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\begin{document}

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\section*{idempotent}

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

\hypertarget{idempotents}{}\paragraph*{{Idempotents}}\label{idempotents}

[[!include idempotents - contents]]

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

[[!include category theory - contents]]

\hypertarget{idempotents_2}{}\section*{{Idempotents}}\label{idempotents_2}

\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{the_algebra_of_idempotents}{The algebra of idempotents}\dotfill \pageref*{the_algebra_of_idempotents} \linebreak
\noindent\hyperlink{the_universal_idempotentsplit_completion}{The universal idempotent-split completion}\dotfill \pageref*{the_universal_idempotentsplit_completion} \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 notion of an \emph{idempotent} [[morphism]] in a [[category]] generalizes the notion of \emph{[[projector]]} in the context of [[linear algebra]]: it is an [[endomorphism]] $e \colon X \to X$ of some [[object]] $X$ that ``squares to itself'' in that the [[composition]] of $e$ with itself is again $e$:

\begin{displaymath}
e \circ e = e
  \,.
\end{displaymath}
Accordingly, given any idempotent $e \colon X \to X$ it is of interest to ask what [[subobject]] $A \stackrel{i}{\hookrightarrow} X$ of $X$ it is the projector onto, in that there is a [[projection]] $X \stackrel{p}{\to} A$ such that the idempotent is the composite of this projection followed by including $A$ back into $X$:

\begin{displaymath}
e \colon X \stackrel{p}{\to} A \stackrel{i}{\hookrightarrow} X
  \,.
\end{displaymath}
As opposed to the case of linear algebra, in general such a factorization into a projection onto a subobject $A$ need not actually exists for an idempotent $e$ in a generic category. If it exists, one says that $e$ is a \emph{[[split idempotent]]}.

Accordingly, one is interested in those categories for which every idempotent is split. These are called \emph{[[idempotent complete categories]]} or \emph{[[Cauchy complete categories]]}. If a category is not yet idempotent complete it can be completed to one that is: its \emph{[[Karoubi envelope]]} or \emph{[[Cauchy completion]]}.

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

An [[endomorphism]] $e\colon B \to B$ in a [[category]] is an \textbf{idempotent} if the [[composition]] with itself equals itself

\begin{displaymath}
e \circ  e = e
  \,.
\end{displaymath}
A \textbf{splitting} of an idempotent $e$ consists of morphisms $s\colon A \to B$ and $r\colon B \to A$ such that $r \circ s = 1_A$ and $s \circ r = e$. In this case $A$ is a [[retract]] of $B$, and we call $e$ a [[split idempotent]].

Of course, we can simply consider the \textbf{idempotent elements} of any [[monoid]].

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

\hypertarget{the_algebra_of_idempotents}{}\subsubsection*{{The algebra of idempotents}}\label{the_algebra_of_idempotents}

Given an [[abelian monoid]] $R$, the idempotent elements form a [[submonoid]] $Idem(R)$.

Given a [[commutative ring]] $R$, the idempotent elements of $R$ form a [[Boolean algebra]] $Idem(R)$ with these operations:

\begin{itemize}%
\item $\top \coloneqq 1$,
\item $P \wedge Q \coloneqq P Q$,
\item $\bot \coloneqq 0$,
\item $P \vee Q \coloneqq P - P Q + Q$,
\item $\neg{P} \coloneqq 1 - P$.

\end{itemize}
This is important in [[measure theory]]; if $R$ is the ring $L^\infty(X,\mathcal{M},\mathcal{N})$ of [[essentially bounded function|essentially bounded]] [[real number|real]]-valued [[measurable functions]] on some [[measurable space]] $(X,\mathcal{M})$ modulo an ideal $\mathcal{N}$ of [[null sets]], then $Idem(R)$ is the Boolean algebra of [[characteristic functions]] of [[measurable sets]] modulo null sets, which is [[isomorphic]] to the Boolean algebra $\mathcal{M}/\mathcal{N}$ of measurable sets modulo null sets itself.

If $R$ is a commutative $*$-[[star-ring|ring]], then we may restrict to the [[self-adjoint element|self-adjoint]] idempotent elements to get the Boolean algebra $Proj(R)$. In measure theory, if $R$ is the [[complex number|complex]]-valued version of $L^\infty(X,\mathcal{M},\mathcal{N})$, then $Proj(R)$ will still reconstruct $\mathcal{M}/\mathcal{N}$. In [[operator algebra]] theory, the self-adjoint idempotent elements of an operator algebra are called [[projection operator]]s, which is the origin of the notation $Proj$. (Sometimes one requires projection operators to be \emph{proper}: to have norm $1$; the only projection operator that is not proper is $0$.)

The projection operators of a commutative $W^\star$-[[W-star-algebra|algebra]] give the link between operator algebra theory and measure theory; in fact, the [[categories]] of commutative $W^\star$-algebras and of [[localisable measurable spaces]] (or [[measurable locales]]) are [[dual equivalence|dual]], and $W^\star$-algebra theory in general may be thought of as noncommutative measure theory. In noncommutative measure theory, the projection operators are still important, but they no longer form a Boolean algebra.

\hypertarget{the_universal_idempotentsplit_completion}{}\subsubsection*{{The universal idempotent-split completion}}\label{the_universal_idempotentsplit_completion}

Given a [[category]] $\mathcal{C}$ one may ask for the [[universal construction|universal]] category obtained from $\mathcal{C}$ subject to the constraint that all idempotents are turned into [[split idempotents]]. This is called the \emph{[[Karoubi envelope]]} of $\mathcal{C}$. More generally, in [[enriched category theory]] it is called the \emph{[[Cauchy completion]]} of $\mathcal{C}$.

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

\begin{itemize}%
\item [[retract]], [[section]]


\item [[Cauchy complete category]]


\item [[idempotent complete (infinity,1)-category]]


\item [[idempotent semiring]]



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

Formalization in [[homotopy type theory]]:

\begin{itemize}%
\item [[Mike Shulman]], \emph{\href{http://homotopytypetheory.org/2014/12/08/splitting-idempotents/}{Splitting idempotents}}

\end{itemize}
[[!redirects idempotent]] [[!redirects idempotents]] [[!redirects idempotent element]] [[!redirects idempotent elements]] [[!redirects idempotent operator]] [[!redirects idempotent operators]] [[!redirects idempotent morphism]] [[!redirects idempotent morphisms]]



\end{document}