t-structure

t-structure (also written T-structure) is a concept in the branch of mathematics called homological algebra. It provides a way to axiomatize the properties of an abelian subcategory of a derived category. The notion was introduced by Beilinson, Bernstein, Deligne, and Gabber in their work on perverse sheaves.

Definition

Let 𝒟 be a triangulated category with translation functor [1]. A t-structure on 𝒟 is a pair (𝒟^≤0, 𝒟^≥0) of full subcategories, each stable under isomorphism, satisfying three axioms:

1. Vanishing of Homs: If X is an object of 𝒟^≤0 and Y is an object of 𝒟^≥0, then Hom_𝒟(X, Y[−1]) = 0.
2. Closure under shift: If X ∈ 𝒟^≤0, then X[1] ∈ 𝒟^≤0. If Y ∈ 𝒟^≥0, then Y[−1] ∈ 𝒟^≥0.
3. Existence of truncation: For any object A ∈ 𝒟, there exists a distinguished triangle X → A → Y → X[1] with X ∈ 𝒟^≤0 and Y ∈ 𝒟^≥1.

For any integer n, one defines 𝒟^≤n = 𝒟^≤0[−n] and 𝒟^≥n = 𝒟^≥0[−n].

Heart of a t-structure

The heart (or core) of a t-structure is the intersection:

𝒟^♡ = 𝒟^≤0 ∩ 𝒟^≥0

A key result is that the heart of a t-structure is an abelian category (whereas a triangulated category is additive but almost never abelian). The heart is stable under extensions.

Truncation Functors

Given a t-structure, there exist unique truncation functors:

• τ_≤n: 𝒟 → 𝒟^≤n (right adjoint to the inclusion)
• τ_≥n: 𝒟 → 𝒟^≥n (left adjoint to the inclusion)

These satisfy natural compatibility conditions and produce the distinguished triangle from axiom 3.

Cohomology Functors

The nth cohomology functor is defined as:

Hⁿ = τ_≤0 ∘ τ_≥0 ∘ [n]

For any distinguished triangle, these yield a long exact sequence, making Hⁿ a cohomological functor in the usual sense.

Key Examples

1. Natural t-structure on a derived category: For an abelian category 𝒜, the derived category D(𝒜) carries the natural t-structure where 𝒟^≤0 consists of complexes with vanishing cohomology in positive degrees, and 𝒟^≥0 consists of complexes with vanishing cohomology in negative degrees. The heart is equivalent to 𝒜.

2. Perverse sheaves: The category of perverse sheaves is defined as the heart of the perverse t-structure on the derived category of sheaves on a complex analytic space or algebraic variety. This t-structure was introduced by Beilinson, Bernstein, and Deligne.

3. Spectra: The category of spectra carries a t-structure where 𝒮𝓅_≥0 is the category of connective spectra (those with vanishing negative homotopy groups).

4. Motives: The conjectural motivic t-structure is closely related to standard conjectures on algebraic cycles.

t-Exact Functors

An exact functor F: 𝒟₁ → 𝒟₂ between triangulated categories with t-structures is:

• Left t-exact if F(𝒟₁^≥0) ⊆ 𝒟₂^≥0
• Right t-exact if F(𝒟₁^≤0) ⊆ 𝒟₂^≤0
• t-exact if both hold

On Stable ∞-Categories

The theory extends naturally to stable ∞-categories. A t-structure on a stable ∞-category 𝒞 is defined as a t-structure on its homotopy category h𝒞 (which is triangulated). The truncation functors become adjoint functors between ∞-categories, and the heart is an abelian ∞-category equivalent to the nerve of its homotopy category.

Related Concepts

• Co-t-structure (or weight structure): A variant where the inclusion direction in axiom 2 is reversed.
• t-localization: A bijection exists between t-structures and certain localization functors called t-localizations.