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\section*{homotopy factorization system}

\hypertarget{homotopy_factorization_systems}{}\section*{{Homotopy factorization systems}}\label{homotopy_factorization_systems}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{the_enriched_case}{The enriched case}\dotfill \pageref*{the_enriched_case} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{the_unenriched_case}{The unenriched case}\dotfill \pageref*{the_unenriched_case} \linebreak
\noindent\hyperlink{definition_2}{Definition}\dotfill \pageref*{definition_2} \linebreak
\noindent\hyperlink{relation_between_definitions}{Relation between definitions}\dotfill \pageref*{relation_between_definitions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{homotopy factorization system} in a [[model category]] is a presentation of an [[orthogonal factorization system in an (∞,1)-category|orthogonal factorization system]] in its [[simplicial localization|underling]] [[(∞,1)-category]].

\hypertarget{the_enriched_case}{}\subsection*{{The enriched case}}\label{the_enriched_case}

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

Let $V$ be a [[monoidal model category]] (with cofibrant unit object) and $M$ a $V$-[[enriched model category]]. In the case $V =$ [[SSet]], the following definition is found in \hyperlink{Bousfield}{Bousfield, section 6}.

$\backslash$begin\{definition\} A \textbf{homotopy factorization system} in $M$ is a pair $(L,R)$ of classes of maps such that:

\begin{enumerate}%
\item Every map in $L$ is a cofibration, and every map in $R$ is a fibration.
\item If $i:A\to B$ is in $L$ and $p:X\to Y$ is in $R$, then the induced [[pullback power]]\begin{displaymath}
[i,p] : [B,X] \to [A,X] \times_{[A,Y]} [B,Y]
\end{displaymath}
is an acyclic fibration in $V$.


\item Every morphism in $M$ factors as a map in $L$ followed by a map in $R$.
\item $L$ and $R$ are closed under [[retracts]].

\end{enumerate}
$\backslash$end\{definition\}

\hypertarget{remarks}{}\subsubsection*{{Remarks}}\label{remarks}

It follows that $(L,R)$ is in fact a [[weak factorization system]]. For on the one hand; the underlying-set functor $V(I,-) : V\to Set$ takes acyclic fibrations to surjections since $I$ is cofibrant; thus $(L,R)$ have the lifting property. And on the other hand, if $i$ has the left lifting property against $R$, then factoring it and applying the retract argument implies $i\in L$, and dually.

Note that $[i,p]$ is automatically a fibration, since $i$ is a cofibration and $p$ a fibration; thus the content of assertion (2) is that this map is a weak equivalence. If $A$ (hence also $B$) is cofibrant and $Y$ (hence also $X$) is fibrant, then the pullback $[A,X] \times_{[A,Y]} [B,Y]$ is pullback of two fibrations between fibrant objects and thus a [[homotopy pullback]]; thus in this case the condition is equivalent to asking that the square $\backslash$begin\{center\} $\backslash$begin\{tikzcd\} \{\}B,X $\backslash$arr $\backslash$ard \& \{\}A,X $\backslash$ard$\backslash$ \{\}B,Y $\backslash$arr \& \{\}A,Y $\backslash$end\{tikzcd\} $\backslash$end\{center\} is a homotopy pullback square. If $V$ is [[right proper]], then the condition for the pullback to be a homotopy pullback can be weakened to ``$A$ (hence also $B$) is cofibrant OR $Y$ (hence also $X$) is fibrant'', and thus becomes automatic if either all objects of $M$ are fibrant or all objects of $M$ are cofibrant.

\hypertarget{the_unenriched_case}{}\subsection*{{The unenriched case}}\label{the_unenriched_case}

\hypertarget{definition_2}{}\subsubsection*{{Definition}}\label{definition_2}

A hierarchy of notions of ``homotopy factorization system'' for unenriched model categories can be found in \hyperlink{Joyal}{Joyal, Appendix F}. Let $M$ be a model category and $(L,R)$ a pair of classes of maps. Write $C$ for the class of cofibrations, $F$ for the class of fibrations, $M_{c}$ for the subcategory of cofibrant objects, $M_f$ for the subcategory of fibrant objects, $M_{c f}$ for the subcategory of fibrant and cofibrant objects, $C_{c f}$ for the class of cofibrations between fibrant and cofibrant objects, etc.

$\backslash$begin\{definition\}

\begin{itemize}%
\item $(L,R)$ is a \textbf{weak homotopy factorization system} if\begin{enumerate}%
\item $L$ and $R$ are closed under weak equivalence in the [[arrow category]] $M^\to$, and
\item $(L\cap C_{c f}, R\cap F_{c f})$ is a [[weak factorization system]] in $M_{c f}$.

\end{enumerate}

\item $(L,R)$ is a \textbf{homotopy factorization system} if it is a weak homotopy factorization system and in addition\begin{enumerate}%
\item If $f\in L$ and $g f\in L$, then $g\in L$.
\item If $g\in R$ and $g f\in R$, then $f\in R$.

\end{enumerate}

\item $(L,R)$ is a \textbf{strong homotopy factorization system} if it is a homotopy factorization system and in addition\begin{enumerate}%
\item $(L\cap C, R\cap F)$ is a weak factorization system in $M$.

\end{enumerate}


\end{itemize}
$\backslash$end\{definition\}

\hypertarget{relation_between_definitions}{}\subsection*{{Relation between definitions}}\label{relation_between_definitions}

The relation between the enriched and unenriched notions is unclear to the author of this page, but here are some things that can be said.

$\backslash$begin\{proposition\} Suppose either every object of $M$ is fibrant and cofibrant, or $V$ is right proper and either every object of $M$ is fibrant or every object of $M$ is cofibrant. Then given an enriched hfs, by closing $L$ and $R$ under weak equivalence in $M^\to$ we obtain an unenriched weak hfs $(L',R')$. $\backslash$end\{proposition\} $\backslash$begin\{proof\} Since $(L,R)$ is a wfs, to show that $(L',R')$ is an unenriched weak hfs, it suffices to show that $L'\cap C_{c f} = L \cap C_{c f}$ and dually. Note that any morphism in $C_{c f}$ is both cofibrant and fibrant in the Reedy model structure on $M^\to$; hence if two such morphisms are weakly equivalent in $M^\to$, there is a single weak equivalence relating them. But the property of being a homotopy pullback square is preserved under weak equivalence; and under the given hypotheses, as remarked above, the homotopy lifting property can be expressed in terms of such a square, and is thus preserved by weak equivalences between cofibrations. The proof for $R$ is dual (using the other Reedy model structure). $\backslash$end\{proof\}

It is unclear whether or under what conditions this weak hfs is a hfs or a strong hfs.

In the converse direction, the following are proven by \hyperlink{Joyal}{Joyal}:

\begin{itemize}%
\item An unenriched weak hfs $(L,R)$ is determined by $(L\cap C_{c f}, R\cap F_{c f})$ (called its \textbf{center}).
\item If $(L,R)$ is an unenriched weak hfs, then $L\cap C_C$ has the left lifting property against $R\cap F_f$.
\item If $(L,R)$ is an unenriched weak hfs, then every morphism from a cofibrant object to a fibrant one factors as a map in $L\cap C_c$ followed by one in $R\cap F_f$.

\end{itemize}
$\backslash$begin\{proposition\} If $(L,R))$ is an unenriched weak hfs, the following are equivalent:

\begin{enumerate}%
\item If $f\in L$ and $g f\in L$, then $g\in L$.
\item If $g\in R$ and $g f\in R$, then $f\in R$.
\item The codiagonal of any map in $L\cap C_c$ belongs to $L$.
\item The diagonal of any map in $R\cap F_f$ belongs to $R$.
\item if $f\in L\cap C_{c f}$ and $g f \in L\cap C_{c f}$ and $g\in C$, then $g\in L$.
\item if $g\in R\cap F_{c f}$ and $g f \in R\cap F_{c f}$ and $f\in F$, then $f\in R$.

\end{enumerate}
$\backslash$end\{proposition\}

The closure under diagonals and codiagonals suggests that some kind of homotopy orthogonality should exist, using [[simplicial resolutions]] rather than enrichment.

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

\begin{itemize}%
\item [[A. K. Bousfield]], \emph{Constructions of factorization systems in categories}, Journal of Pure and Applied Algebra 9 (1977) 207-220, \href{http://web.math.rochester.edu/people/faculty/doug/otherpapers/Bousfield_Fact.pdf}{pdf}


\item [[Andre Joyal]], \emph{Notes on quasi-categories}, \href{http://www.math.uchicago.edu/~may/IMA/Joyal.pdf}{pdf}



\end{itemize}
[[!redirects homotopy factorization system]] [[!redirects homotopy factorization systems]] [[!redirects weak homotopy factorization system]] [[!redirects weak homotopy factorization systems]] [[!redirects strong homotopy factorization system]] [[!redirects strong homotopy factorization systems]] [[!redirects homotopy factorization system in a model category]] [[!redirects homotopy factorization systems in a model category]] [[!redirects factorization system in a model category]] [[!redirects factorization systems in a model category]] [[!redirects orthogonal factorization system in a model category]] [[!redirects orthogonal factorization systems in a model category]]



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