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\section*{idempotent complete (infinity,1)-category}

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

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

[[!include idempotents - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{coherent_vs_incoherent_idempotents}{Coherent vs incoherent idempotents}\dotfill \pageref*{coherent_vs_incoherent_idempotents} \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}

An ordinary [[category]] is \emph{[[idempotent complete category|idempotent complete]]}, aka \emph{[[Karoubi envelope|Karoubi complete]]} or \emph{[[Cauchy completion|Cauchy complete]]} , if every [[idempotent]] splits. Since the splitting of an idempotent is a limit \emph{or} colimit of that idempotent, any category with [[finitely complete category|all finite limits]] or [[finitely cocomplete category|all finite colimits]] is idempotent complete.

In an [[(∞,1)-category]] the idea is the same, except that the notion of \emph{idempotent} is more complicated. Instead of just requiring that $e\circ e = e$, we need an [[equivalence]] $e\circ e \simeq e$, together with higher coherence data saying that, for instance, the two derived equivalences $e\circ e\circ e \simeq e$ are equivalent, and so on up. In particular, \emph{being idempotent} is no longer a [[stuff, structure, property|property]] of a morphism, but \emph{structure} on it.

It is still true that a splitting of an idempotent in an $(\infty,1)$-category is a limit or colimit of that idempotent (now regarded as a diagram with all its higher coherence data), but this limit is no longer a finite limit; thus an $(\infty,1)$-category can have all finite limits without being idempotent-complete.

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

\begin{defn}
\label{}\hypertarget{}{}
Let $Idem^+$ be the [[nerve]] of the [[walking structure|free 1-category containing a]] [[retraction]], with $e:X\to X$ the idempotent, $r:X\to Y$ the retraction, and $i:Y\to X$ the section (and $e = i r$ and $r i = 1_Y$). Let $Idem$ be the similar nerve of the free 1-category containing an idempotent, which is the full sub-simplicial set of $Idem^+$ spanned by the object $X$. Let $Ret$ be the image in $Idem^+$ of the 2-simplex $\Delta^2 \to Idem^+$ exhibiting the composite $r i = 1_Y$; thus $Ret$ is also the quotient of $\Delta^2$ that collapses the 1-face to a point.

\end{defn}
(\hyperlink{Lurie}{Lurie, 4.4.5.2 p.304})

\begin{defn}
\label{}\hypertarget{}{}
Let C be an $\infty$-category, incarnated as a [[quasi-category]].

\begin{enumerate}%
\item An \textbf{idempotent morphism} in C is a map of simplicial sets $Idem \to C$. We will refer to $Fun(Idem, C)$ as the \textbf{$(\infty,1)$-category of idempotents} in $C$.


\item A \textbf{weak retraction diagram} in $C$ is a [[homomorphism]] of [[simplicial sets]] $Ret \to C$. We refer to $Fun(Ret, C)$ as the \textbf{$(\infty,1)$-category of weak retraction diagrams} in $C$.


\item A \textbf{strong retraction diagram} in $C$ is a map of simplicial sets $Idem^+ \to C$. We will refer to $Fun(Idem+, C)$ as the \textbf{$(\infty,1)$-category of strong retraction diagrams} in $C$.



\end{enumerate}
\end{defn}
(\hyperlink{Lurie}{Lurie, 4.4.5.4 p.304})

\begin{defn}
\label{}\hypertarget{}{}
An idempotent $F \colon Idem \to C$ is \textbf{effective} if it extends to a map $Idem^+ \to C$.

\end{defn}
(\hyperlink{Lurie}{Lurie, above corollary 4.4.5.14})

\begin{prop}
\label{}\hypertarget{}{}
An idempotent diagram $F \colon Idem \to C$ is effective precisely if it admits an [[(∞,1)-limit]], equivalently if it admits an [[(∞,1)-colimit]].

\end{prop}
By (\hyperlink{Lurie}{Lurie, lemma 4.3.2.13}).

\begin{defn}
\label{}\hypertarget{}{}
$C$ is called an \textbf{idempotent complete $(\infty,1)$} if every idempotent is effective.

\end{defn}
(\hyperlink{Lurie}{Lurie, above corollary 4.4.5.14})

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

The following properties generalize those of idempotent-complete 1-categories.

\begin{theorem}
\label{}\hypertarget{}{}
A [[small (∞,1)-category]] is idempotent-complete if and only if it is [[accessible (∞,1)-category|accessible]].

\end{theorem}
This is [[Higher Topos Theory|HTT, 5.4.3.6]].

\begin{theorem}
\label{}\hypertarget{}{}
For $C$ a [[small (∞,1)-category]] and $\kappa$ a [[regular cardinal]], the [[(∞,1)-Yoneda embedding]] $C \to C' \hookrightarrow Ind_\kappa(C)$ with $C'$ the full subcategory on $\kappa$-[[compact object]]s exhibits $C'$ as the idempotent completion of $C$.

\end{theorem}
This is [[Higher Topos Theory|HTT, lemma 5.4.2.4]].

\hypertarget{coherent_vs_incoherent_idempotents}{}\subsection*{{Coherent vs incoherent idempotents}}\label{coherent_vs_incoherent_idempotents}

We may also ask how idempotent-completeness of $C$ is related to that of its [[homotopy category]] $h C$. An idempotent in $h C$ is an ``incoherent idempotent'' in $C$, i.e. a map $e:X\to X$ such that $e \sim e^2$, but without any higher coherence conditions. In this case we have:

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{([[Higher Algebra|HA]] Lemma 1.2.4.6)} If $C$ is [[stable (infinity,1)-category|stable]], then $C$ is idempotent-complete (i.e. every coherent idempotent is effective) if and only if $h C$ is (as a 1-category).

\end{theorem}
However, if $C$ is not stable, this is false. The following counterexample in [[?Gpd]] is constructed in Warning 1.2.4.8 of [[Higher Algebra|HA]]. Let $\lambda : G \to G$ be an injective but non-bijective group homomorphism such that $\lambda$ and $\lambda^2$ are conjugate. (One such is obtained by letting $G$ be the group of endpoint-fixing homeomorphisms of $[0,1]$, with $\lambda(g)$ acting as a scaled version of $g$ on $[0,\frac 1 2]$ and the identity on $[\frac 1 2,1]$. Then $\lambda(g) \circ h = h \circ \lambda^2(g)$ for any $h$ such that $h(t) = 2t$ for $t \in [0,\frac 1 4]$.)

Then $B\lambda : B G \to B G$ is homotopic to $B\lambda^2$, hence idempotent in the homotopy category. If it could be lifted to a coherent idempotent, then the colimit of the diagram

\begin{displaymath}
B G \xrightarrow{B \lambda} B G \xrightarrow{B \lambda} BG \to \cdots
\end{displaymath}
would be its splitting, and hence the map $B G \to \colim (B G\xrightarrow{B \lambda} B G \to\cdots)$ would have a [[section]]. Passing to fundamental groups, $G \to \colim (G\xrightarrow{\lambda} G \to\cdots)$ would also have a section; but this is impossible as $\lambda$ is injective but not surjective.

However, we do have the following:

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{([[Higher Algebra|HA]] Lemma 7.3.5.14)} A morphism $e$ in an $(\infty,1)$-category $C$ is idempotent (i.e. $e:\Delta^1 \to C$ extends to $Idem$) if and only if there is a homotopy $h : e \sim e^2$ such that $h\circ 1 \sim 1\circ h$.

\end{theorem}
In other words, an incoherent idempotent can be fully coherentified as soon as it admits one additional coherence datum.

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

\begin{itemize}%
\item [[Karoubi envelope]]


\item [[Cauchy completion]]


\item \href{noncommutative+motive#AsUniversalAditiveInvariant}{noncommutative motives -- as universal additive invariants}



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

Section 4.4.5 of

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

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

\begin{itemize}%
\item [[Mike Shulman]], \emph{Idempotents in intensional type theory}, \href{https://arxiv.org/abs/1507.03634}{arXiv:1507.03634}, \href{http://homotopytypetheory.org/2014/12/08/splitting-idempotents/}{blog post}

\end{itemize}
[[!redirects idempotent complete (∞,1)-category]] [[!redirects idempotent-complete (infinity,1)-category]] [[!redirects idempotent-complete (∞,1)-category]]

[[!redirects idempotent-complete quasi-category]] [[!redirects idempotent-complete quasi-categories]]

[[!redirects idempotent-completion of an (∞,1)-category]] [[!redirects idempotent-completion of an (infinity,1)-category]]

[[!redirects idempotent complete (infinity,1)-categories]]



\end{document}