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
such that
for all $X \in C_{\geq 0}$ and $Y \in C_{\leq 0}$ the hom object is the zero object: $Hom_{C}(X, Y[-1]) = 0$;
the subcategories are closed under suspension/desuspension: $C_{\geq 0}[1] \subset C_{\geq 0}$ and $C_{\leq 0}[-1] \subset C_{\leq 0}$.
For all objects $X \in C$ there is a fiber sequence $Y \to X \to Z$ with $Y \in C_{\geq 0}$ and $Z \in C_{\leq 0}[-1]$.
Given a t-structure, its heart is the intersection
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
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:
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:
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
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 $\mathfrak{t}_{1,\dots, n}$:
(…)
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)
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).
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
For reflective factorization systems and normal torsion theories in stable $\infty$-categories
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).
Last revised on March 25, 2021 at 08:30:04. See the history of this page for a list of all contributions to it.