# nLab t-structure

### Context

#### Stable Homotopy theory

stable homotopy theory

# Contents

## Definition

###### Definition

Let $\mathcal{D}$ be a triangulated category. A t-structure on $\mathcal{D}$ is a pair of full subcategories

$\mathcal{D}_{\geq 0}, \mathcal{D}_{\leq 0} \hookrightarrow \mathcal{D}$

such that

1. for all $X \in \mathcal{D}_{\geq 0}$ and $Y \in \mathcal{D}_{\leq 0}$ the hom object is the zero object: $Hom_{\mathcal{D}}(X, Y[-1]) = 0$;

2. the subcategories are closed under suspension/desuspension: $\mathcal{D}_{\geq 0}[1] \subset \mathcal{D}_{\geq 0}$ and $\mathcal{D}_{\leq 0}[-1] \subset \mathcal{D}_{\leq 0}$.

3. For all objects $X \in \mathcal{D}$ there is a fiber sequence $Y \to X \to Z$ with $Y \in \mathcal{D}_{\geq 0}$ and $Z \in \mathcal{D}_{\leq 0}[-1]$.

###### Definition

Given a t-structure, its heart is the intersection

$\mathcal{D}_{\geq 0} \cap \mathcal{D}_{\leq 0} \hookrightarrow \mathcal{D} \,.$
###### Definition

A t-structure on a stable (∞,1)-category is a t-structure on its homotopy category of an (∞,1)-category, according to def. 1.

## Properties

### General

###### Proposition

The heart of a stable $(\infty,1)$-category is an abelian category.

### Application to spectral sequence

If a the heart of a t-structure on a stable (∞,1)-category with sequential limits is an abelian category, then the spectral sequence of a filtered stable homotopy type converges (see there).

## References

Related $n$Lab entries include Bridgeland stability?.

For triangulated categories

• D. Abramovich, A. Polishchuk, Sheaves of t-structures and valuative criteria for stable complexes, J. reine angew. Math. 590 (2006), 89–130
• A. L. Gorodentsev, S. A. Kuleshov, A. N. Rudakov, t-stabilities and t-structures on triangulated categories, Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004), no. 4, 117–150
• A. Polishchuk, Constant families of t-structures on derived categories of coherent sheaves, Moscow Math. J. 7 (2007), 109–134
• John Collins, Alexander Polishchuk, Gluing stability conditions, arxiv/0902.0323

Revised on November 19, 2013 03:38:32 by Zoran Škoda (161.53.130.104)