\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{composition law for factorizations}

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

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

[[!include factorization systems - contents]]

\hypertarget{composition_laws_for_factorizations}{}\section*{{Composition laws for factorizations}}\label{composition_laws_for_factorizations}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{relation_to_awfs}{Relation to awfs}\dotfill \pageref*{relation_to_awfs} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{composition law} for a [[functorial factorization]] is a functorial way to compose lifting structures for its algebraic right maps. A functorial factorization whose pointed endofunctor extends to a monad over $cod$ is an [[algebraic weak factorization system]] if and only if it has a composition law. Moreover, subject to smallness conditions, any functorial factorization with a composition law freely generates an algebraic weak factorization system.

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

Suppose $\mathcal{C}$ is a category equipped with a [[functorial factorization]], sending every arrow $f:A\to B$ to a factorization $A \xrightarrow{f_L} E f \xrightarrow{f_R} B$. As noted at [[functorial factorization]], a functorial factorization is equivalent to a pointed endofunctor $R$ on $\mathcal{C}^{\mathbf{2}}$ over $cod$, which maps each morphism $f$ (regarded as an object of the [[arrow category]] $\mathcal{C}^{\mathbf{2}}$) to its right factor $f_R$, the point being given by the left factor $f_L$ and the identity:

\begin{displaymath}
\itexarray{ & \xrightarrow{f_L} & \\ ^f \downarrow & & \downarrow^{f_R} \\ & \xrightarrow{id}&. }
\end{displaymath}
As with any pointed endofunctor, we can consider the category of [[algebra over a pointed endofunctor|algebras]] for $R$. Such an $R$-algebra is an arrow $f:A\to B$ equipped with a map $s : E f \to A$ such that $s \circ f_L = id_A$ and $f \circ s = f_R$. Equivalently, it is a diagonal lifting in the square

\begin{displaymath}
\itexarray{ A & \xrightarrow{id} & A \\ ^{f_L}\downarrow & & \downarrow^{f} \\ E f & \xrightarrow{f_R} & B.}
\end{displaymath}
In particular, this means that if $(L,R)$ is a factorization for a [[weak factorization system]], then the arrows of $\mathcal{C}$ that admit some structure of $R$-algebra are precisely those in the right class of the weak factorization system.

The morphisms of $R$-algebras are commuting squares $g \circ h = k \circ f$ that additionally commute with the actions, i.e. $h \circ s_f = s_g \circ E(h,k)$. This defines a category $R Alg$ with a forgetful functor $U : R Alg \to \mathcal{C}^{\mathbf{2}}$.

If it should happen that the pointed endofunctor $R$ is actually a [[monad]] over $cod$, i.e. it also has a multiplication $R R \to R$ that is also the identity on codomains, then we can also consider the smaller category $\mathbb{R} Alg$ of [[algebra over a monad|monad algebras]], the $R$-algebras as above such that $s$ also satisfies an associativity condition.

\begin{udefn}
A \textbf{right weak composition law} for a functorial factorization is a functor $R Alg \times_{\mathcal{C}} R Alg \to R Alg$, where the pullback is over $dom \circ U : R Alg \to \mathcal{C}$ and $cod \circ U : R Alg \to \mathcal{C}$, lying over the composition functor $\mathcal{C}^{\mathbf{2}} \times \mathcal{C}^{\mathbf{2}}\to \mathcal{C}$. If $R$ is a monad over $cod$, then a \textbf{right strong composition law} is defined analogously using $\mathbb{R} Alg$ instead.

\end{udefn}
More explicitly, this means that

\begin{enumerate}%
\item whenever $(f, s)$ and $(g, t)$ are $R$-algebras (resp. $\mathbb{R}$-algebras) such that $cod(f) = dom(g)$, we have a specified $R$-algebra structure (resp. $\mathbb{R}$-algebra structure) $t \bullet s$ for $g f$, such that
\item for any morphisms of $R$-algebras (resp. $\mathbb{R}$-algebras) $(u, v) : (f, s) \to (f' , s')$ and $(v, w) : (g, t) \to (g' , t')$ between composable pairs $(f, s), (g, t)$ and $(f' , s' ),(g' , t' )$, the pasted square $(u, w) : (g f, t \bullet s) \to (g ' f' , t' \bullet s ' )$ is also a map of $R$-algebras (resp. $\mathbb{R}$-algebras).

\end{enumerate}
In other words, a composition law is an operation with the requisite shape to be the vertical composition in a [[double category]] whose vertical arrows are algebras, whose horizontal arrows are arbitrary arrows, and whose 2-cells are commutative squares. Associativity is not assumed, but as noted below it often comes for free.

\hypertarget{relation_to_awfs}{}\subsection*{{Relation to awfs}}\label{relation_to_awfs}

\begin{utheorem}
Suppose $(L,R)$ is a functorial factorization whose underlying pointed endofunctor $R$ over $cod$ has the structure of a [[monad]] on $\mathcal{C}^{\mathbf{2}}$ over $cod$. Then $(L,R)$ is an [[algebraic weak factorization system]] if and only if it admits a right strong composition law.

\end{utheorem}
\begin{proof}
See \hyperlink{Garner09}{Garner 09}, \hyperlink{Garner10}{Garner 10}, \hyperlink{Riehl11}{Riehl 11}, and \hyperlink{BR13}{Barthel-Riehl 13} for proofs in varying degrees of explicitness.

\end{proof}
\begin{utheorem}
Suppose $(L,R)$ is a functorial factorization with a right weak composition law, and that the [[algebraically free monad]] on the pointed endofunctor $R$ exists and is over $cod$ (for instance if it can be constructed by a [[transfinite construction of free algebras]]). Then the latter monad has a right strong composition law, hence underlies an algebraic weak factorization system whose right-monad-algebras coincide with the $R$-pointed-endofunctor-algebras, including their natural composition law.

\end{utheorem}
\begin{proof}
By definition, the algebraically-free monad $\mathbb{F}(R)$ satisfies $\mathbb{F}(R) Alg = R Alg$. Thus, the weak composition law for $R$ extends to a strong one for $\mathbb{F}(R)$; now we can apply the previous theorem.

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

\begin{itemize}%
\item [[Richard Garner]]. \emph{Understanding the small object argument}. Appl. Categ. Structures. 17(3) (2009) 247--285.


\item [[Richard Garner]]. \emph{Homomorphisms of higher categories}. Adv. Math. 224(6) (2010), 2269--2311.


\item [[Emily Riehl]]. \emph{Algebraic model structures}. New York J. Math. 17 (2011) 173-231.


\item [[Tobias Barthel]] and [[Emily Riehl]]. \emph{On the construction of functorial factorizations for model categories}. Algebr. Geom. Topol. Volume 13, Number 2 (2013), 1089-1124. \href{https://projecteuclid.org/euclid.agt/1513715550}{projecteuclid}



\end{itemize}
[[!redirects composition law for factorization]] [[!redirects composition law for factorizations]] [[!redirects composition laws for factorization]] [[!redirects composition laws for factorizations]]

[[!redirects composition law for a functorial factorization]] [[!redirects composition law for functorial factorizations]] [[!redirects composition laws for a functorial factorization]] [[!redirects composition laws for functorial factorizations]]

[[!redirects composition law for a factorization system]] [[!redirects composition law for factorization systems]] [[!redirects composition laws for a factorization system]] [[!redirects composition laws for factorization systems]]

[[!redirects composition law for a weak factorization system]] [[!redirects composition law for weak factorization systems]] [[!redirects composition laws for a weak factorization system]] [[!redirects composition laws for weak factorization systems]]

[[!redirects composition law for an algebraic weak factorization system]] [[!redirects composition law for algebraic weak factorization systems]] [[!redirects composition laws for an algebraic weak factorization system]] [[!redirects composition laws for algebraic weak factorization systems]]

[[!redirects composition law for wfs]] [[!redirects composition laws for wfs]] [[!redirects composition law for awfs]] [[!redirects composition laws for awfs]]

[[!redirects weak composition law for factorization]] [[!redirects weak composition law for factorizations]] [[!redirects weak composition laws for factorization]] [[!redirects weak composition laws for factorizations]]

[[!redirects weak composition law for a functorial factorization]] [[!redirects weak composition law for functorial factorizations]] [[!redirects weak composition laws for a functorial factorization]] [[!redirects weak composition laws for functorial factorizations]]

[[!redirects weak composition law for a factorization system]] [[!redirects weak composition law for factorization systems]] [[!redirects weak composition laws for a factorization system]] [[!redirects weak composition laws for factorization systems]]

[[!redirects strong composition law for factorization]] [[!redirects strong composition law for factorizations]] [[!redirects strong composition laws for factorization]] [[!redirects strong composition laws for factorizations]]

[[!redirects strong composition law for a functorial factorization]] [[!redirects strong composition law for functorial factorizations]] [[!redirects strong composition laws for a functorial factorization]] [[!redirects strong composition laws for functorial factorizations]]

[[!redirects strong composition law for a factorization system]] [[!redirects strong composition law for factorization systems]] [[!redirects strong composition laws for a factorization system]] [[!redirects strong composition laws for factorization systems]]

[[!redirects right weak composition law for factorization]] [[!redirects right weak composition law for factorizations]] [[!redirects right weak composition laws for factorization]] [[!redirects right weak composition laws for factorizations]]

[[!redirects right weak composition law for a functorial factorization]] [[!redirects right weak composition law for functorial factorizations]] [[!redirects right weak composition laws for a functorial factorization]] [[!redirects right weak composition laws for functorial factorizations]]

[[!redirects right weak composition law for a factorization system]] [[!redirects right weak composition law for factorization systems]] [[!redirects right weak composition laws for a factorization system]] [[!redirects right weak composition laws for factorization systems]]

[[!redirects right strong composition law for factorization]] [[!redirects right strong composition law for factorizations]] [[!redirects right strong composition laws for factorization]] [[!redirects right strong composition laws for factorizations]]

[[!redirects right strong composition law for a functorial factorization]] [[!redirects right strong composition law for functorial factorizations]] [[!redirects right strong composition laws for a functorial factorization]] [[!redirects right strong composition laws for functorial factorizations]]

[[!redirects right strong composition law for a factorization system]] [[!redirects right strong composition law for factorization systems]] [[!redirects right strong composition laws for a factorization system]] [[!redirects right strong composition laws for factorization systems]]

[[!redirects left weak composition law for factorization]] [[!redirects left weak composition law for factorizations]] [[!redirects left weak composition laws for factorization]] [[!redirects left weak composition laws for factorizations]]

[[!redirects left weak composition law for a functorial factorization]] [[!redirects left weak composition law for functorial factorizations]] [[!redirects left weak composition laws for a functorial factorization]] [[!redirects left weak composition laws for functorial factorizations]]

[[!redirects left weak composition law for a factorization system]] [[!redirects left weak composition law for factorization systems]] [[!redirects left weak composition laws for a factorization system]] [[!redirects left weak composition laws for factorization systems]]

[[!redirects left strong composition law for factorization]] [[!redirects left strong composition law for factorizations]] [[!redirects left strong composition laws for factorization]] [[!redirects left strong composition laws for factorizations]]

[[!redirects left strong composition law for a functorial factorization]] [[!redirects left strong composition law for functorial factorizations]] [[!redirects left strong composition laws for a functorial factorization]] [[!redirects left strong composition laws for functorial factorizations]]

[[!redirects left strong composition law for a factorization system]] [[!redirects left strong composition law for factorization systems]] [[!redirects left strong composition laws for a factorization system]] [[!redirects left strong composition laws for factorization systems]]



\end{document}