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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{karoubi_envelope}{Karoubi envelope}\dotfill \pageref*{karoubi_envelope} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{Karoubian category} or \emph{pseudo-abelian category} (also: pseudoabelian). is a [[pre-additive category]] $C$ such that every [[idempotent]] morphism $p: A \to A$ in $C$ has a [[kernel]], and hence (one can easily show) also a [[cokernel]].

This is stronger than [[pre-additive category|pre-additivity]] but weaker than [[abelian category|abelianness]], which requires that \emph{every} morphism has a [[kernel]] and [[cokernel]].

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

Let $C$ be a category and $p : X \to X$ an [[idempotent]] [[endomorphism]] of an object $X$. One says that $p$ \textbf{admits an image} if the functor $Ker(id_X, p)$ is [[representable]], and the representing object is called the image of $p$. Here $Ker(id_X, p)$ is the functor $C^{op} \to \underline{Set}$ mapping

\begin{displaymath}
Y \mapsto Ker(Hom(Y, X) \rightrightarrows Hom(Y, X)); \qquad (\ast)
\end{displaymath}
in other words the image of $p$ is the [[equaliser|difference kernel]] of $(id_X, p)$, when it exists.

Now $C$ is called \textbf{Karoubian} if every idempotent $p$ admits an image. Since $p: X \to X$ is idempotent iff $id_X - p$ is idempotent, this is the same as saying every idempotent has a kernel.

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

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

One can show that for any idempotent $p$, $Ker(id_X, p)$ is representable if and only if $Coker(id_X, p)$ is, and that in fact their representing objects are canonically isomorphic.

Recall that one says $p$ \textbf{[[split idempotent|splits]]} if there exists an object $Y$, and morphisms $f : X \to Y$, $g : Y \to X$, such that $f \circ g = id_Y$ and $g \circ f = p$. Observe that when $p$ admits an image $K$, it splits: by definition there are functorial isomorphisms $\Phi_Y$ for all $Y$ between the image of the functor $(\ast)$ and $\Hom(Y, K)$; now take $f : X \to K$ the morphism corresponding to $p$ via $\Phi_X$, $g : K \to X$ the morphism corresponding to $id_K \in Hom(K, K)$ via $\Phi_K$. Conversely, if $p$ splits via a pair $(f, g)$, then $g: Y \to X$ is a difference kernel of $(id_X, p)$: we have $g = g \circ f \circ g = p \circ g$, and if $h: Z \to X$ satisfies $h = p \circ h = g \circ f \circ h$, then $h$ clearly factors through $g$, and uniquely so since sections $g$ are monomorphisms.

\hypertarget{karoubi_envelope}{}\subsubsection*{{Karoubi envelope}}\label{karoubi_envelope}

There is a [[reflective subcategory|universal functor]] from the category of (say, small) [[preadditive categories]] to the category of Karoubian categories, the \textbf{[[Karoubinization functor]]}; its value on a preadditive category $C$ is also called the \textbf{[[Karoubian envelope]]} or the \textbf{[[pseudo-abelian completion]]} of $C$.

In more detail, there exists a Karoubian category $kar(C)$ associated to any category $C$, and a fully faithful functor $\varphi : C \to kar(C)$, which is [[universal property | universal]] in the sense that for any Karoubian category $C'$, the functor

\begin{displaymath}
\underline{Hom}(kar(C), C') \to \underline{Hom}(C, C')
\end{displaymath}
taking a functor $F : kar(C) \to C'$ to the composite $F \circ \varphi$ is an [[equivalence]] of categories. $kar(C)$ is called the \textbf{[[Karoubi envelope]]} of $C$ (aka the \emph{[[Cauchy completion]]}, or the \emph{[[idempotent-splitting completion]]}). It can be realized explicitly by taking as objects pairs $(X, p)$, with $p$ idempotent, and as morphisms $(X, p) \to (Y, q)$ the morphisms $f : X \to Y$ that satisfy $f = q \circ f \circ p$.

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

The requirement that, say, a [[dg-category]] or a [[triangulated category]] be Karoubian is a natural requirement in a number of contexts.

The Karoubian envelope is also used in the construction of the [[category of pure motives]], and in [[K-theory]].

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

\begin{itemize}%
\item [[Alexandre Grothendieck]], [[Jean-Louis Verdier]]. Exercice 7.5 in \emph{Topos}, Expos\'e{} IV of [[SGA 4]], volume 1.

\end{itemize}
[[!redirects pseudo-abelian category]] [[!redirects pseudoabelian category]]

[[!redirects pseudo-abelian categories]] [[!redirects pseudoabelian categories]]



\end{document}