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\begin{document}

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\section*{recollement}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection}

[[!include modalities - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{recollement} situation is a [[diagram]] of six [[additive functors]]

\begin{displaymath}
\mathcal{A}' 
    \stackrel{\overset{i^*}{\longleftarrow}}{\stackrel{\overset{i_*}{\longrightarrow}}{\underset{i^!}{\longleftarrow}}}
  \mathcal{A}\stackrel{\overset{j_!}{\longleftarrow}}{\stackrel{\overset{j^*}{\longrightarrow}}{\underset{j_*}{\longleftarrow}}}\mathcal{A}''
\end{displaymath}
among three [[abelian category|abelian]] or [[triangulated categories]] satisfying a strong list of [[exact functor|exactness]] and [[adjoint functor|adjointness]] axioms.

The paradigmatic situation is about the categories of [[abelian sheaves]] $\mathcal{A}' = Sh(C)$, $\mathcal{A} = Sh(X)$, $\mathcal{A}'' = Sh(U)$, where $U\subset X$ is an [[open subset]] of a [[topological space]], $C = X\backslash U$ the [[complement]], and the functors among the [[category of sheaves|sheaf categories]] are induced by the [[open embedding]] $j \colon U\hookrightarrow X$ and [[closed embedding]] $i \colon C\hookrightarrow X$. As suggested by this example, recollement may in fact be regarded as the additive or triangulated version of [[Artin gluing]].

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

A modern treatment for the recollement of abelian categories is in (\hyperlink{FranjouPirashvili04}{Franjou-Pirashvili 04}), where the following axioms are listed:

(i) $j_!\dashv j^*\dashv j_*$

(ii) the [[unit of an adjunction|unit]] $Id_{\mathcal{A}''}\to j^* j_!$ and the [[counit of an adjunction|counit]] $j^* j_*\to Id_{\mathcal{A}''}$ are [[isomorphism|iso]] (\href{adjoint%20functor#FullyFaithfulAndInvertibleAdjoints}{hence} $j_\ast$ and $j_!$ are [[fully faithful functor|fully faithful]])

(iii) $i^*\dashv i_*\dashv i^!$

(iv) the unit $Id_{\mathcal{A}'}\to i^! i_*$ and the counit $i^* i_*\to Id_{\mathcal{A}'}$ are iso

(v) the functor $i_*:\mathcal{A}'\to Ker(j^*)$ is an equivalence of categories.

In fact (i) and (ii) for $j^*:\mathcal{A}\to\mathcal{A}''$ enable one to define $\mathcal{A}'$ as the full subcategory of $\mathcal{A}$ whose objects $a$ satisfy $j^* a = 0$ such that one satisfies the recollement situation.

A standard treatment for the sequence of triangulated functors

\begin{displaymath}
\mathcal{D}' 
   \overset{i_*}{\to}
  \mathcal{D}\overset{j^*}{\to}\mathcal{D}''
\end{displaymath}
is in (\hyperlink{BeilinsonBernsteinDeligne82}{Beilinson-Bernstein-Deligne 82}) where in 1.4.3 the following axioms are listed

(a) $i_* = i_!$ admits a triangulated left adjoint $i^*$ and triangulated right adjoint $i^!$

(b) $j^* = j^!$ admits a triangulated left adjoint $j_*$ and triangulated right adjoint $j_!$

(c) $j^* i_* = 0$ (hence by adjointness, also $i^*j_! = 0$ and $i^! j_*=0$)

(d) given $d\in Ob{\mathcal{D}}$, there exist (necessarily unique) distinguished triangles

\begin{displaymath}
i_! i^! d \to d\to j_* j^* d\to (i_! i^! d) [1]
\end{displaymath}
\begin{displaymath}
j_! j^! d \to d\to i_* i^* d\to (j_! j^! d) [1]
\end{displaymath}
(e) $i_*, j_*, j_!$ are full embeddings.

Again in good situations, less data is needed to provide the recollement.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The [[forgetful functor]] from [[global equivariant stable homotopy theory]] to plain [[stable homotopy theory]] exhibits a recollement,

\end{itemize}
See at \emph{\href{global+equivariant+stable+homotopy+theory#RelationToPlainStableHomotopyTheory}{global equivariant stable homotopy theory -- Relation to plain stable homotopy theory}}.

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

\begin{itemize}%
\item [[adjoint modality]]

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

\begin{itemize}%
\item V. Franjou, [[T. Pirashvili]], \emph{Comparison of abelian categories recollements}, Doc. Math. 9 (2004), 41--56, \href{http://www.ams.org/mathscinet-getitem?mr=2054979}{MR2005c:18008}, \href{http://www.math.uni-bielefeld.de/documenta/vol-09/03.pdf}{pdf}


\item A.A. Beilinson, J. Bernstein, [[Pierre Deligne]], \emph{Faisceaux pervers. Analysis and topology on singular spaces}, I (Luminy, 1981), 5--171, Aste'risque 100, Soc. Math. France, Paris 1982.



\end{itemize}
In references

\begin{itemize}%
\item E. Cline, B. Parshall, L. Scott, \emph{Finite dimensional algebras and highest weight categories}, J. Reine Angew. Math, 1988, 391: 85---99, \href{http://www.ams.org/mathscinet-getitem?mr=961165}{MR90d:18005}, \href{http://resolver.sub.uni-goettingen.de/purl?GDZPPN002205971}{goettingen}
\item E. Cline, B. Parshall, L. Scott, \emph{Algebraic stratification in representative categories}, J. of Algebra 117, 1988, 504---521.

\end{itemize}
one studies the following kind of sources of recollement situations for triangulated categories: $k$ is a commutative [[field]], $A$ a finite dimensional unital associative $k$-algebra, $e$ an idempotent, and $D^b(A)$ the bounded derived category of right $A$-modules. Suppose $eA(1-e) = 0$ and the global dimension of $A$ is finite. Then there is a recollement of triangulated categories involving $D^b(eAe)$, $D^b(A)$ and $D^b((1-e)A(1-e))$.

\begin{itemize}%
\item S. Koenig, \emph{Tilting complexes, perpendicular categories and recollements of derived module categories of rings.}, \href{http://www.ams.org/mathscinet-getitem?mr=1124785}{MR92k:18009}, 
\item Qinghua Chen,Yanan Lin, \emph{Recollements of extension algebras}, Science in China 46, 4, 2003 \href{http://www.springerlink.com/content/h185372128726855/fulltext.pdf}{pdf}

\end{itemize}
Another source of examples is due MacPherson and Vilonen

\begin{itemize}%
\item Kari Vilonen, \emph{Perverse sheaves and finite dimensional algebras}, Trans. A.M.S. 341 (1994), 665--676, \href{http://www.ams.org/mathscinet-getitem?mr=1135104}{MR94d:16012}, \href{http://dx.doi.org/10.2307/2154577}{doi}


\item Michael Artin, Grothendieck Topologies. Harvard University, 1962.


\item Yuri Berest, Oleg Chalykh, Farkhod Eshmatov, \emph{Recollement of deformed preprojective algebras and the Calogero-Moser correspondence}, Mosc. Math. J. 8 (2008), no. 1, 21--37, 183, \href{http://arxiv.org/abs/0706.3006}{arxiv/0706.3006}, \href{http://www.ams.org/mathscinet-getitem?mr=2422265}{MR2009h:16030}


\item Roy Joshua, \href{http://www.math.osu.edu/~joshua.1/GV.pdf}{pdf}


\item Yang Han, \emph{Recollements and Hochschild theory}, \href{http://arxiv.org/abs/1101.5697}{arxiv/1101.5697}



\end{itemize}
For treatment in the setting of $\infty$-categories:

\begin{itemize}%
\item [[Jacob Lurie]], section A.8 of \emph{[[Higher Algebra]]},


\item [[Clark Barwick]], [[Saul Glasman]], \emph{A note on stable recollements} (\href{https://arxiv.org/abs/1607.02064}{arXiv:1607.02064})



\end{itemize}


\end{document}