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\begin{document}

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\section*{retract}

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

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

[[!include category theory - contents]]

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

[[!include idempotents - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{to_the_point}{To the point}\dotfill \pageref*{to_the_point} \linebreak
\noindent\hyperlink{of_simplices}{Of simplices}\dotfill \pageref*{of_simplices} \linebreak
\noindent\hyperlink{in_arrow_categories}{In arrow categories}\dotfill \pageref*{in_arrow_categories} \linebreak
\noindent\hyperlink{RetractsOfDiagrams}{Retracts of diagrams}\dotfill \pageref*{RetractsOfDiagrams} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

An [[object]] $A$ in a [[category]] is called a \textbf{retract} of an object $B$ if there are [[morphisms]] $i\colon A\to B$ and $r \colon B\to A$ such that $r \circ i = id_A$. In this case $r$ is called a \textbf{retraction} of $B$ onto $A$.

\begin{displaymath}
id 
    \;\colon\; 
  A \underoverset{section}{i}{\to} B \underoverset{retraction}{r}{\to} A
  \,.
\end{displaymath}
Here $i$ may also be called a \emph{[[section]]} of $r$. (In particular if $r$ is thought of as exhibiting a [[bundle]]; the terminology originates from [[topology]].)

Hence a \textbf{retraction} of a [[morphism]] $i  \;\colon\;  A \to B$ is a \emph{left-[[inverse]]}.

In this situation, $r$ is a [[split epimorphism]] and $i$ is a [[split monomorphism]]. The entire situation is said to be a \emph{splitting of the [[idempotent]]}

\begin{displaymath}
B \stackrel{r}{\longrightarrow} A \stackrel{i}{\longrightarrow} B
  \,.
\end{displaymath}
Accordingly, a \textbf{[[split monomorphism]]} is a morphism that \emph{has} a retraction; a \textbf{[[split epimorphism]]} is a morphism that \emph{is} a retraction.

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

\begin{lemma}
\label{LeftInverseWithLeftInverseIsLeftInverse}\hypertarget{LeftInverseWithLeftInverseIsLeftInverse}{}
\textbf{([[left inverse]] with [[left inverse]] is [[inverse]])}

Let $\mathcal{C}$ be a [[category]], and let $f$ and $g$ be [[morphisms]] in $\mathcal{C}$, such that $g$ is a [[left inverse]] to $f$:

\begin{displaymath}
g \circ f = id
  \,.
\end{displaymath}
If $g$ itself has a left inverse $h$

\begin{displaymath}
h \circ g = id
\end{displaymath}
then $h = f$ and $g = f^{-1}$ is an actual (two-sided) [[inverse morphism]] to $f$.

\end{lemma}
\begin{proof}
Since [[inverse morphisms]] are unique if they exists, it is sufficient to show that

\begin{displaymath}
f \circ g = id
  \,.
\end{displaymath}
Compute as follows:

\begin{displaymath}
\begin{aligned}
    f \circ g
    & =  \underset{ = id}{\underbrace{h \circ g}} \circ f \circ g
    \\
    & =
    h \circ \underset{= id}{\underbrace{g \circ f}} \circ g
    \\
    & = h \circ g
    \\
    & = id
  \end{aligned}
\end{displaymath}
\end{proof}
\begin{remark}
\label{RetractsPreservedByFunctor}\hypertarget{RetractsPreservedByFunctor}{}
Retracts are clearly preserved by any [[functor]].

\end{remark}
\begin{remark}
\label{SplitEpisAndMonos}\hypertarget{SplitEpisAndMonos}{}
A [[split epimorphism]] $r; B \to A$ is the strongest of various notions of [[epimorphism]] (e.g., it is a [[regular epimorphism]], in fact an [[abolute limit|absolute]] [[coequalizer]], being the coequalizer of a pair $(e, 1_B)$ where $e = i \circ r: B \to B$ is idempotent). Dually, a [[split monomorphism]] is the strongest of various notions of monomorphism.

\end{remark}
\begin{prop}
\label{RetractOfObjectWithLLP}\hypertarget{RetractOfObjectWithLLP}{}
If an object $B$ has the [[left lifting property]] against a morphism $X \to Y$, then so does every of its retracts $A \to B$:

\begin{displaymath}
\left(
  \itexarray{
     && Y
     \\
     & {}^{\mathllap{\exists}}\nearrow& \downarrow
     \\
     A &\to& Y
  }
  \right)
  \;\;\;\;
  :=
  \;\;\;\;
  \left(    
  \itexarray{
     && && &&  Y
     \\
     &&& {}^{\mathllap{\exists}}\nearrow& && \downarrow
     \\
     A &\to& B &\to& A &\to& Y
  }
  \right)
\end{displaymath}
\end{prop}
\begin{prop}
\label{RetractOfRepresentable}\hypertarget{RetractOfRepresentable}{}
Let $C$ be a [[category]] with [[split idempotent]]s and write $PSh(C) = [C^{op}, Set]$ for its [[presheaf category]]. Then a retract of a [[representable functor]] $F = PSh(C)$ is itself representable.

\end{prop}
This appears as (\hyperlink{Borceux}{Borceux, lemma 6.5.6})

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

\hypertarget{to_the_point}{}\subsubsection*{{To the point}}\label{to_the_point}

\begin{itemize}%
\item In a category with [[terminal object]] $*$ every morphism of the form $* \to X$ is a section, and the unique morphism $X \to *$ is the corresponding retraction.

\end{itemize}
\hypertarget{of_simplices}{}\subsubsection*{{Of simplices}}\label{of_simplices}

The inclusion of standard topological [[horn]]s into the topological [[simplex]] $\Lambda^n_k \hookrightarrow \Delta^n$ is a retract in [[Top]].

\hypertarget{in_arrow_categories}{}\subsubsection*{{In arrow categories}}\label{in_arrow_categories}

Let $\Delta[1] = \{0 \to 1\}$ be the [[interval category]]. For every category $C$ the [[functor category]] $[\Delta[1], C]$ is the [[arrow category]] of $C$.

In the theory of [[weak factorization systems]] and [[model categories]], an important role is played by retracts in $C^{\Delta[1]}$, the [[arrow category]] of $C$. Explicitly spelled out in terms of the original category $C$, a morphism $f:X\to Y$ is a retract of a morphism $g:Z\to W$ if we have commutative squares

\begin{displaymath}
\itexarray{
     id_X \colon & X & \to & Z & \to & X
     \\
     & f \downarrow  & & g \downarrow & & \downarrow f
     \\
     id_Y \colon & Y & \to & W & \to & Y
  }
\end{displaymath}
such that the top and bottom rows compose to identities.

\begin{prop}
\label{RetractsOfMorphismWithLiftingProperty}\hypertarget{RetractsOfMorphismWithLiftingProperty}{}
Classes of morphisms in a category $C$ that are given by a left or right [[lifting property]] are preserved under retracts in the [[arrow category]] $[\Delta[1],C]$. In particular the defining classes of a [[model category]] are closed under retracts.

\end{prop}
This is fairly immediate, a proof is made explicit \href{ClosurePropertiesOfInjectiveAndProjectiveMorphisms}{here}.

This implies:

\begin{prop}
\label{RetractOfIso}\hypertarget{RetractOfIso}{}
In every category $C$ the class of [[isomorphisms]] is preserved under retracts in the [[arrow category]] $[\Delta[1], C]$

\end{prop}
\begin{proof}
This is also checked directly: for

\begin{displaymath}
\itexarray{
    id \colon & a_1 &\to& a_2 &\to& a_1
    \\
    & \downarrow && \downarrow && \downarrow
    \\
    id \colon & b_1 &\to& b_2 &\to& b_1    
  }
\end{displaymath}
a retract diagram and $a_2 \to b_2$ an isomorphism, the inverse to $a_1 \to b_1$ is given by the composite

\begin{displaymath}
\itexarray{
    &  & & a_2 &\to& a_1
    \\
    & && \uparrow && 
    \\
    & b_1 &\to& b_2 &&     
  }
  \,,
\end{displaymath}
where $b_2 \to a_2$ is the inverse of the middle morphism.

\end{proof}
\hypertarget{RetractsOfDiagrams}{}\subsubsection*{{Retracts of diagrams}}\label{RetractsOfDiagrams}

For the following, let $C$ and $J$ be categories and write $J^{\triangleleft}$ for the [[join of quasi-categories|join]] of $J$ with a single [[initial object]], so that [[functor]]s $J^{\triangleleft} \to C$ are precisely [[cone]]s over functors $J \to C$. Write

\begin{displaymath}
i : J \to J^{\triangleleft}
\end{displaymath}
for the canonical inclusion and hence $i^* F$ for the underlying diagram of a cone $F : J^{\triangleleft} \to C$. Finally, write $[J^{\triangleleft}, C]$ for the [[functor category]].

\begin{prop}
\label{RetractsOfLimits}\hypertarget{RetractsOfLimits}{}
If $Id: F_1 \hookrightarrow F_2 \to F_1$ is a [[retract]] in the category $[J^{\triangleleft}, C]$ and $F_2 : J^{\triangleleft} \to C$ is a [[limit]] [[cone]] over the [[diagram]] $i^* F_2 : J \to C$, then also $F_1$ is a limit cone over $i^* F_1$.

\end{prop}
\begin{proof}
We give a direct and a more abstract argument.

\textbf{Direct argument}. We can directly check the [[universal property]] of the limit: for $G$ any other [[cone]] over $i^* F_1$, the composite $i^* G = i^* F_1 \to i^* F_2$ exhibits $G$ also as a cone over $i^* F_2$. By the pullback property of $F_2$ this extends to a morphism of cones $G \to F_2$. Postcomposition with $F_2 \to F_1$ makes this a morphism of cones $G \to F_1$. By the injectivity of $F_1 \to F_2$ and the universality of $F_2$, any two such cone morphisms are equals.

\textbf{More abstract argument}. The limiting cone over a diagram $D : J \to C$ may be regarded as the right [[Kan extension]] $i_* D := Ran_i D$ along $i$

\begin{displaymath}
\itexarray{   
    J &\stackrel{D}{\to}& C
    \\
    {}^{\mathllap{i}}\downarrow & \nearrow_{i_* D}
    \\
    J^{\triangleleft}
  }
  \,.
\end{displaymath}
Therefore a cone $F : J^{\triangleleft} \to C$ is limiting precisely if the $(i^* \dashv i_*)$-[[unit of an adjunction|unit]]

\begin{displaymath}
F \stackrel{}{\to} i_* i^* F
\end{displaymath}
is an [[isomorphism]]. Since this unit is a [[natural transformation]] it follows that applied to the retract diagram

\begin{displaymath}
Id : F_1 \hookrightarrow F_2 \to F_1
\end{displaymath}
it yields the retract diagram

\begin{displaymath}
\itexarray{
    Id : & F_1 &\to& F_2 &\to& F_1
    \\ 
    & \downarrow && \downarrow && \downarrow
    \\
    Id : & i_* i^* F_1 &\to& i_* i^* F_2 &\to& i_* i^* F_1
  }
\end{displaymath}
in $[\Delta[1], [J^{\triangleleft}, C]]$. Here by assumption the middle morphism is an isomorphism. Since isomorphisms are stable under retract, by prop. \ref{RetractOfIso}, also the left and right vertical morphism is an isomorphism, hence also $F_1$ is a limiting cone.

\end{proof}
This argument generalizes form limits to [[homotopy limit]]s.

For that, let now $C$ be a [[category with weak equivalences]] and write $Ho(C) : Diagram^{op} \to Cat$ for the corresponding [[derivator]]: $Ho(C)(J) := [J,C](W^J)^{-1}$ is the [[homotopy category]] of $J$-diamgrams in $C$, with respect to the degreewise weak equivalences in $C$.

\begin{cor}
\label{RetractOfHomotopyLimits}\hypertarget{RetractOfHomotopyLimits}{}
Let

\begin{displaymath}
Id : F_1 \to F_2 \to F_1
\end{displaymath}
be a retract in $Ho(C)(J^{\triangleleft})$. If $F_2$ is a [[homotopy limit]] cone over $i^* F_2$, then also $F_1$ is a homotopy limit cone over $i^* F_1$.

\end{cor}
\begin{proof}
By the discussion at [[derivator]] we have that

\begin{enumerate}%
\item $i_* : Ho(C)(J) \to Ho(C)(J^{\triangleleft})$ forms [[homotopy limit]] cones;


\item $F \to i_* i^* F$ is an isomorphism precisely if $F$ is a homotopy limit cone.



\end{enumerate}
With this the claim follows as in prop. \ref{RetractsOfLimits}.

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

\begin{itemize}%
\item [[deformation retract]], [[neighbourhood retract]]


\item [[absolute retract]]



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

\begin{itemize}%
\item [[Karol Borsuk]], \emph{Theory of retracts}


\item [[Sibe Mardešić]], \emph{Absolute Neighborhood Retracts and Shape Theory} (\href{https://www.maths.ed.ac.uk/~v1ranick/papers/mardesic.pdf}{pdf})


\item [[Francis Borceux]], Def. 1.7.3 and Sec. 6.5 of: \emph{[[Handbook of Categorical Algebra]] I}



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Retract}{Retract}}

\end{itemize}
[[!redirects retracts]]

[[!redirects retraction]] [[!redirects retractions]]

[[!redirects left inverse]] [[!redirects left inverses]] [[!redirects left-inverse]] [[!redirects left-inverses]]



\end{document}