nLab t-structure




Originally, t-structures were defined

These typically arise as homotopy categories of t-structures

On triangulated categories


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

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

such that

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

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

  3. For all objects XCX \in C there is a fiber sequence (i.e. an exact triangle) YXZY \to X \to Z with YC 0Y \in C_{\geq 0} and ZC 0[1]Z \in C_{\leq 0}[-1].


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

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

On stable \infty-categories


(t-structure in a stable \infty-category)
A t-structure on a stable (∞,1)-category 𝒞\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.

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


In this situation (Def. )

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

    𝒞 nτ n𝒞 \mathcal{C}_{\leq n} \underoverset {\underset{}{\hookrightarrow}} {\overset{\tau_{\leq n}}{\longleftarrow}} {\;\; \bot \;\;} \mathcal{C}
  2. the 𝒞 n\mathcal{C}_{\geq n} are coreflective sub-(∞,1)-categories;

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

(Lurie, Higher Algebra, Def., Cor., Ntn.




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

(BBD 82, Higher Algebra, remark, 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, tt-structures arise as torsion/torsionfree classes associated with suitable factorization systems on stable ∞-categories CC.

  • 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 CC.

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

  • A bireflective factorization system (E,M)(E,M) on a stable \infty-category CC is called normal if the diagram SxxRxS x\to x\to R x obtained from the reflection R:CM/0R\colon C\to M/0 and the coreflection S:C*/ES\colon C\to *\!/E (where the category M/*={A(0A)M}M/\!* =\{A\mid (0\to A)\in M\} is obtained as Ψ(E,M)\Psi(E,M) under the adjunction ΦΨ\Phi \dashv \Psi described at reflective factorization system and in CHK85; see also FL16, §1.1) is exact, meaning that the square in

    0 SX X 0 RX 0 \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 XX; see FL16, Def 3.5 and Prop. 3.10 for equivalent conditions for normality.


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.


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

  • On the one side, given a normal, bireflective factorization system (E,M)(E,M) on CC we define the two classes (C 0(𝔽),C <0(𝔽))(C_{\ge0}(\mathbb{F}), C_{\lt 0}(\mathbb{F})) of a tt-structure 𝔱(𝔽)\mathfrak{t}(\mathbb{F}) to be the torsion and torsionfree classes (*/E,M/*)(*\!/E, M/\!*) associated to the factorization (E,M)(E,M).

  • On the other side, given a tt-structure on CC we set

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

This is FL16, Theorem 3.13


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

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


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

  1. 𝔱[1]=𝔱\mathfrak{t}[1]=\mathfrak{t}, i.e. C 1=C 0C_{\geq 1}= C_{\geq 0};

  2. C 0=*/EC_{\geq 0}=*\!/E is a stable \infty-category;

  3. the class EE is closed under pullback.

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

This is FLM19, Theorem 6.3

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

Two “extremal” choices of \mathbb{Z}-chains of tt-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 𝔱 1,,n\mathfrak{t}_{1,\dots, n} to denote the tuple 𝔱 1𝔱 2𝔱 n\mathfrak{t}_1\preceq \mathfrak{t}_2\preceq\cdots\preceq \mathfrak{t}_n, each of the 𝔱 i\mathfrak{t}_i being a tt-structure ((C i) 0,(C i) <0)((C_i)_{\ge 0}, (C_i)_{\lt 0}) on CC, and we denote similarly 𝔱 ω\mathfrak{t}_\omega. Then

  • In the stable case the tuple t 1,,nt_{1,\dots, n} is endowed with a (monotone) \mathbb{Z}-action, and the map {0<1<n}TS(C)\{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<1<n}\{0\lt 1\cdots\lt n\} forces each t it_i to be stable.
  • In the orbit case we consider an infinite family t ωt_\omega of tt-structures on CC, obtained as the orbit of a fixed (E 0,M 0)TS(C)(E_0, M_0)\in TS( C ) with respect to the natural \mathbb{Z}-action.


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


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


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

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

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

(eg. Gelfand & Manin (1996), IV.4 §3)


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

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

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


For triangulated categories:

the notion is due to

Further development:

For stable (∞,1)-categories:

On reflective factorization systems:

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

Last revised on April 20, 2023 at 06:33:40. See the history of this page for a list of all contributions to it.