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\begin{document}

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\section*{orthogonal factorization system}

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

\hypertarget{factorization_systems}{}\paragraph*{{Factorization systems}}\label{factorization_systems}

[[!include factorization systems - contents]]

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

[[!include category theory - contents]]

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

\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{orthogonal_factorization_systems}{Orthogonal factorization systems}\dotfill \pageref*{orthogonal_factorization_systems} \linebreak
\noindent\hyperlink{prefactorization_systems}{Prefactorization systems}\dotfill \pageref*{prefactorization_systems} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{closure_properties}{Closure properties}\dotfill \pageref*{closure_properties} \linebreak
\noindent\hyperlink{CancellationProperties}{Cancellation properties}\dotfill \pageref*{CancellationProperties} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \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}

\hypertarget{orthogonal_factorization_systems}{}\subsubsection*{{Orthogonal factorization systems}}\label{orthogonal_factorization_systems}

\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be a [[category]] and let $(E,M)$ be two [[classes]] of [[morphisms]] in $C$. We say that $(E,M)$ is an \textbf{orthogonal factorization system} if $(E,M)$ is a [[weak factorization system]] in which solutions to [[lifting problems]] are unique.

\end{defn}
We spell out several equivalent explicit formulation of what this means.

\begin{defn}
\label{EquivalentDefinitions}\hypertarget{EquivalentDefinitions}{}
$(E,M)$ is an \textbf{orthogonal factorization system} if every morphism $f$ in $C$ factors $f = r\circ \ell$ as a morphism $\ell \in E$ followed by a morphism $r \in M$; and the following equivalent conditions hold

\begin{enumerate}%
\item We have:

a. $E$ is precisely the class of morphisms that are left [[orthogonality|orthogonal]] to every morphism in $M$;

b. $M$ is precisely the class of morphisms that are right [[orthogonality|orthogonal]] to every morphism in $E$.


\item We have:

a. The factorization is unique up to unique [[isomorphism]].

b. $E$ and $M$ both contain all [[isomorphisms]] and are closed under composition.


\item We have:

a. $E$ and $M$ are [[replete subcategories]] of the [[arrow category]] $C^I$.

b. Every morphism in $E$ is left [[orthogonality|orthogonal]] to every morphism in $M$.



\end{enumerate}
\end{defn}
OFS's are traditionally called just \textbf{factorization systems}. See the \emph{[[joyalscatlab:Factorization systems|Catlab]]} for more of the theory.

An orthogonal factorization system is called \textbf{proper} if every morphism in $E$ is an [[epimorphism]] and every morphism in $M$ is a [[monomorphism]].

\hypertarget{prefactorization_systems}{}\subsubsection*{{Prefactorization systems}}\label{prefactorization_systems}

For any class $E$ of morphisms in $C$, we write $E^\perp$ for the class of all morphisms that are right orthogonal to every morphism in $E$. Dually, given $M$ we write ${}^\perp M$ for the class of all morphisms that are left orthogonal to every morphism in $M$. The second condition in the definition of an OFS then says that $E= {}^\perp M$ and $M= E^\perp$.

In general, $(-)^\perp$ and ${}^\perp(-)$ form a [[Galois connection]] on the [[poset]] of classes of morphisms in $C$. A pair $(E,M)$ such that $E= {}^\perp M$ and $M= E^\perp$ is sometimes called a \textbf{prefactorization system}. Note that by generalities about Galois connections, for any class $A$ of maps we have prefactorization systems $({}^\perp(A^\perp),A^\perp)$ and $({}^\perp A, ({}^\perp A)^\perp)$. We call these \emph{generated} and \emph{cogenerated} by $A$, respectively.

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

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

\begin{prop}
\label{}\hypertarget{}{}
The different characterization in def. \ref{EquivalentDefinitions} are indeed all equivalent.

\end{prop}
\begin{proof}
(\ldots{})

For the moment see (\hyperlink{Joyal}{Joyal}).

(\ldots{})

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
A [[weak factorization system]] $(L,R)$ is an orthogonal factorization system precisely if $L \perp R$.

\end{prop}
\begin{proof}
(\ldots{})

For the moment see (\hyperlink{Joyal}{Joyal}).

(\ldots{})

\end{proof}
\begin{prop}
\label{IsomorphismsAreIntersection}\hypertarget{IsomorphismsAreIntersection}{}
For $(L,R)$ an orthogonal factorization system in a category $C$, the intersection $L \cap R$ is precisely the class of [[isomorphisms]] in $C$.

\end{prop}
\begin{proof}
If is clear that every isomorphism is in $L \cap R$. Conversely, let $f : A \to B$ be a morphism in $L \cap R$. This implies that the two trivial factorizations

\begin{displaymath}
f = A \stackrel{id_A}{\to} A \stackrel{f}{\to} B
\end{displaymath}
and

\begin{displaymath}
f = A \stackrel{f}{\to} B \stackrel{id_B}{\to} B
\end{displaymath}
are both $(L,R)$-factorization. Therefore there is a unique morphism $\tilde f$ in the [[commuting diagram]]

\begin{displaymath}
\itexarray{
    A &\stackrel{id_A}{\to}& A
    \\
    \downarrow^{\mathrlap{f}}
     &\nearrow_{\bar f}&
    \downarrow^{\mathrlap{f}}
    \\
    B &\stackrel{id_B}{\to}& B
  }
  \,.
\end{displaymath}
This says precisely that $\bar f$ is a left and right [[inverse]] of $f$.

\end{proof}
\hypertarget{closure_properties}{}\subsubsection*{{Closure properties}}\label{closure_properties}

A prefactorization system $(E,M)$ (and hence, also, a factorization system) satisfies the following closure properties. We state them for $M$, but $E$ of course satisfies the dual property.

\begin{itemize}%
\item $M$ contains the isomorphisms and is closed under composition and [[pullback]] (insofar as pullbacks exist in $C$).
\item If a composite $f g$ is in $M$, and $f$ is either in $M$ or a [[monomorphism]], then $g$ is in $M$.
\item $M$ is closed under all [[limits]] in the [[arrow category]] $Arr(C)$.

\end{itemize}
If $C$ is a [[locally presentable category]], then for any \emph{small set} of maps $A$, the prefactorization system $({}^\perp(A^\perp),A^\perp)$ is actually a factorization system. The argument is by a transfinite construction similar to the [[small object argument]].

On the other hand, if $(E,M)$ is any prefactorization system for which $M$ consists of monomorphisms and $C$ is [[complete category|complete]] and [[well-powered category|well-powered]], then $(E,M)$ is actually a factorization system. (Of course, there is a dual statement as well.) In fact something slightly more general is true; see [[M-complete category]] for this and other related ways to construct factorization systems.

\hypertarget{CancellationProperties}{}\subsubsection*{{Cancellation properties}}\label{CancellationProperties}

\begin{prop}
\label{}\hypertarget{}{}
For $(L,R)$ an orthogonal factorization system. Let

\begin{displaymath}
\itexarray{
    && Y
    \\
    & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
    \\
    X
    &&
    \stackrel{g \circ f}{\to}
    &&
    Z
  }
\end{displaymath}
be two composable morphisms. Then

\begin{itemize}%
\item If $f$ and $g \circ f$ are in $L$, then so is $g$.


\item If $g$ and $g\circ f$ are in $R$, then so is $f$.



\end{itemize}
\end{prop}
\begin{proof}
Consider the first case. The second is directly analogous.

Choose an $(L,R)$-factorization of $g$

\begin{displaymath}
g : Y \stackrel{\ell}{\to} I \stackrel{r}{\to} Z
  \,.
\end{displaymath}
With this we have lifting diagrams of the form

\begin{displaymath}
\itexarray{
    X &\stackrel{g \circ f}{\to}& Z
    \\
    \downarrow^{\mathrlap{f}} &&  \downarrow^{id_Z}
    \\
    Y & \nearrow_r&
    \\
    \downarrow^{\mathrlap{\ell}} && \downarrow^{id_Z}
    \\
    I &\stackrel{r}{\to}& Z
  }
  \;\;\;\;\;\;\;
  \;\;\;\;\;\;\;
  \itexarray{
    X &\stackrel{f}{\to}& Y &\stackrel{\ell}{\to}& I
    \\
    {}^{\mathllap{g \circ f}}\downarrow
     & & \nearrow_{r^{-1}}& &
    \downarrow^{\mathrlap{r}}
    \\
    Z &\underset{id_Z}{\to}& &\underset{id_Z}{\to}& Z 
  }
\end{displaymath}
exhibiting an inverse of $r$. Therefore $r$ is an isomorphism, hence is in $L$, by prop. \ref{IsomorphismsAreIntersection}, hence so is the composite $f = r \circ \ell$.

\end{proof}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

Several classical examples of OFS $(E,M)$:

\begin{itemize}%
\item in any [[topos]] or [[pretopos]], $E$ = class of all epis, $M$ = class of all monos: the [[(epi, mono) factorization system]];


\item more generally, in any [[regular category]], $E$ = class of all [[regular epimorphisms]], $M$ = class of all monos


\item in any [[quasitopos]], $E$ = all epimorphisms, $M$ = all [[strong monomorphisms]]


\item In [[Cat]], $E$ = [[bo functors]], $M$ = [[fully faithful functors]]: the [[bo-ff factorization system]]


\item (Street) in [[Cat]], $E$ = 0-[[final functors]], $M$ = [[discrete fibrations]]


\item (Street) in $\mathrm{Cat}$, $M$ = 0-[[initial functors]], $M$ = [[discrete opfibration]]s


\item in $\mathrm{Cat}$, $M$ = [[conservative functor]]s, $E$ = left orthogonal of $M$ (``iterated strict localizations'' after A. Joyal)


\item in the category of small categories where morphisms are functors which are [[exact functor|left exact]] and have [[right adjoint]]s, $E$ = class of all such functors which are also localizations, $M$ = class of all such functors which are also conservative


\item if $F\to C$ is a [[fibered category]] in the sense of Grothendieck, then $F$ admits a factorization system $(E,M)$ where $E$ = arrows whose projection to $C$ is invertible, $M$ = cartesian arrows in $F$


\item See the (\hyperlink{Joyal}{catlab}) for more examples.



\end{itemize}
\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

There is a [[categorification|categorified]] notion of a [[factorization system on a 2-category]], in which lifts are only required to exist and be unique up to isomorphism. Some examples include:

\begin{itemize}%
\item In [[Cat]], [[(eso, fully faithful) factorization system]]


\item In [[Cat]], [[(eso+full, faithful) factorization system]]



\end{itemize}
Similarly, we can have a [[factorization system in an (∞,1)-category]], and so on; see the links below for other generalizations.

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

\begin{itemize}%
\item [[factorization system]]

\begin{itemize}%
\item [[weak factorization system]]


\item \textbf{orthogonal factorization system}


\item [[reflective factorization system]]


\item [[stable factorization system]]



\end{itemize}

\item [[factorization system in a 2-category]]


\item [[factorization system in an (∞,1)-category]]

\begin{itemize}%
\item [[orthogonal factorization system in an (∞,1)-category]]


\item [[orthogonal factorization system in a derivator]]



\end{itemize}

\item [[factorization structure for sinks]]



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

\begin{itemize}%
\item [[André Joyal]] \emph{[[joyalscatlab:Factorization systems]]}

\end{itemize}
[[!redirects orthogonal factorization system]] [[!redirects orthogonal factorization systems]] [[!redirects orthogonal factorisation system]] [[!redirects orthogonal factorisation systems]] [[!redirects unique factorization system]] [[!redirects unique factorization systems]] [[!redirects unique factorisation system]] [[!redirects unique factorisation systems]] [[!redirects prefactorization system]] [[!redirects prefactorization systems]]



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