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\section*{thick subcategory}

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

\hypertarget{notions_of_subcategory}{}\paragraph*{{Notions of subcategory}}\label{notions_of_subcategory}

[[!include notions of subcategory]]

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

[[!include stable homotopy theory - contents]]

\hypertarget{thick_subcategories_and_serre_quotient_categories}{}\section*{{Thick subcategories and Serre quotient categories}}\label{thick_subcategories_and_serre_quotient_categories}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{serre_quotient_category}{Serre quotient category}\dotfill \pageref*{serre_quotient_category} \linebreak
\noindent\hyperlink{localizing_subcategories}{Localizing subcategories}\dotfill \pageref*{localizing_subcategories} \linebreak
\noindent\hyperlink{thick_subcategories_and_saturation}{Thick subcategories and saturation}\dotfill \pageref*{thick_subcategories_and_saturation} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{link}{Link}\dotfill \pageref*{link} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A [[full subcategory|full]] [[triangulated subcategory]] is \textbf{thick} (or \textbf{\'e{}paisse}) if it is closed under [[extension]]s.

Sometimes the same definition is used in [[abelian categories]] as well. However, for many authors, including [[Pierre Gabriel]], in abelian categories, this term denotes the stronger notion of a [[topologizing subcategory]] closed under extensions; in other words, a nonempty full subcategory $T$ of an abelian category $A$ is \textbf{thick} (in the strong sense) iff for every exact sequence

\begin{displaymath}
0 \longrightarrow M\longrightarrow M''\longrightarrow M'\longrightarrow 0
\end{displaymath}
in $A$, the object $M''$ is in $T$ iff $M$ and $M'$ are in $T$.

For some authors the thick subcategory (strong version) is called a \emph{[[Serre subcategory]]} (in a weak sense), the term which we reserve for (generally) a stronger notion.

For any subcategory of an [[abelian category]] $A$ one denotes by $\bar{T}$ the full subcategory of $A$ generated by all objects $N$ for which any (nonzero) [[subquotient]] of $N$ in $T$ has a (nonzero) [[subobject]] from $T$. This becomes an idempotent operation on the class of subcategories of $A$ where $T\subset \bar{T}$ iff $T$ is [[topologizing subcategory|topologizing]]. Moreover $\bar{T}$ is always thick in the stronger sense. [[Serre subcategory|Serre subcategories]] in the strong sense are those (nonempty) subcategories which are stable under the operation $T\mapsto\bar{T}$.

\hypertarget{serre_quotient_category}{}\subsection*{{Serre quotient category}}\label{serre_quotient_category}

Following the extensions of an early work of Serre by Grothendieck and Gabriel, for a thick subcategory $T$ in an [[abelian category]] $A$, one defines the (Serre) quotient category $A/T$ as the one having the same [[objects]] as $A$ and [[hom-sets]] given by

\begin{displaymath}
(A/T)(X,Y) := colim A(X',Y/Y')
\end{displaymath}
where the [[colimit]] runs through all [[subobjects]] $X'\subset X$, $Y'\subset Y$ such that $X/X' \in Ob T$, $Y'\in Ob T$. The quotient functor $Q \colon A\to A/T$ is obvious.

Notice that the set of morphisms is indeed [[small set|small]], so that the Serre quotient category exists as a [[locally small category]]. On the other hand, one can construct an equivalent [[localization]] by the Gabriel-Zisman localizing at the class $\Sigma$ of all morphisms whose kernel and cokernel are in $T$. Although $\Sigma$ admits the [[calculus of fractions]], this method does not guarantee the existence in general.

\hypertarget{localizing_subcategories}{}\subsubsection*{{Localizing subcategories}}\label{localizing_subcategories}

A thick subcategory (here always in the strong sense) is said to be [[localizing subcategory|localizing]] if $T$ is thick and the canonical functor $Q$ admits a right adjoint $A/T\to A$, often called the \textbf{section functor}. In other words $A/T$ is a reflective subcategory of $A$. Every coreflective thick subcategory $T$ admits a section functor, and the converse holds if $A$ has injective envelopes. A thick subcategory $T\subset A$ is a coreflective iff $(T,F)$ is a [[torsion theory]] where

\begin{displaymath}
F := \{X\in Ob A\,|\,A(T,X) = 0\}
\end{displaymath}
\hypertarget{thick_subcategories_and_saturation}{}\subsection*{{Thick subcategories and saturation}}\label{thick_subcategories_and_saturation}

Recall that a class $\Sigma$ of morphisms with category of fractions $\mathcal{C}_\Sigma$ is called \emph{saturated} if $\Sigma$ coincides with the class of morphisms inverted by the canonical functor $\mathcal{C}\to \mathcal{C}_\Sigma$.

Let $A$ be an abelian category, $T$ be a thick subcategory in the strong sense and let $\Sigma_T$ be the class of morphisms in $A$ such that their kernel and cokernel are in $T$. Then $\Sigma_T$ has a right and a left calculus of fractions and is saturated for the canonical functor $p:A\to A/T$. Furthermore, $p$ maps to zero objects precisely the objects in $T$.

Conversely, let $\Sigma$ be a saturated class of morphisms of $A$ with a calculus of fractions on the right and on the left. Then the full subcategory on the objects that are mapped to zero by the canonical $p:A\to A_\Sigma$ is thick. In other words, there is a bijection between thick subcategories in the strong sense and saturated classes of morphisms with a calculus of fractions on the right and on the left.

(For this material see \hyperlink{Schubert70b}{Schubert 1970}, pp.105-107).

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

\begin{itemize}%
\item [[thick subcategory theorem]]

\end{itemize}
\hypertarget{link}{}\subsection*{{Link}}\label{link}

\begin{itemize}%
\item Springer eom: \href{http://eom.springer.de/l/l060290.htm}{localization of categories}

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

\begin{itemize}%
\item [[Pierre Gabriel]], \emph{Des cat\'e{}gories ab\'e{}liennes}, Bulletin de la Soci\'e{}t\'e{} Math\'e{}matique de France, 90 (1962), p. 323-448 (\href{http://www.numdam.org/item?id=BSMF_1962__90__323_0}{numdam})


\item [[Jacob Lurie]], \emph{[[Chromatic Homotopy Theory]]}, lecture 26, \emph{Thick subcategories} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture26.pdf}{pdf})


\item [[A. L. Rosenberg]], \emph{Noncommutative algebraic geometry and representations of quantized algebras}, MIA \textbf{330}, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9


\item H. Schubert, \emph{Kategorien II} , Springer Heidelberg 1970.



\end{itemize}
[[!redirects thick subcategories]] [[!redirects epaisse subcategory]] [[!redirects epaisse subcategories]] [[!redirects épaisse subcategory]] [[!redirects épaisse subcategories]] [[!redirects Serre quotient category]]

[[!redirects thick ideal]] [[!redirects thick ideals]]



\end{document}