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\begin{document}

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\section*{t-structure}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_stable_categories}{In stable $\infty$-categories}\dotfill \pageref*{in_stable_categories} \linebreak
\noindent\hyperlink{towers}{Towers}\dotfill \pageref*{towers} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{application_to_spectral_sequence}{Application to spectral sequence}\dotfill \pageref*{application_to_spectral_sequence} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{tStruc}\hypertarget{tStruc}{}
Let $C$ be a [[triangulated category]]. A \emph{t-structure} on $C$ is a [[pair]] $\mathfrak{t}=(C_{\ge 0}, C_{\le 0})$ of [[strictly full subcategories]]

\begin{displaymath}
C_{\geq 0}, C_{\leq 0} \hookrightarrow C
\end{displaymath}
such that

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


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


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



\end{enumerate}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Given a t-structure, its \emph{heart} is the intersection

\begin{displaymath}
C_{\geq 0} \cap C_{\leq 0}
  \hookrightarrow
  C
  \,.
\end{displaymath}
\end{defn}
\hypertarget{in_stable_categories}{}\subsubsection*{{In stable $\infty$-categories}}\label{in_stable_categories}

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

\begin{itemize}%
\item 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 $C$;


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


\item A bireflective factorization system $(E,M)$ on a stable $\infty$-category $C$ is called \emph{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 \hyperlink{CHK}{CHK}; see also \hyperlink{FL0}{FL0, \S{}1.1}) is \emph{exact}, meaning that the square in

\begin{displaymath}
\begin{array}{cccccc} 
0 &\to& S X &\to& X\\
&& \downarrow&&\downarrow\\
&& 0 &\to& R X\\
&& && \downarrow\\
&& && 0
\end{array}
\end{displaymath}
is a fiber sequence for any object $X$; see \hyperlink{FL0}{FL0, Def 3.5 and Prop. 3.10} for equivalent conditions for normality.



\end{itemize}
\textbf{Remark.} \hyperlink{CHK}{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 \hyperlink{FL0}{FL0, Thm 3.11}.

\textbf{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:

\begin{itemize}%
\item 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 $(E,M)$.
\item On the other side, given a $t$-structure on $C$ we set\begin{displaymath}
E(t)=\{f\in C^{\Delta[1]} \mid \tau_{\lt 0}(f) \;\text{ is an equivalence}\};
\end{displaymath}
\begin{displaymath}
M(t)=\{f\in C^{\Delta[1]} \mid \tau_{\geq0}(f) \;\text{ is an equivalence}\}.
\end{displaymath}


\end{itemize}
\emph{Proof.} This is \hyperlink{FL0}{FL0, Theorem 3.13}

\textbf{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 \emph{families} of $t$-structures $\{\mathbb{F}_i\}_{i\in I}$; more precisely, we are led to study \emph{$\mathbb{Z}$-equivariant} [[k-ary factorization system|multiple factorization systems]] $J\to TS( C )$.

\textbf{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:

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

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

\emph{Proof.} This is \hyperlink{FL1}{FL1, Theorem 2.16}

This results allows us to recognize \emph{$t$-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: \emph{Harder-Narashiman filtrations} and \emph{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

\begin{itemize}%
\item In the \emph{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.
\item In the \emph{orbit case} we consider an \emph{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.

\end{itemize}
\hypertarget{towers}{}\subsubsection*{{Towers}}\label{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 \emph{tower} associated to a tuple $\mathfrak{t}_{1,\dots, n}$:

(\ldots{})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{prop}
\label{}\hypertarget{}{}
The heart of a stable $(\infty,1)$-category is an [[abelian category]].

\end{prop}
(\hyperlink{BBD82}{BBD 82}, [[Higher Algebra|Higher Algebra, remark 1.2.1.12]], \hyperlink{FL0}{FL0, Ex. 4.1} and \hyperlink{FL1}{FL1, \S{}3.1})

\hypertarget{application_to_spectral_sequence}{}\subsubsection*{{Application to spectral sequence}}\label{application_to_spectral_sequence}

If a 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).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Bridgeland stability]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

For triangulated categories

\begin{itemize}%
\item S. I. Gelfand, [[Yuri Manin]], \emph{Methods of homological algebra}, Nauka 1988, Springer 1998, 2003


\item [[Donu Arapura]], \emph{Triangulated categories and $t$-structures} (\href{http://www.math.purdue.edu/~dvb/preprints/perv2.pdf}{pdf})


\item [[Alexander Beilinson]], [[Joseph Bernstein]], [[Pierre Deligne]], \emph{Faisceaux pervers}, Asterisque \textbf{100}, Volume 1, 1982



\end{itemize}
\begin{itemize}%
\item D. Abramovich, A. Polishchuk, \emph{Sheaves of t-structures and valuative criteria for stable complexes}, J. reine angew. Math. \textbf{590} (2006), 89--130
\item A. L. Gorodentsev, S. A. Kuleshov, A. N. Rudakov, \emph{t-stabilities and t-structures on triangulated categories}, Izv. Ross. Akad. Nauk Ser. Mat. \textbf{68} (2004), no. 4, 117--150
\item A. Polishchuk, \emph{Constant families of t-structures on derived categories of coherent sheaves}, Moscow Math. J. \textbf{7} (2007), 109--134
\item John Collins, [[Alexander Polishchuk]], \emph{Gluing stability conditions}, \href{http://arxiv.org/abs/0902.0323}{arxiv/0902.0323}

\end{itemize}
For [[stable (∞,1)-categories]]

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Algebra]]}

\end{itemize}
For reflective factorization systems and normal torsion theories in stable $\infty$-categories

\begin{itemize}%
\item Cassidy and H\'e{}bert and [[Max Kelly|Kelly]], ``Reflective subcategories, localizations, and factorization systems''. \emph{J. Austral. Math Soc. (Series A)} 38 (1985), 287--329 (\href{http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ1_38_03%2FS1446788700023624a.pdf&code=5796045be8904c5183c2e95bce65491e}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Jiri Rosicky]], [[Walter Tholen]], \emph{Factorization, Fibration and Torsion}, Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 295-314 (\href{http://arxiv.org/abs/0801.0063}{arXiv:0801.0063}, \href{http://www.emis.de/journals/JHRS/volumes/2007/n2a14/}{publisher})

\end{itemize}
\begin{itemize}%
\item [[Domenico Fiorenza]] and [[Fosco Loregian]], ``$t$-structures are normal torsion theories'' (\href{http://arxiv.org/abs/1408.7003}{arxiv}).


\item [[Domenico Fiorenza]] and [[Fosco Loregian]], ``Hearts and Postnikov towers in stable $\infty$-categories'' (in preparation).



\end{itemize}
[[!redirects t-structures]]

[[!redirects t-structure on a stable (∞,1)-category]] [[!redirects t-structures on a stable (∞,1)-category]]

[[!redirects t-structure on a stable (infinity,1)-category]] [[!redirects t-structures on a stable (infinity,1)-category]]

[[!redirects heart]] [[!redirects hearts]]

[[!redirects heart of a t-structure]] [[!redirects hearts of t-structures]]

[[!redirects heart of a stable (∞,1)-category]] [[!redirects hearts of stable (∞,1)-categories]]

[[!redirects heart of a stable (infinity,1)-category]] [[!redirects hearts of stable (infinity,1)-categories]]



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