Domenico Fiorenza somethingsomething

$t$-structures are normal torsion theories

Idea

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

• A bireflective factorization system on a $\infty$-category $C$ consists of a factorization system? $(E,M)$ 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 $M/*$ 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; 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 $t(\mathbb{F})$ to be the torsion and torsionfree classes $(0/\E, \M/0)$ 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 $t \in TS(C)$ and $\mathbb{F}=(E,M)$ correspond each other under the above bijection; then the following conditions are equivalent:

1. $t[1]=t$, 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 $t$ 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 $t_{1,\dots, n}$ to denote the tuple $t_1\preceq t_2\preceq\cdots\preceq t_n$, each of the $t_i$ being a $t$-structure $((C_i)_{\ge 0}, (C_i)_{\lt 0})$ on $C$, and we denote similarly $t_\omega$. 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 stem from the same construction:

(…)

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References

• 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)

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

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