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\begin{document}

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\section*{accessible weak factorization system}

\hypertarget{accessible_weak_factorization_systems}{}\section*{{Accessible weak factorization systems}}\label{accessible_weak_factorization_systems}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{equivalent_forms}{Equivalent forms}\dotfill \pageref*{equivalent_forms} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{left_and_right_lifting}{Left and right lifting}\dotfill \pageref*{left_and_right_lifting} \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}

A [[weak factorization system]] on a [[locally presentable category]] is \textbf{accessible} if it admits a [[functorial factorization]] that is an [[accessible functor]].

\hypertarget{equivalent_forms}{}\subsection*{{Equivalent forms}}\label{equivalent_forms}

$\backslash$begin\{theorem\} For a weak factorization system $(L,R)$ on a locally presentable category $M$, the following are equivalent.

\begin{enumerate}%
\item $(L,R)$ is accessible.
\item There is a small category $J \to M^\to$ over the [[arrow category]] such that $R$ consists of the morphisms with coherent right lifting functions relative to $J$.
\item $(L,R)$ can be generated by the [[algebraic small object argument]].
\item $(L,R)$ can be equipped with the structure of an accessible [[algebraic weak factorization system]].

\end{enumerate}
$\backslash$end\{theorem\}

$\backslash$begin\{proof\} Properties of the algebraic small object argument yield (2)$\Rightarrow$(3)$\Rightarrow$(4), and (4)$\Rightarrow$(1) since an algebraic wfs is in particular a functorial factorization. The remaining implication is the following lemma, which is Remark 3.1.8 of \hyperlink{HKRS15}{HKRS15}, which relies on Theorem 4.3 of \hyperlink{Rosicky17}{Rosicky17}. $\backslash$end\{proof\}

$\backslash$begin\{lemma\} Suppose $E$ is an accessible functorial factorization on a locally presentable category $M$, realizing a weak factorization system. Then there is an accessible algebraic weak factorization system realizing the same weak factorization system. $\backslash$end\{lemma\} $\backslash$begin\{proof\} Let $Coalg(L)$ be the category of [[algebra over an endofunctor|coalgebras]] for the [[pointed endofunctor|copointed endofunctor]] of the [[arrow category]] $M^\to$ induced by $E$, i.e. morphisms equipped with a section of their $E$-factorization exhibiting them as a retract of the first factor. Then $Coalg(L)$ is locally presentable (being complete and a [[PIE limit]] construction from $M$). Thus, it has a small dense subcategory $X$. We can then apply [[Garner's small object argument]] to generate an algebraically-free algebraic weak factorization system from $X$. The algebraic right-maps in this awfs are the morphisms with coherent lifting functions against $X$, which by density is the same as having a coherent lifting function against all of $Coalg(L)$, which is the same as being an algebra for the pointed endofunctor of $M^\to$ corresponding to $E$. Thus this is an awfs with the same right-maps, hence the same underlying weak factorization system. $\backslash$end\{proof\}

Regarding point (4), note that being accessible is a \emph{property} of a wfs, while being algebraic is a \emph{structure} on it. A given accessible wfs can admit many different algebraic realizations, and not all of them (even the accessible ones) may be produced by the ordinary algebraic small object argument (although they can all be produced by the fancier version involving a double category of maps as input).

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

\hypertarget{left_and_right_lifting}{}\subsubsection*{{Left and right lifting}}\label{left_and_right_lifting}

Any accessible weak factorization system can be right-lifted along a right adjoint, or also left-lifted along a left adjoint, between locally presentable categories. That is, if $U:A\to B$ is a functor between locally presentable categories and $(L,R)$ is an accessible weak factorization system of $B$, then:

\begin{enumerate}%
\item If $U$ is a right adjoint, then there is an accessible wfs $(L',R')$ on $A$ such that $R' = U^{-1}(R)$.


\item If $U$ is a left adjoint, then there is an accessible wfs $(L',R')$ on $A$ such that $L' = U^{-1}(L)$.



\end{enumerate}
See \hyperlink{HKRS15}{HKRS} and its correction in \hyperlink{GKR18}{GKR} for details. In particular, this is useful for the construction of [[transferred model structures]].

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

\begin{itemize}%
\item [[algebraic weak factorization system]]
\item [[accessible model structure]]

\end{itemize}
[[!include algebraic model structures - table]]

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

\begin{itemize}%
\item [[Jiri Rosicky]], \emph{Accessible model categories}, Appl Categor Struct (2017) 25: 187. \href{https://doi.org/10.1007/s10485-015-9419-6}{doi}, \href{https://arxiv.org/abs/1503.05010}{arxiv}


\item [[Kathryn Hess]], Magdalena Kdziorek, [[Emily Riehl]], [[Brooke Shipley]], \emph{A necessary and sufficient condition for induced model structures} (\href{http://arxiv.org/abs/1509.08154}{arXiv:1509.08154}). This paper contains an error, corrected by:


\item [[Richard Garner]], Magdalena Kedziorek, [[Emily Riehl]], \emph{Lifting accessible model structures}, \href{https://arxiv.org/abs/1802.09889}{arXiv:1802.09889}



\end{itemize}
[[!redirects accessible weak factorization systems]] [[!redirects accessible weak factorisation system]] [[!redirects accessible weak factorisation systems]] [[!redirects accessible wfs]] [[!redirects accessible WFS]]



\end{document}