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\begin{document}

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\section*{Quillen exact category}

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

\hypertarget{additive_and_abelian_categories}{}\paragraph*{{Additive and abelian categories}}\label{additive_and_abelian_categories}

[[!include additive and abelian categories - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{via_embedding}{Via embedding}\dotfill \pageref*{via_embedding} \linebreak
\noindent\hyperlink{via_exact_structure}{Via exact structure}\dotfill \pageref*{via_exact_structure} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{quillengabriel_embedding_theorem}{Quillen-Gabriel embedding theorem}\dotfill \pageref*{quillengabriel_embedding_theorem} \linebreak
\noindent\hyperlink{relation_to_waldhausen_categories_and_algebraic_ktheory}{Relation to Waldhausen categories and algebraic K-theory}\dotfill \pageref*{relation_to_waldhausen_categories_and_algebraic_ktheory} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{via_embedding}{}\subsection*{{Via embedding}}\label{via_embedding}

A full [[additive category|additive]] [[subcategory]] $A$ of an [[abelian category]] $B$ is called \textbf{Quillen exact category} if it is closed under [[extensions]] (if in extension $0\to X\stackrel{j}\to Y\stackrel{p}\to Z\to 0$, $X$ and $Z$ are in $A$ then $Y$ is in $A$). It is viewed as a pair $(A,E)$ where $E$ is a class of all short exact sequences in $A$ which are [[exact sequence|exact]] in $B$.

All $j$ which appear as $j$ in an exact sequence as above are called \textbf{inflation}s or \textbf{admissible monomorphism}s. All $p$ which appear in an exact sequence as above are called \textbf{deflation}s or \textbf{admissible epimorphism}s.

\hypertarget{via_exact_structure}{}\subsection*{{Via exact structure}}\label{via_exact_structure}

A \textbf{Quillen exact category} is a pair $(A,E)$ of an [[additive category]] $A$ and a class of sequences $E$ called `exact'. The following axioms are required for $(A,E)$:

(QE1) The class of `exact' sequences is closed under [[isomorphism]]s and it contains all split extensions. For any `exact' sequence the deflation is the [[cokernel]] of inflation and the inflation is the [[kernel]] of the deflation.

(QE2) The class of deflations is closed under [[composition]] and [[base change]] by arbitrary maps. The class of inflations is closed under compositions and [[cobase change]] by arbitrary maps.

(QE3) If a morphism $M\to M'$ having a [[kernel]] can factor a deflation $N\to M'$ as $N\to M\to M'$ then it is a deflation. If a morphism $I\to I'$ having a cokernel can factor an inflation $I\to J$ as $I\to I'\to J$ then it is also an inflation.

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

\hypertarget{quillengabriel_embedding_theorem}{}\subsubsection*{{Quillen-Gabriel embedding theorem}}\label{quillengabriel_embedding_theorem}

For every small exact category in the sense of a pair $(A,E)$, there is an embedding $A\hookrightarrow B$ into an [[abelian category]] such that $E$ is a class of all sequences which are (short) exact in $B$.

\hypertarget{relation_to_waldhausen_categories_and_algebraic_ktheory}{}\subsubsection*{{Relation to Waldhausen categories and algebraic K-theory}}\label{relation_to_waldhausen_categories_and_algebraic_ktheory}

Every Quillen exact category can be made into a [[Waldhausen category]]. However some information is lost in the process. Moreover, not every Waldhausen category comes from a Quillen exact category. Both Quillen exact categories and Waldhausen categories are devised in order to do [[algebraic K-theory]]. The K-theory spectrum based on [[Quillen's Q-construction]] and an exact category agrees with the K-theory spectrum based on the [[Waldhausen S-construction]] of the K-theory spectrum from its associated [[Waldhausen category]].

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

Quillen introduced exact categories in above sense in the article

\begin{itemize}%
\item [[Daniel Quillen]], ``Higher algebraic K-theory'', in Higher K-theories, pp. 85--147, Proc. Seattle 1972, Lec. Notes Math. 341, Springer 1973.

\end{itemize}
[[Alexander Rosenberg]]  one sided generalizations of Quillen exact categories: right `exact' categories involving deflations, and left `exact' categories involving inflations. One of the motivations an alternative definition of higher K-theory of (right exact) categories not involving spectra. In this setup the K-theory is an example of a derived functor in nonabelian homological algebra utilizing roughly the left `exact' structure on the category of essentially small right `exact' categories. It is not known if this K-theory when restricted to the category of essentially small Quillen exact categories agrees with Quillen K-theory. But it has the standard properties of Quillen K-theory (devissage, exactness and so on). The one-sided generalization inspired by ideas introduced by Keller and Vossieck in the build up of the theory of [[suspended category|suspended categories]].

A right `exact' category is a category with an initial object and a Grothendieck pretopology consisting of single maps which are [[strict epimorphism]]s. The distinguisheed class of strict epimorphisms is called a right `exact' structure, or the class of \emph{deflations}. The construction of derived functors in this generality involves a version of [[satellite|satellites]].

\begin{itemize}%
\item Dmitry Kaledin, Wendy Lowen, \emph{Cohomology of exact categories and (non-)additive sheaves}, \href{http://arxiv.org/abs/1102.5756}{arxiv/1102.5756}

\end{itemize}
[[!redirects Quillen exact categories]] [[!redirects Quillen exactness]]



\end{document}