nLab t-structure

Definition

Definition

Let $C$ be a triangulated category. A t-structure on $C$ is a pair $\mathfrak{t}=(C_{\ge 0}, C_{\le 0})$ of strictly full subcategories

$$C_{\geq 0}, C_{\leq 0} \hookrightarrow C$$

such that

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

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

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

Definition

Given a t-structure, its heart is the intersection

$$C_{\geq 0} \cap C_{\leq 0} \hookrightarrow C \,.$$

In stable $\infty$-categories

In the infinity-categorical setting $t$-structures arise as torsion/torsionfree classes associated to suitable factorization systems on a stable infinity-category $C$.

• In a stable setting, the subcategories are closed under de/suspension simply because they are co/reflective and reflective and these operations are co/limits. Co/reflective subcategories of $C$ arise from co/reflective factorization systems on $C$;

• A bireflective factorization system on a $\infty$-category $C$ consists of a factorization system $\mathbb{F}=(E,M)$ where both classes satisfy the two-out-of-three property.

• A bireflective factorization system $(E,M)$ on a stable $\infty$-category $C$ is called normal if the diagram $S x\to x\to R x$ obtained from the reflection $R\colon C\to M/0$ and the coreflection $S\colon C\to *\!/E$ (where the category $M/\!* =\{A\mid (0\to A)\in M\}$ is obtained as $\Psi(E,M)$ under the adjunction $\Phi\dashv \Psi$ described at reflective factorization system and in CHK; see also FL0, §1.1) is exact, meaning that the square in

  $$\begin{array}{cccccc} 0 &\to& S X &\to& X\\ && \downarrow&&\downarrow\\ && 0 &\to& R X\\ && && \downarrow\\ && && 0 \end{array}$$

  is a fiber sequence for any object $X$; see FL0, Def 3.5 and Prop. 3.10 for equivalent conditions for normality.

Remark. CHK established a hierarchy between the three notions of simple, semi-exact and normal factorization system: in the setting of stable $\infty$-category the three notions turn out to be equivalent: see FL0, Thm 3.11.

Theorem. There is a bijective correspondence between the class $TS( C )$ of $t$-structures and the class of normal torsion theories on a stable $\infty$-category $C$, induced by the following correspondence:

• On the one side, given a normal, bireflective factorization system $(E,M)$ on $C$ we define the two classes $(C_{\ge0}(\mathbb{F}), C_{\lt 0}(\mathbb{F}))$ of a $t$-structure $\mathfrak{t}(\mathbb{F})$ to be the torsion and torsionfree classes $(*\!/E, M/\!*)$ associated to the factorization $(E,M)$.
• On the other side, given a $t$-structure on $C$ we set
  $$E(t)=\{f\in C^{\Delta[1]} \mid \tau_{\lt 0}(f) \;\text{ is an equivalence}\};$$
  $$M(t)=\{f\in C^{\Delta[1]} \mid \tau_{\geq0}(f) \;\text{ is an equivalence}\}.$$

Proof. This is FL0, Theorem 3.13

Theorem. There is a natural monotone action of the group $\mathbb{Z}$ of integers on the class $TS( C )$ (now confused with the class $FS_\nu( C )$ of normal torsion theories on $C$) given by the suspension functor: $\mathbb{F}=(E,M)$ goes to $\mathbb{F}[1] = (E[1], M[1])$.

This correspondence leads to study families of $t$-structures $\{\mathbb{F}_i\}_{i\in I}$; more precisely, we are led to study $\mathbb{Z}$-equivariant multiple factorization systems $J\to TS( C )$.

Theorem. Let $\mathfrak{t} \in TS(C)$ and $\mathbb{F}=(E,M)$ correspond each other under the above bijection; then the following conditions are equivalent:

1. $\mathfrak{t}[1]=\mathfrak{t}$, i.e. $C_{\geq 1}= C_{\geq 0}$;
2. $C_{\geq 0}=*\!/E$ is a stable $\infty$-category;
3. the class $E$ is closed under pullback.

In each of these cases, we say that $\mathfrak{t}$ or $(E,M)$ is stable.

Proof. This is FL1, Theorem 2.16

This results allows us to recognize $t$-structures with stable classes precisely as those which are fixed in the natural $\mathbb{Z}$-action on $TS( C )$.

Two “extremal” choices of $\mathbb{Z}$-chains of $t$-structures draw a connection between two apparently separated constructions in the theory of derived categories: Harder-Narashiman filtrations and semiorthogonal decompositions on triangulated categories: we adopt the shorthand $\mathfrak{t}_{1,\dots, n}$ to denote the tuple $\mathfrak{t}_1\preceq \mathfrak{t}_2\preceq\cdots\preceq \mathfrak{t}_n$, each of the $\mathfrak{t}_i$ being a $t$-structure $((C_i)_{\ge 0}, (C_i)_{\lt 0})$ on $C$, and we denote similarly $\mathfrak{t}_\omega$. Then

• In the stable case the tuple $t_{1,\dots, n}$ is endowed with a (monotone) $\mathbb{Z}$-action, and the map $\{0\lt 1\cdots\lt n\}\to TS( C )$ is equivariant with respect to this action; the absence of nontrivial $\mathbb{Z}$-actions on $\{0\lt 1\cdots\lt n\}$ forces each $t_i$ to be stable.
• In the orbit case we consider an infinite family $t_\omega$ of $t$-structures on $C$, obtained as the orbit of a fixed $(E_0, M_0)\in TS( C )$ with respect to the natural $\mathbb{Z}$-action.

Towers

The HN-filtration induced by a $t$-structure and the factorization induced by a semiorthogonal decomposition on $C$ both are the byproduct of the tower associated to a tuple $\mathfrak{t}_{1,\dots, n}$:

(…)

Properties

General

Proposition

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

(BBD 82, Higher Algebra, remark 1.2.1.12, FL0, Ex. 4.1 and FL1, §3.1)

Application to spectral sequence

If 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).

Related concepts

• Bridgeland stability

References

For triangulated categories

• S. I. Gelfand, Yuri Manin, Methods of homological algebra, Nauka 1988, Springer 1998, 2003

• Donu Arapura, Triangulated categories and $t$-structures (pdf)

• Alexander Beilinson, Joseph Bernstein, Pierre Deligne, Faisceaux pervers, Asterisque 100, Volume 1, 1982

• 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

For stable (∞,1)-categories

• Jacob Lurie, Higher Algebra

For reflective factorization systems and normal torsion theories in stable $\infty$-categories

• Cassidy and Hébert and Kelly, “Reflective subcategories, localizations, and factorization systems”. J. Austral. Math Soc. (Series A) 38 (1985), 287–329 (pdf)

• Jiri Rosicky, Walter Tholen, Factorization, Fibration and Torsion, Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 295-314 (arXiv:0801.0063, publisher)

• Domenico Fiorenza and Fosco Loregian, “$t$-structures are normal torsion theories” (arxiv).

• Domenico Fiorenza and Fosco Loregian, “Hearts and Postnikov towers in stable $\infty$-categories” (in preparation).