\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Karoubi envelope}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{AbstractDefinition}{Abstract definition}\dotfill \pageref*{AbstractDefinition} \linebreak
\noindent\hyperlink{InComponents}{In components}\dotfill \pageref*{InComponents} \linebreak
\noindent\hyperlink{UnderTheYonedaEmbedding}{Under the Yoneda embedding}\dotfill \pageref*{UnderTheYonedaEmbedding} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Finality}{Finality of the completion}\dotfill \pageref*{Finality} \linebreak
\noindent\hyperlink{monadicity_over_semicategories}{Monadicity over semicategories}\dotfill \pageref*{monadicity_over_semicategories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{the_category_of_smooth_manifolds}{The category of smooth manifolds}\dotfill \pageref*{the_category_of_smooth_manifolds} \linebreak
\noindent\hyperlink{projective_modules_and_vector_bundles}{Projective modules and vector bundles}\dotfill \pageref*{projective_modules_and_vector_bundles} \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 \emph{Karoubi envelope} or \emph{idempotent completion} of a [[category]] is the [[universal construction|universal]] enlargement of the category with the property that every [[idempotent]] is a [[split idempotent]]. This is the [[Set]]-enriched version of the more general notion of \emph{[[Cauchy completion]]} of an [[enriched category]].

A category in which all idempotents split is called \emph{Karoubi complete} or \emph{Cauchy complete} or \emph{idempotent-complete}. Thus, the Karoubi envelope is a [[completion]] operation into such categories.

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

There is an

\begin{itemize}%
\item \emph{\hyperlink{AbstractDefinition}{Abstract definition}}

\end{itemize}
that characterizes idempotent completions. In particular the idempotent completion always exists and is unique up to [[equivalence of categories]]. Explicit constructions include:

\begin{itemize}%
\item \emph{\hyperlink{InComponents}{Construction in components}}


\item \emph{\hyperlink{UnderTheYonedaEmbedding}{Construction via the Yoneda embedding}}



\end{itemize}
For more constructions and equivalent characterizations see at \emph{[[Cauchy complete category]]} in the section \emph{\href{Cauchy+complete+category#InOrdinaryCategoryTheory}{In ordinary category theory}}.

\hypertarget{AbstractDefinition}{}\subsubsection*{{Abstract definition}}\label{AbstractDefinition}

\begin{defn}
\label{IdempotentCompletionAbstractly}\hypertarget{IdempotentCompletionAbstractly}{}
For $\mathcal{C}$ a [[category]], a [[functor]] $\mathcal{C} \to \tilde \mathcal{C}$ exhibits $\tilde \mathcal{C}$ as an \textbf{idempotent completion} of $\mathcal{C}$ if

\begin{itemize}%
\item $\tilde \mathcal{C}$ is an [[idempotent complete category]];


\item $\mathcal{C} \to \tilde \mathcal{C}$ is a [[full and faithful functor]];


\item every object in $\tilde \mathcal{C}$ is the [[retract]] of an object in $\mathcal{C}$, under this embedding.



\end{itemize}
\end{defn}
See e.g. (\hyperlink{Lurie}{Lurie, def. 5.1.4.1}).

\begin{lemma}
\label{def2}\hypertarget{def2}{}
For a fully faithful embedding $i \colon \mathcal{C} \to \mathcal{D}$ to exhibit an idempotent(-splitting) completion of $\mathcal{C}$, it suffices that

\begin{itemize}%
\item $i(p)$ splits in $\mathcal{D}$ for every idempotent $p$ in $\mathcal{C}$, and


\item every object in $\mathcal{D}$ is the retract of an object in $\mathcal{C}$ under $i$.



\end{itemize}
\end{lemma}
\begin{proof}
We must show that these conditions imply that every idempotent $e \colon D \to D$ in $\mathcal{D}$ splits. Write $D$ as a retract of some $i(C)$, say $r: i(C) \to D$ with right inverse $s$ ($r s = 1_D$). Then $p = s e r \colon i(C) \to i(C)$ is idempotent, and we may split $p$, say as $p = \sigma \pi$ with $\pi \sigma = 1_E$ for some $E$. We claim that the pair

\begin{displaymath}
\pi s \colon D \to E, \qquad r \sigma \colon E \to D
\end{displaymath}
provides a splitting of $e$. Certainly we have

\begin{displaymath}
(r \sigma)(\pi s) = r s e r s = e,
\end{displaymath}
and we also have

\begin{displaymath}
\sigma(\pi s)(r \sigma) = p s r \sigma = s e r s r \sigma = s e r \sigma = p \sigma = \sigma \pi \sigma = \sigma
\end{displaymath}
whence

\begin{displaymath}
(\pi s)(r \sigma) = \pi \sigma (\pi s)(r \sigma) = \pi \sigma = 1_E,
\end{displaymath}
as desired.

\end{proof}
\hypertarget{InComponents}{}\subsubsection*{{In components}}\label{InComponents}

Let $C$ be a [[category]]. We give an elementary construction of the \textbf{Karoubi envelope} $\bar{C}$ which formally splits [[idempotents]] in $C$.

The objects of $\bar{C}$ are pairs $(c, e: c \to c)$ where $e$ is an idempotent on an object $c$ of $C$. Morphisms $(c, e) \to (d, f)$ are morphisms $\phi: c \to d$ in $C$ such that $f \circ \phi = \phi = \phi \circ e$ (or equivalently, such that $\phi = f \circ \phi \circ e$). NB: the identity on $(c, e)$ in $\bar{C}$ is the morphism $e: c \to c$.

There is a functor

\begin{displaymath}
E: C \to \bar{C}
\end{displaymath}
which maps an object $c$ to $(c, 1_c)$. This functor is [[fully faithful functor|full and faithful]]: it fully embeds $C$ in $\bar{C}$. If $e: c \to c$ is an idempotent in $C$, then in $\bar{C}$ there are maps

\begin{displaymath}
p: (c, 1_c) \to (c, e), \, j: (c, e) \to (c, 1_c),
\end{displaymath}
both given by $e: c \to c$. It is clear that $p \circ j$ is the identity $e: (c, e) \to (c, e)$, and that $j \circ p$ is the idempotent $E(e): E(c) \to E(c)$. Thus the pair $(p, j)$ formally splits the idempotent $e: c \to c$. The same argument shows that every idempotent $\phi: (c, e) \to (c, e)$ in $\bar{C}$ splits. Actually this formal construction does more: it gives a \emph{choice} of splitting for every idempotent.

Let $D$ be any category in which every idempotent $h: d \to d$ has a chosen splitting $(p_h: d \to d_h, j_h: d_h \to d)$ (using identities to split identities), and let $F: C \to D$ be a functor. Define an extension

\begin{displaymath}
\bar{F}: \bar{C} \to D
\end{displaymath}
by sending $(c, e: c \to c)$ to the underlying object $F(c)_{F(e)}$ of the splitting of $F(e): F(c) \to F(c)$ in $D$. For morphisms $\phi: (c, e) \to (c', e')$, define $\bar{F}(\phi)$ to be the composite

\begin{displaymath}
F(c)_{F(e)} \overset{F(j_{F(e)})}{\to} F(c) \overset{F(\phi)}{\to} F(c') \overset{F(p_{F(e')})}{\to} F(c')_{F(e')}
\end{displaymath}
Then $\bar{F}$ is the unique extension of $F$ which preserves chosen splittings. Thus the Karoubi envelope is universal among functors from $C$ into categories $D$ in which every idempotent has a chosen splitting.

If $D$ is a category in which every idempotent splits, then we can choose a splitting for each idempotent using the [[axiom of choice]] (AC); the extension $\bar{F}$ depends on how we do this but is unique up to unique [[natural isomorphism]]. Alternatively, we can define $\bar{F}$ as an [[anafunctor]]; then no AC is needed, and we still have $\bar{F}$ unique up to unique natural isomorphism. (It is key here that a splitting of an idempotent is unique up to a coherent isomorphism.)

Essentially the same argument shows that for any $D$ in which idempotents split, the restriction functor $[E, D]: [\bar{C}, D] \to [C, D]$ is an equivalence. The details are spelled out \href{http://ncatlab.org/toddtrimble/published/Karoubi+envelope}{here}.

\hypertarget{UnderTheYonedaEmbedding}{}\subsubsection*{{Under the Yoneda embedding}}\label{UnderTheYonedaEmbedding}

\begin{prop}
\label{}\hypertarget{}{}
For $\mathcal{C}$ a [[small category]], write $PSh(\mathcal{C})$ for its [[category of presheaves]] and write $\tilde \mathcal{C} \hookrightarrow PSh(\mathcal{C})$ for the [[full subcategory]] on those presheaves which are [[retracts]] of objects in $\mathcal{C}$, under the [[Yoneda embedding]]. Then the Yoneda embedding

\begin{displaymath}
\mathcal{C} \to \tilde \mathcal{C}
\end{displaymath}
exhibits $\tilde \mathcal{C}$ as [[generalized the|the]] idempotent completion of $\mathcal{C}$.

\end{prop}
For instance (\hyperlink{Lurie}{Lurie, proof of prop. 5.1.4.2}).

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

\hypertarget{Finality}{}\subsubsection*{{Finality of the completion}}\label{Finality}

\begin{defn}
\label{}\hypertarget{}{}
A functor $\mathcal{C}\to \tilde \mathcal{C}$ exhibiting an idempotent completion, def. \ref{IdempotentCompletionAbstractly}, is a [[final functor]].

\end{defn}
For instance (\hyperlink{Lurie}{Lurie, lemma 5.1.4.6}).

\hypertarget{monadicity_over_semicategories}{}\subsubsection*{{Monadicity over semicategories}}\label{monadicity_over_semicategories}

The functor that forms idempotent completion is the [[monad]] induced from the [[adjunction]] between categories and [[semicategories]] given by the [[forgetful functor]] $Cat \to SemiCat$ and its [[right adjoint]]. More details on this are at \emph{\href{semicategory#RelationToCategories}{Semicategory - Relation to categories}}.

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

\begin{itemize}%
\item [[category of Chow motives]]

\end{itemize}
\hypertarget{the_category_of_smooth_manifolds}{}\subsubsection*{{The category of smooth manifolds}}\label{the_category_of_smooth_manifolds}

Let $Man$ be the category of [[smooth manifolds]] and [[smooth maps]], where by a ``smooth manifold'', we mean a finite-dimensional, second-countable, Hausdorff, $C^\infty$ [[manifold]] without boundary. Let $i: Open \hookrightarrow Man$ be the [[full subcategory]] whose objects are the [[open subspaces]] of finite-dimensional [[Cartesian spaces]].

\begin{theorem}
\label{}\hypertarget{}{}
The subcategory $i: Open \hookrightarrow Man$ exhibits $Man$ as an idempotent-splitting completion of $Open$.

\end{theorem}
\begin{proof}
By lemma \ref{def2}, it suffices to prove that

\begin{itemize}%
\item Every smooth manifold is a smooth retract of an open set in Euclidean space;


\item If $p : U \to U$ is a smooth idempotent on an open set $U \subseteq \mathbb{R}^n$, then the subset $Fix(p) \hookrightarrow U$ is an embedded submanifold.



\end{itemize}
For the first statement, we use the fact that any manifold $M$ can be realized as a closed submanifold of some $\mathbb{R}^n$, and every closed submanifold has a [[tubular neighborhood theorem|tubular neighborhood]] $U \subseteq \mathbb{R}^n$. In this case $U$ carries a structure of vector bundle over $M$ in such a way that the inclusion $M \hookrightarrow U$ is identified with the zero section, so that the bundle projection $U \to M$ provides a retraction, with right inverse given by the zero section.

For the second statement, assume that the origin $0$ is a fixed point of $p$, and let $T_0(U) \cong \mathbb{R}^n$ be its tangent space (observe the presence of a \emph{canonical} isomorphism to $\mathbb{R}^n$). Thus we have idempotent linear maps $d p(0), Id-d p(0): T_0(U) \to T_0(U)$ where the latter factors through the inclusion $\ker \; d p(0) \hookrightarrow T_0(U)$ via a projection map $\pi: T_0(U) \to \ker \; d p(0)$. We have a map $f: U \to \mathbb{R}^n$ that takes $x \in U$ to $x - p(x)$; let $g$ denote the composite

\begin{displaymath}
U \stackrel{f}{\to} \mathbb{R}^n \cong T_0(U) \stackrel{\pi}{\to} \ker\; d p(0).
\end{displaymath}
Now we make some easy observations:

\begin{enumerate}%
\item $Fix(p) \subseteq g^{-1}(0)$.


\item The map $p: U \to U$ restricts to a map $p: g^{-1}(0) \to g^{-1}(0)$, by idempotence of $p$.


\item The derivative $d g(0): T_0(U) \to T_0(\ker \; d p(0)) \cong \ker \; d p(0)$ is $\pi$ again since $Id - d p(0)$ is idempotent. Thus $d g(0)$ has full rank ($m$ say), and so the restriction of $g$ to some neighborhood $V$ has $0$ as a regular value, and $g^{-1}(0) \cap V$ is a manifold of dimension $m$ by the [[implicit function theorem]]. The tangent space $T_0(g^{-1}(0) \cap V)$ is canonically identified with $im(d p(0))$.


\item There are smaller neighborhoods $V'' \subseteq V' \subseteq V$ so that $p$ restricts to maps $p_1, p_2$ as in the following diagram ($i, i', i''$ are inclusion maps, all taking a domain element $x$ to itself):

\begin{displaymath}
\itexarray{
g^{-1}(0) \cap V'' & \stackrel{i''}{\hookrightarrow} & g^{-1}(0) \\ 
_\mathllap{p_2} \downarrow & & \downarrow _\mathrlap{p} \\ 
g^{-1}(0) \cap V' & \stackrel{i'}{\hookrightarrow} & g^{-1}(0) \\ 
_\mathllap{p_1} \downarrow & & \downarrow _\mathrlap{p} \\ 
g^{-1}(0) \cap V & \stackrel{i}{\hookrightarrow} & g^{-1}(0)
}
\end{displaymath}
and such that $p_1, p_2$ are diffeomorphisms by the [[implicit function theorem|inverse function theorem]] (noting here that $d p_i(0): im(d p(0)) \to im(d p(0))$ is the identity map, by idempotence of $p$).


\item Letting $q: g^{-1}(0) \cap V' \to g^{-1}(0) \cap V''$ denote the smooth inverse to $p_2$, we calculate $i' = p \circ i'' \circ q$, and

\begin{displaymath}
i p_1 = p i' = p p i''q = p i'' q = i',
\end{displaymath}
so that $p_1(x) = x$ for every $x \in g^{-1}(0) \cap V'$. Hence $g^{-1}(0) \cap V' \subseteq Fix(p)$.



\end{enumerate}
From all this it follows that $Fix(p) \cap V' = g^{-1}(0) \cap V'$, meaning $Fix(p)$ is locally diffeomorphic to $\mathbb{R}^m$, and so $Fix(p)$ is an embedded submanifold of $\mathbb{R}^n$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
\hyperlink{Law}{Lawvere} comments on this fact as follows: ``For example, if $\mathbf{C}$ is the category of all smooth maps between all open subsets of all Euclidean spaces, then $\widebar{\mathbf{C}}$ $[$the Karoubi envelope$]$ is the category of all smooth manifolds. This powerful theorem justifies bypassing the complicated considerations of charts, coordinate transformations, and atlases commonly offered as a ''basic`` definition of the concept of manifold. For example the 2-sphere, a manifold but not an open set of any Euclidean space, may be fully specified with its smooth structure by considering any open set $A$ in 3-space $E$ which contains it but not its center (taken to be $0$) and the smooth idempotent endomap of $A$ given by $e(x) = x/{|x|}$. All general constructions (i.e., functors into categories which are Cauchy complete) on manifolds now follow easily (without any need to check whether they are compatible with coverings, etc.) provided they are known on the opens of Euclidean spaces: for example, the tangent bundle on the sphere is obtained by splitting the idempotent $e'$ on the tangent bundle $A \times V$ of $A$ ($V$ being the vector space of translations of $E$) which is obtained by differentiating $e$. The same for cohomology groups, etc.''

\end{remark}
\hypertarget{projective_modules_and_vector_bundles}{}\subsubsection*{{Projective modules and vector bundles}}\label{projective_modules_and_vector_bundles}

The category of projective modules over any ring is the Karoubi envelope of its full subcategory of free modules.

The category of (locally trivial, finite dimensional) vector bundles over any fixed paracompact space is the Karoubi envelope of its full subcategory of trivial bundles.

Both examples are related by the [[Serre-Swan theorem]]. In fact both these facts together with the observation that the global sections functor is an equivalence from the category trivial bundles over $X$ to the category of free modules over $C(X)$ prove the Serre-Swan theorem itself.

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

\begin{itemize}%
\item [[Karoubian category]], [[pseudo-abelian category]]


\item [[Cauchy completion]]


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



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

A classical account is for instance in

\begin{itemize}%
\item [[Francis Borceux]] and D. Dejean, \emph{Cauchy completion in category theory} Cahiers Topologie G\'e{}om. Diff\'e{}rentielle Cat\'e{}goriques, 27:133--146, (1986) (\href{http://www.numdam.org/item?id=CTGDC_1986__27_2_133_0}{numdam})

\end{itemize}
For more classical references see the references at \emph{[[Cauchy complete category]]}.

Karoubi envelopes for [[(infinity,1)-category|(∞,1)-categories]] are discussed in section 4.4.5 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
Some discussion of the stable version is in section 4.1.2 of

\begin{itemize}%
\item [[David Ben-Zvi]], John Francis, David Nadler, \emph{Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry} (\href{http://arxiv.org/abs/0805.0157}{arXiv}, [[geometric infinity-function theory|nLab entry]])

\end{itemize}
and section 2.3 of

\begin{itemize}%
\item [[David Ben-Zvi]], David Nadler, \emph{The Character Theory of a Complex Group} (\href{http://arxiv.org/abs/0904.1247}{arXiv})

\end{itemize}
In section 3.1.2 of latter are also given references (to Neeman and Lurie) for an important result of Neeman's about Karoubi closure and compact generators.

The Karoubi envelope for the additive case (see also [[additive envelope]]) is covered at

\begin{itemize}%
\item [[Dror Bar-Natan]], [[Scott Morrison]], \emph{The Karoubi envelope and Lee's degeneration of Khovanov homology} (\href{http://arxiv.org/abs/math/0606542}{arXiv})

\end{itemize}
Discussion for [[triangulated categories]] is in

\begin{itemize}%
\item Paul Balmer, Marco Schlichting, \emph{Idempotent completion of triangulated categories} (\href{http://www.math.ucla.edu/~balmer/research/Pubfile/IdempCompl.pdf}{pdf})

\end{itemize}
The proof that idempotents split in the category of manifolds was adapted from this MO answer:

\begin{itemize}%
\item Zack (http://mathoverflow.net/users/300/zack), Idempotents split in category of smooth manifolds?, URL (version: 2014-04-06): http://mathoverflow.net/q/162556 (\href{http://mathoverflow.net/a/162556/2926}{web})

\end{itemize}
Which provides a solution to exercise 3.21 in

\begin{itemize}%
\item F. W. Lawvere, \emph{Perugia Notes - Theory of Categories over a Base Topos} , Ms. Universit\`a{} di Perugia 1973.

\end{itemize}
The accompanying above remark of Lawvere appears on page 267 of

\begin{itemize}%
\item [[F. William Lawvere]], \emph{Qualitative distinctions between some toposes of generalized graphs}, Contemporary Mathematics 92 (1989), 261-299. (\href{http://www.ams.org/books/conm/092/1003203/conm092-1003203.pdf}{pdf})

\end{itemize}
A comparison of the Karoubi envelope to other completions can be found here:

\begin{itemize}%
\item [[Marta Bunge]], \emph{Tightly Bounded Completions} , TAC \textbf{28} no. 8 (2013) pp.213-240. (\href{http://www.tac.mta.ca/tac/volumes/28/8/28-08.pdf}{pdf})

\end{itemize}
Formalization in [[homotopy type theory]]:

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

\end{itemize}
A generalization of the Karoubi envelope for [[n-categories]] is in

\begin{itemize}%
\item [[Davide Gaiotto]], [[Theo Johnson-Freyd]], \emph{Condensations in higher categories}, (\href{https://arxiv.org/abs/1905.09566}{arXiv:1905.09566})

\end{itemize}
[[!redirects Karoubi completion]] [[!redirects Karoubi complete category]] [[!redirects idempotent-complete category]] [[!redirects idempotent complete category]]

[[!redirects idempotent completion]] [[!redirects idempotent completions]]

[[!redirects idempotent-splitting completion]] [[!redirects idempotent-splitting completions]]

[[!redirects pseudo-abelian completion]] [[!redirects Karoubian envelope]] [[!redirects Karoubianization]] [[!redirects Karoubianization functor]] [[!redirects Karoubinization functor]]



\end{document}