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\begin{document}

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\section*{functorial factorization}

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

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

[[!include factorization systems - 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{concrete_explicit}{Concrete explicit}\dotfill \pageref*{concrete_explicit} \linebreak
\noindent\hyperlink{abstract_version}{Abstract version}\dotfill \pageref*{abstract_version} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{enriched_functorial_factorizations}{Enriched functorial factorizations}\dotfill \pageref*{enriched_functorial_factorizations} \linebreak
\noindent\hyperlink{equivalence_to_pointed_endofunctors}{Equivalence to pointed endofunctors}\dotfill \pageref*{equivalence_to_pointed_endofunctors} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{functorial factorization} is a structure on a category that factors any morphism into a composite of two morphisms, in a way that depends functorially on commutative squares.

Functorial factorizations play a prominent role in [[model category]] theory. On the one hand, many constructions there do rely on the factorizations into (acyclic) [[cofibrations]] and (acyclic) [[fibrations]] to be functorial, while on the other hand via the [[small object argument]] many examples of model categories do in fact carry a functorial factorization. (As a result, some authors include functorial factorization in the axioms of a model category right away.)

Functorial factorizations also play an important role as an ingredient in [[algebraic weak factorization systems]].

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

\hypertarget{concrete_explicit}{}\subsubsection*{{Concrete explicit}}\label{concrete_explicit}

\begin{udefn}
A \textbf{functorial factorization} on a category $\mathcal{C}$ is a way of assigning to any arrow $f$ in $\mathcal{C}$ a pair of composable arrows $f_L, f_R$ such that $f = f_R \circ f_L$, together with for any [[commuting square]]

\begin{displaymath}
\itexarray{
    & \overset{h}{\longrightarrow}
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g}}
    \\
    & \overset{k}{\longrightarrow}
  }
\end{displaymath}
a morphism $E(h,k)$ completing their factorizations $f = f_R \circ f_L$ and $g = g_R \circ g_L$ to a further [[commuting diagram]]

\begin{displaymath}
\itexarray{
    & \overset{h}{\longrightarrow}
    \\
    {}^{\mathllap{f_L}}\downarrow && \downarrow^{\mathrlap{g_L}}
    \\
    & \overset{E(h,k)}{\longrightarrow}
    \\
    {}^{\mathllap{f_R}}\downarrow && \downarrow^{\mathrlap{g_R}}
    \\
    & \overset{k}{\longrightarrow}
  }
  \,,
\end{displaymath}
in a way that depends functorially on the given commutative square, i.e. $E(h_1\circ h_2,k_1\circ k_2) = E(h_1,k_1) \circ E(h_2,k_2)$.

\end{udefn}
\hypertarget{abstract_version}{}\subsubsection*{{Abstract version}}\label{abstract_version}

Write $\Delta[1] = \{0 \to 1\}$ and $\Delta[2] = \{0 \to 1 \to 2\}$ for the [[ordinal numbers]], regarded as [[posets]] and hence as [[categories]]. The [[arrow category]] $Arr(\mathcal{C})$ is equivalently the [[functor category]] $\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C})$, while $\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C})$ has as objects pairs of composable morphisms in $\mathcal{C}$. There are three injective functors $\delta_i \colon [1] \rightarrow [2]$, where $\delta_i$ omits the index $i$ in its image. By precomposition, this induces [[functors]] $d_i  \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}$. Here

\begin{itemize}%
\item $d_1$ sends a pair of composable morphisms to their [[composition]];


\item $d_2$ sends a pair of composable morphisms to the first morphism;


\item $d_0$ sends a pair of composable morphisms to the second morphism.



\end{itemize}
\begin{defn}
\label{FunctorialFactorization}\hypertarget{FunctorialFactorization}{}
For $\mathcal{C}$ a [[category]], a \textbf{functorial factorization} of the morphisms in $\mathcal{C}$ is a [[functor]]

\begin{displaymath}
fact \;\colon\; \mathcal{C}^{\Delta[1]} \longrightarrow \mathcal{C}^{\Delta[2]}
\end{displaymath}
which is a [[section]] of the [[composition]] functor $d_1 \;\colon\; \mathcal{C}^{\Delta[2]}\to \mathcal{C}^{\Delta[1]}$.

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

\begin{itemize}%
\item Not all [[weak factorization systems]] are functorial, although most are. This includes those produced by the [[small object argument]], with due care, and also all [[algebraic weak factorization systems]].


\item All [[orthogonal factorization systems]] are automatically functorial.



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

\hypertarget{enriched_functorial_factorizations}{}\subsubsection*{{Enriched functorial factorizations}}\label{enriched_functorial_factorizations}

Sufficient conditions in [[enriched category theory]] (in particular [[enriched model category]]-theory) for functorial factorization to exist as an [[enriched functor]] is discussed in \hyperlink{Hirschhorn02}{Hirschhorn 02, Theorem 4.3.8}, \hyperlink{Shulman06}{Shulman 06, Prop. 24.2}

\hypertarget{equivalence_to_pointed_endofunctors}{}\subsubsection*{{Equivalence to pointed endofunctors}}\label{equivalence_to_pointed_endofunctors}

\begin{uprop}
The following are equivalent:

\begin{enumerate}%
\item A functorial factorization on $\mathcal{C}$.
\item A [[pointed endofunctor]] $R$ of $\mathcal{C}^{\Delta[1]}$ such that $cod : \mathcal{C}^{\Delta[1]} \to \mathcal{C}$ is a strict morphism of pointed endofunctors from $R$ to $Id_{\mathcal{C}}$, i.e. $cod \circ R = cod$ and $cod$ maps the point $Id_{\mathcal{C}^{\Delta[1]}}\to R$ to the identity map of $Id_{\mathcal{C}}$. (This is called a \textbf{pointed endofunctor over $cod$}.)
\item Dually, a copointed endofunctor $L$ of $\mathcal{C}^{\Delta[1]}$ under $dom$.

\end{enumerate}
\end{uprop}
\begin{proof}
A endofunctor $R$ of $\mathcal{C}^{\Delta[1]}$ maps every morphism $f$ in $\mathcal{C}$ to a morphism $f_R$, in a functorial way. A point of such an endofunctor assigns to each $f$ a pair of morphisms $f_L, f_M$ such that $f_R \circ f_L = f_M\circ f$, naturally with respect to commutative squares. To say $cod \circ R = cod$ means that $f_R$ has the same codomain as $f$, and to say that $cod$ respects the points says that $f_M$ is an identity. Thus, $f = f_R \circ f_L$ is a factorization. For functoriality, the functoriality of $R$ gives the commutative square $k \circ f = f_R \circ E(h,k)$ and the functoriality of $E(-,-)$, while the naturality of $(-)_L$ gives the commutative square $E(h,k) \circ f = f_L \circ h$. The converse and dual are straightforward.

\end{proof}
An [[algebraic weak factorization system]] is a functorial factorization together with compatible enhancements of these endofunctors to a monad and comonad. This can often be detected with the help of a [[composition law for factorizations]].

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

\begin{itemize}%
\item [[Philip Hirschhorn]], \emph{Model Categories and Their Localizations}, AMS Math. Survey and Monographs Vol 99 (2002) (\href{http://www.ams.org/bookstore?fn=20&arg1=whatsnew&item=SURV-99}{AMS}, \href{http://www.gbv.de/dms/goettingen/360115845.pdf}{pdf toc}, \href{http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf}{pdf})


\item [[Michael Shulman]], \emph{Homotopy limits and colimits and enriched homotopy theory} (\href{https://arxiv.org/abs/math/0610194}{arXiv:math/0610194})



\end{itemize}
[[!redirects functorial factorizations]]

[[!redirects functorial factorization system]] [[!redirects functorial factorization systems]]



\end{document}