nLab t-structure

Definition

Originally, t-structures were defined

• on triangulated categories

These typically arise as homotopy categories of t-structures

• on stable $\infty$-categories.

On triangulated categories

Definition

(t-structure on a triangulated category)
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 (i.e. an exact triangle) $Y \to X \to Z$ with $Y \in C_{\geq 0}$ and $Z \in C_{\leq 0}[-1]$.

Definition

Given a t-structure (Def. ), its heart is the intersection

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

On stable $\infty$-categories

Definition

(t-structure in a stable $\infty$-category)
A t-structure on a stable (∞,1)-category $\mathcal{C}$ is a t-structure in the above sense (Def. ) on its underlying homotopy category (which is triangulated, see there).

(Lurie, Higher Algebra, Def. 1.2.1.4)

Therefore, a t-structure on a stable $\infty$-category $\mathcal{C}$ is a system of full sub-(∞,1)-categories $\mathcal{C}_{\geq n}$, $\mathcal{C}_{\leq n}$, $n \in \mathbb{Z}$.

Proposition

In this situation (Def. )

1. the $\mathcal{C}_{\leq n}$ are reflective sub-(∞,1)-categories

   $$\mathcal{C}_{\leq n} \underoverset {\underset{}{\hookrightarrow}} {\overset{\tau_{\leq n}}{\longleftarrow}} {\;\; \bot \;\;} \mathcal{C}$$

2. the $\mathcal{C}_{\geq n}$ are coreflective sub-(∞,1)-categories;

   $$\mathcal{C} \underoverset {\underset{\tau_{\geq n}}{\longrightarrow}} {\hookleftarrow} {\;\; \bot \;\;} \mathcal{C}_{\leq n}$$

(Lurie, Higher Algebra, Def. 1.2.1.5, Cor. 1.2.1.6, Ntn. 1.2.1.7)

Properties

General

Proposition

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

(BBD 82, Higher Algebra, remark 1.2.1.12, FL16, Ex. 4.1 and FLM19, §3.1)

Relation to spectral sequences

If the heart (Def. ) 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).

Relation to normal torsion theories

In the (∞,1)-category theory, $t$-structures arise as torsion/torsionfree classes associated with suitable factorization systems on stable ∞-categories $C$.

• In stable ∞-category-theory, the relevant sub-(∞,1)-categories are closed under de/suspension simply because they are (co-)reflective, arising 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 CHK85; see also FL16, §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 FL16, Def 3.5 and Prop. 3.10 for equivalent conditions for normality.

Remark

CHK85 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 FL16, Thm 3.11.

Proposition

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

  $$E(t)=\{f\in C^{\Delta[1]} \mid \tau_{\lt 0}(f) \;\text{ is an equivalence}\};$$
  $$M(t)=\{f\in C^{\Delta[1]} \mid \tau_{\geq0}(f) \;\text{ is an equivalence}\}.$$

This is FL16, Theorem 3.13

Proposition

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 $J\to TS( C )$.

Proposition

Let $\mathfrak{t} \in TS(C)$ and $\mathbb{F}=(E,M)$ correspond each other under the above bijection (Prop. ); 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.

This is FLM19, Theorem 6.3

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 $\mathfrak{t}_{1,\dots, n}$:

Examples

The archetypical and historically motivating example (cf. Gelfand & Manin (1996), IV.4 §1) is the following:

Example

(canonical t-structure on the derived category of an abelian category)
For $\mathcal{A}$ an abelian category, its unbounded derived category $\mathcal{D}_\bullet(\mathcal{A})$

1. carries a t-structure (Def. ) for which $\mathcal{D}(\mathcal{A})_{\geq n}$ (rep. $\mathcal{D}(\mathcal{A})_{\leq n}$) is the full subcategory of objects presented by chain complexes $V_\bullet$ whose chain homology-groups are trivial in degrees $\lt n$ (resp. $\gt n$);

2. whose heart (Def. ) is equivalent to $\mathcal{A}$ (embedded as the chain complexes which are concentrated in degree 0).

(eg. Gelfand & Manin (1996), IV.4 §3)

Example

(canonical t-structure on spectra)
The stable (infinity,1)-category of spectra, $Spectra$, carries a canonical t-structure for which

• $Spectra_{\geq 0}$ is the sub-category of connective spectra, with $\tau_{\geq 0} \colon Spectra \to Spectra_{\geq 0}$ the connective cover-construction.

• …

(e.g. Lurie, Higher Algebr, pp. 150)

Related concepts

• Bridgeland stability

• n-connective object

• connective cover

• theorem of the heart

References

For triangulated categories:

the notion is due to

• Alexander Beilinson, Joseph Bernstein, Pierre Deligne, Faisceaux pervers, Astérisque 100 (1982) [ISBN:978-2-85629-878-7, pdf, MR86g:32015]

  (otherwise introducing perverse sheaves)

Further development:

• Sergei Gelfand, Yuri Manin, Section IV.4 of: Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [doi:10.1007/978-3-662-12492-5]

• Donu Arapura, Triangulated categories and $t$-structures [pdf]

• Dan Abramovich, Alexander Polishchuk, Sheaves of t-structures and valuative criteria for stable complexes, J. reine angew. Math. 590 (2006) 89-130 [arXiv:math/0309435, doi:10.1515/CRELLE.2006.005]

• 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

• Alexander Polishchuk, Constant families of t-structures on derived categories of coherent sheaves, Moscow Math. J. 7 (2007) 109-134 [arXiv:math/0606013]

• John Collins, Alexander Polishchuk, Gluing stability conditions [arxiv/0902.0323]

For stable (∞,1)-categories:

• Jacob Lurie, Section 1.2.1 in: Higher Algebra

On reflective factorization systems:

• C. Cassidy, M. Hébert, Max Kelly, Reflective subcategories, localizations, and factorization systems, J. Austral. Math Soc. (Series A) 38 (1985) 287-329 [doi:10.1017/S1446788700023624]

• 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]

and on normal torsion theories in stable $\infty$-categories:

• Domenico Fiorenza, Fosco Loregian, $t$-Structures are normal torsion theories, Appl Categor Struct 24 (2016) 181–208 [arxiv:1408.7003, doi:10.1007/s10485-015-9393-z]

• Domenico Fiorenza, Fosco Loregian, Giovanni Marchetti, Hearts and towers in stable $\infty$-categories, J. Homotopy Relat. Struct. 14 (2019) 993–1042 [arXiv:1501.04658, doi:10.1007/s40062-019-00237-0]

• Fosco Loregian, Simone Virili Triangulated factorization systems and t-structures, Journal of Algebra 550 (2020) 219-241 [doi:10.1016/j.jalgebra.2019.12.021]