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\section*{reflective factorization system}

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

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

[[!include category theory - contents]]

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

[[!include factorization systems - contents]]

\hypertarget{reflective_factorization_systems}{}\section*{{Reflective factorization systems}}\label{reflective_factorization_systems}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{characterization}{Characterization}\dotfill \pageref*{characterization} \linebreak
\noindent\hyperlink{construction_of_factorizations}{Construction of factorizations}\dotfill \pageref*{construction_of_factorizations} \linebreak
\noindent\hyperlink{Localization}{Relation to localizations}\dotfill \pageref*{Localization} \linebreak
\noindent\hyperlink{reflective_stable_factorization_systems}{Reflective stable factorization systems}\dotfill \pageref*{reflective_stable_factorization_systems} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{reflective factorization system} is an [[orthogonal factorization system]] $(E,M)$ that is determined by the [[reflective subcategory]] $M/1$.

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

Let $C$ be a [[category]] with a [[terminal object]] $1$. If $(E,M)$ is an (orthogonal) [[orthogonal factorization system|factorization system]] on $C$, then the [[full subcategory]] $M/1 \subseteq C$ (consisting of those objects $X$ for which $X\to 1$ is in $M$) is [[reflective subcategory|reflective]]. The reflection of $Y\in C$ is obtained by the $(E,M)$-factorization $Y \xrightarrow{e} \ell Y \xrightarrow{m} 1$. (e.g. (\hyperlink{RosickyTholen08}{Rosicky-Tholen 08, 2.10}))

In fact, in this we do not need $(E,M)$ to be a factorization system; only a [[prefactorization system]] with the property that any morphism with [[terminal object|terminal]] [[codomain]] admits an $(E,M)$-factorization. For the nonce, let us call such a prefactorization system \emph{favorable}.

Conversely, suppose that $A\hookrightarrow C$ is a [[reflective subcategory]], and define $E$ to be the class of morphisms inverted by the [[reflector]] $\ell\colon C\to A$, and define $M = E^\perp$. Then $(E,M)$ is a favorable prefactorization system. In this way we obtain an [[adjunction]]

\begin{displaymath}
\Phi : \text{reflective subcategories} \; \rightleftarrows \; \text{favorable prefactorization systems} : \Psi.
\end{displaymath}
Here subcategories form a (possibly [[large category|large]]) [[poset]] ordered by inclusion, and prefactorization systems form a poset ordered by inclusion of the right classes $M$.

The [[unit of an adjunction|unit]] of this adjunction is easily seen to be an [[isomorphism]]. That is, given a reflective subcategory $A$, if we construct $(E,M)$ from it as above, then $A \simeq M/1$. Therefore, the adjunction allows us to identify reflective subcategories with certain favorable prefactorization systems.

The prefactorization systems arising in this way --- equivalently, those for which $(E,M) =  \Phi \Psi(E,M)$ --- are called the \textbf{reflective prefactorization systems}. A \textbf{reflective factorization system} is a reflective prefactorization system which happens to be a factorization system.

More generally, for any favorable factorization system $(E,M)$, we have a reflective prefactorization system $\Phi \Psi(E,M)$, called the \textbf{reflective interior} of $(E,M)$. Dualizing, it also has a \textbf{coreflective closure}.

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

\hypertarget{characterization}{}\subsubsection*{{Characterization}}\label{characterization}

The following is Theorem 2.3 in \hyperlink{CHK}{CHK}.

\begin{theorem}
\label{}\hypertarget{}{}
Let $(E',M')$ be the reflective interior of $(E,M)$. Then:

\begin{enumerate}%
\item $f\in E'$ precisely when there exists a $g\in E$ such that $g f \in E$.
\item $(E,M)$ is reflective precisely when $g f\in E$ and $g\in E$ together imply $f\in E$.

\end{enumerate}
\end{theorem}
\begin{proof}
That (1) implies (2) is obvious, so we prove (1).

Since $E'$ is, by definition, the class of maps inverted by the reflector into $M/1$, it satisfies the [[2-out-of-3 property]]. Since $E\subseteq E'$, it follows that $f g\in E$ and $g\in E$ imply $f\in E'$.

Conversely, if $f\colon X\to Y$ is in $E'$, then we have $\eta_Y \circ f = \ell(f) \circ \eta_X$ by naturality, where $\ell$ is the reflector into $M/1$ and $\eta$ its unit. But by construction of $\ell$, $\eta_Y$ and $\eta_X$ are in $E$, and by assumption, $\ell(f)$ is invertible; hence we can take $g = \eta_Y$.

\end{proof}
Note that the left class in any orthogonal factorization system is automatically closed under composition, contains the isomorphisms, and satisfies the property that $g f \in E$ and $f\in E$ together imply $g\in E$. Therefore, $(E,M)$ is reflective precisely when $E$ is a system of [[category with weak equivalences|weak equivalences]]. See \hyperlink{Localization}{Relation to Localization}, below.

\hypertarget{construction_of_factorizations}{}\subsubsection*{{Construction of factorizations}}\label{construction_of_factorizations}

The following is a slightly generalized version of Corollary 3.4 from \hyperlink{CHK}{CHK}.

\begin{theorem}
\label{}\hypertarget{}{}
Suppose that $C$ is finitely complete and [[M-complete category|M-complete]] for some factorization system $(E,M)$, where $M$ consists of [[monomorphisms]] and contains the [[split monomorphism|split monics]]. Then any reflective prefactorization system on $C$ is a factorization system.

\end{theorem}
\begin{proof}
This follows directly from \href{/nlab/show/M-complete+category#OFSFromAdjunction}{this theorem} applied to the reflection adjunction.

\end{proof}
The following is a consequence of Theorems 4.1 and 4.3 from \hyperlink{CHK}{CHK}.

\begin{theorem}
\label{SemiLeftExact}\hypertarget{SemiLeftExact}{}
Suppose that $C$ is finitely complete and that $(E,M)$ is a reflective prefactorization system on $C$ such that $E$-morphisms are stable under pullback along $M$-morphisms. Then $(E,M)$ is a factorization system.

\end{theorem}
\begin{proof}
Write $\ell$ for the corresponding reflection. Now given $f\colon A\to B$, let $m$ be the pullback of $\ell(f)$ along $\eta_B\colon B \to \ell B$:

\begin{displaymath}
\itexarray{ Y & \overset{g}{\to} & \ell A \\
  ^m \downarrow & & \downarrow^{\ell(f)}\\
  B & \underset{\eta_B}{\to} & \ell B}
\end{displaymath}
By closure properties of prefactorization systems, any morphism in $M/1$ lies in $M$, so in particular $\ell(f)\in M$. Since $M$ is stable under pullback (being, again, the right class of a prefactorization system), we have $m\in M$.

But $f$ factors through $m$, by the universal property of the pullback applied to the naturality square for $\eta$ at $f$. Thus we have $f = m e$ and it suffices to show $e\in E$. However, we also have $g e = \eta_A$, where $\eta_A\in E$ by definition, and $g\in E$ by assumption (being the pullback of $\eta_B\in E$ along $\ell(f)\in M$). By the characterization theorem above, since $(E,M)$ is reflective this implies $e\in E$, as desired.

\end{proof}
A reflection satisfying the condition of Theorem \ref{SemiLeftExact} is called \textbf{[[semi-left-exact reflection|semi-left-exact]]} (which see, for more equivalent characterizations). Note that saying that $E$-morphisms are stable under \emph{all} pullbacks is equivalent to saying that $\ell$ preserves all pullbacks, hence all finite limits---i.e. it is left-exact. In this case the factorization system is called [[stable factorization system|stable]]. Thus the terminology ``semi-left-exact'' for the weaker assumption.

However, semi-left-exactness is not necessary for the ``one-step'' construction in the proof of Theorem \ref{SemiLeftExact} to work. The necessary and sufficient condition is that the reflection is \textbf{simple}; one characterization of this is that a morphism $f$ is in $M$ if and only if its naturality square for the reflector $\ell:C\to A$ is a pullback. For others, see \hyperlink{CHK}{CHK, Theorem 4.1}.

\hypertarget{Localization}{}\subsubsection*{{Relation to localizations}}\label{Localization}

For any favorable prefactorization system $(E,M)$, it is easy to show that $M/1$ is the [[localization]] of $C$ at $E$. If $(E',M')$ is the reflective interior of $(E,M)$, then since $E'$ is the class of maps inverted by the reflector into $M/1$, it is precisely the \emph{saturation} of $E$ in the sense of localization (the class of maps inverted by the localization at $E$).

\hypertarget{reflective_stable_factorization_systems}{}\subsubsection*{{Reflective stable factorization systems}}\label{reflective_stable_factorization_systems}

A reflective factorization system on a finitely [[complete category]] is a [[stable factorization system]] if and only if its corresponding [[reflector]] preserves [[finite limits]]. A stable reflective factorization system is sometimes called \textbf{local}.

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

Obviously, any [[reflective subcategory]] gives rise to a reflective factorization system. Here are a few examples.

\begin{itemize}%
\item The category of complete [[metric spaces]] is reflective in the category of all metric spaces; the reflector is completion. In the corresponding factorization system, $E$ is the class of [[dense subspace|dense embeddings]].


\item Given a small [[site]] $S$, the [[sheaf topos]] $Sh(S)$ is a reflective subcategory of the [[presheaf topos]] $Psh(S)$. In the corresponding factorization system, $E$ is the class of [[local isomorphisms]].



\end{itemize}
On the other hand, many commonly encountered factorization systems are not reflective.

\begin{itemize}%
\item The factorization system $(Epi, Mono)$ on [[Set]] is not reflective. If $(E',M')$ is its reflective interior, then $E'$ is the class of morphisms $e\colon X\to Y$ such that if $Y$ is [[inhabited set|inhabited]], so is $X$, while $M'$ is the class of morphisms $m\colon X\to Y$ such that if $X$ is inhabited, then $m$ is an isomorphism.

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

\begin{itemize}%
\item [[semi-left-exact reflection]]


\item [[stable factorization system]]


\item [[closure operator]]


\item [[reflective localization]]



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

The basic theory is developed in

\begin{itemize}%
\item Cassidy and H\'e{}bert and [[Max Kelly|Kelly]], ``Reflective subcategories, localizations, and factorization systems''. \emph{J. Austral. Math Soc. (Series A)} 38 (1985), 287--329 (\href{http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ1_38_03%2FS1446788700023624a.pdf&code=5796045be8904c5183c2e95bce65491e}{pdf})

\end{itemize}
\begin{itemize}%
\item Carboni and [[George Janelidze|Janelidze]] and [[Max Kelly|Kelly]] and [[Paré]], ``On localization and stabilization for factorization systems'', \emph{Appl. Categ. Structures} 5 (1997), 1--58

\end{itemize}
Discussion of ``simple'' reflective factorization systems and of simultaneously reflective and coreflective factorization systems is in

\begin{itemize}%
\item [[Jiri Rosicky]], [[Walter Tholen]], \emph{Factorization, Fibration and Torsion}, Journal of Homotopy and Related Structures, Vol. 2(2007), No. 2, pp. 295-314 (\href{http://arxiv.org/abs/0801.0063}{arXiv:0801.0063}, \href{http://www.emis.de/journals/JHRS/volumes/2007/n2a14/}{publisher})

\end{itemize}
[[!redirects reflective factorization system]] [[!redirects reflective factorization systems]] [[!redirects reflective prefactorization system]] [[!redirects reflective prefactorization systems]] [[!redirects coreflective factorization system]] [[!redirects coreflective factorization systems]] [[!redirects coreflective prefactorization system]] [[!redirects coreflective prefactorization systems]] [[!redirects reflective interior]] [[!redirects coreflective closure]] [[!redirects simple reflection]] [[!redirects simple reflections]]



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