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\begin{document}

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\section*{factorization system in a 2-category}

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

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

[[!include factorization systems - contents]]

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{catenriched_factorization_systems}{Cat-enriched factorization systems}\dotfill \pageref*{catenriched_factorization_systems} \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}

In a (weak) [[2-category]], the appropriate notion of an [[orthogonal factorization system]] is suitably weakened up to isomorphism. Specifically, a factorization system in a 2-category $K$ consists of two classes $(E,M)$ of 1-morphisms in $K$ such that:

\begin{enumerate}%
\item Every 1-morphism $f:x\to y$ in $K$ is \emph{isomorphic} to a composite $m\circ e$ where $e\in E$ and $m\in M$, and


\item For any $e:a\to b$ in $E$ and $m:x\to y$ in $M$, the following square

\begin{displaymath}
\itexarray{ K(b,x) & \to & K(b,y) \\
\downarrow & \cong & \downarrow \\
K(a,x) & \to & K(a,y)}
\end{displaymath}
(which commutes up to isomorphism) is a [[2-pullback]] in $Cat$.



\end{enumerate}
This second property is a ``2-categorical orthogonality.'' In particular, it implies that any square

\begin{displaymath}
\itexarray{a & \to & x \\
^e\downarrow & \cong & \downarrow^m \\
b & \to & y}
\end{displaymath}
which commutes up to specified isomorphism, where $e\in E$ and $m\in M$, has a diagonal filler $b\to x$ making both triangles commute up to isomorphisms that are coherent with the given one. It also implies an additional factorization property for 2-cells.

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

\begin{itemize}%
\item The following are all factorization systems in the 2-category $Cat$. Many of them have analogues in more general 2-categories.

\begin{itemize}%
\item $E=$ [[essentially surjective functors]], $M=$ [[fully faithful functors]]. This is the ``ur-example,'' and it generalizes to [[enriched category theory]], [[internal category]] theory, etc. See [[(eso, fully faithful) factorization system]].


\item $E=$ functors $e\colon a\to b$ such that every object of $b$ is a [[retract]] of an object in the image of $a$, and $M=$ fully faithful functors whose image is closed under retracts.


\item $E=$ [[essentially surjective functor|essentially surjective]] and [[full functors]], $M=$ [[faithful functors]]. See [[(eso+full, faithful) factorization system]].


\item $E=$ (possibly [[transfinite compositions|transfinite]]) composites of [[localizations]], $M=$ [[conservative functors]].



\end{itemize}

\item The 2-category [[Topos]] admits several interesting factorization systems.

\begin{itemize}%
\item The [[(geometric surjection, embedding) factorization system]].


\item $E=$ [[hyperconnected geometric morphisms]], $M=$ [[localic geometric morphisms]].



\end{itemize}

\item The [[monadic decomposition]] is a factorization system on a suitable 2-category.



\end{itemize}
\hypertarget{catenriched_factorization_systems}{}\subsection*{{Cat-enriched factorization systems}}\label{catenriched_factorization_systems}

If instead $K$ is a [[strict 2-category]] and we require that

\begin{enumerate}%
\item Every 1-morphism in $K$ is \emph{equal} to a composite of a morphism in $E$ and a morphism in $M$, and


\item The above square (which commutes strictly when $K$ is a strict 2-category) is a strict 2-pullback (i.e. a $Cat$-enriched pullback).



\end{enumerate}
then we obtain the notion of a $Cat$-enriched, or strict 2-categorical, factorization system.

It is important to note that in general, the strict and weak notions of 2-categorical factorization system are incomparable; neither is a special case of the other. For example, on $Cat$ there is a weak 2-categorical factorization system where $E=$ [[essentially surjective functors]] and $M=$ [[fully faithful functors]], and a strict 2-categorical factorization system where $E=$ [[bijective on objects functors]] and $M=$ [[fully faithful functors]].

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

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

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


\item [[orthogonal factorization system]]



\end{itemize}

\item \textbf{factorization system in a 2-category}


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

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

\end{itemize}

\item Factorization systems in a 2-category play an important role in the construction of a [[proarrow equipment]] out of [[codiscrete cofibrations]].


\item Combining the (eso,ff) and (eso+full, faithful) factorization systems into a [[ternary factorization system]] has connections with the theory of [[stuff, structure, property]].



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

For instance

\begin{itemize}%
\item [[Mathieu Dupont]], [[Enrico Vitale]], \emph{Proper factorization systems in 2-categories} (\href{http://breckes.org/dokumenty/propfactsys.pdf}{pdf})

\end{itemize}
[[!redirects factorization system on a 2-category]] [[!redirects factorization systems in a 2-category]] [[!redirects factorization systems on a 2-category]] [[!redirects factorization systems in 2-categories]] [[!redirects factorization systems on 2-categories]] [[!redirects 2-categorical factorization system]] [[!redirects 2-categorical factorization systems]]



\end{document}