0}[-n]$ for any integer $n$. The shift invariance then implies that $\mathcal{D}^{\le n} \subseteq \mathcal{D}^{\le n+1}$ and $\mathcal{D}^{\ge n+1} \subseteq \mathcal{D}^{\ge n}$.
The Heart of a t-structure
The heart of a t-structure $(\mathcal{D}^{\le 0}, \mathcal{D}^{\ge 0})$ is defined as the intersection of the two subcategories: $\mathcal{H} = \mathcal{D}^{\le 0} \cap \mathcal{D}^{\ge 0}$.A fundamental result states that the heart $\mathcal{H}$ of any t-structure is always an [[abelian category]]. The objects in the heart can be thought of as objects that are "concentrated in degree 0" relative to the given t-structure. For an object $X \in \mathcal{H}$, its truncation triangle is $X \to X \to 0 \to X[1]$, meaning $\tau_{\le 0}X = X$ and $\tau_{\ge 1}X = 0$. This implies that objects in the heart are "pure" in a sense, having no non-zero components in $\mathcal{D}^{\le -1}$ or $\mathcal{D}^{\ge 1}$.
Examples
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Standard t-structure on Derived Categories: Let $\mathcal{A}$ be an abelian category (e.g., the category of abelian groups, modules over a ring, or sheaves on a topological space). The [[derived category]] $D(\mathcal{A})$ has a canonical t-structure, often called the standard t-structure.
- $\mathcal{D}^{\le 0}$ consists of complexes whose [[cohomology]] is concentrated in non-positive degrees (i.e., $H^n(X) = 0$ for $n > 0$).
- $\mathcal{D}^{\ge 0}$ consists of complexes whose cohomology is concentrated in non-negative degrees (i.e., $H^n(X) = 0$ for $n < 0$). The heart of this t-structure is precisely the original abelian category $\mathcal{A}$.
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Perverse Sheaves: On a stratified topological space or [[algebraic variety]], the category of [[perverse sheaves]] is defined as the heart of a specific t-structure on the derived category of constructible sheaves. This t-structure is constructed using the geometry of the stratification and is crucial in areas like the geometric Langlands program and the theory of D-modules.
Significance and Applications
T-structures are essential tools in modern homological algebra and its applications, particularly in [[algebraic geometry]], [[representation theory]], and [[mathematical physics]]. They allow mathematicians to:- Connect Triangulated and Abelian Categories: Bridge the gap between the often complex, non-abelian structure of triangulated categories and the more familiar and tractable world of abelian categories.
- Truncation and Cohomology: Provide a systematic way to "truncate" objects in a triangulated category, extracting components concentrated in certain degrees. The standard cohomology functors for complexes can be seen as applications of the truncation functors associated with the standard t-structure.
- Categorify Invariants: In some contexts, t-structures provide the categorical framework for defining and studying important invariants, such as those arising in the theory of mixed Hodge modules or representations of quivers.
- Study Singularities: T-structures are used to define the category of singularities of a scheme, providing a homological perspective on singular geometry.
The concept was introduced by Alexander Beilinson, Joseph Bernstein, and Pierre Deligne in 1982 in their seminal work on perverse sheaves.