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\begin{document}

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\section*{factorization structure}

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

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

[[!include category theory - contents]]

\hypertarget{factorization_structures_for_sinks_and_cosinks}{}\section*{{Factorization structures for sinks and cosinks}}\label{factorization_structures_for_sinks_and_cosinks}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{on_size_and_foundations}{On size and foundations}\dotfill \pageref*{on_size_and_foundations} \linebreak
\noindent\hyperlink{constructions}{Constructions}\dotfill \pageref*{constructions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{_consists_of_monics}{$M$ consists of monics}\dotfill \pageref*{_consists_of_monics} \linebreak


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

A \emph{factorization structure} is a strengthening of an [[orthogonal factorization system]] which allows us to factor, not only single [[morphisms]], but arbitrary [[sinks]] (or cosinks).

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

Let $\{e_i \colon X_i \to Y\}_{i\in I}$ be a [[sink]] and $m\colon A \to B$ a [[morphism]] in some [[category]]. We say that $\{e_i\}$ is \emph{orthogonal} to $m$ if for any sink $\{f_i \colon X_i \to A\}_{i\in I}$ and morphism $g\colon Y\to B$ such that $m f_i = g e_i$ for all $i\in I$:

\begin{displaymath}
\itexarray{X_i & \overset{f_i}{\to} & A\\
  ^{e_i}\downarrow && \downarrow^m\\
  Y & \underset{g}{\to} & B}
\end{displaymath}
there exists a unique arrow $h\colon Y\to A$ such that $h e_i = f_i$ for all $i$ and $m h = g$. Clearly, if ${|I|}=1$ this reduces to the usual notion of orthogonality for morphisms.

Let $(E,M)$ be an [[orthogonal factorization system|(orthogonal) factorization system]]; we say it extends to a \textbf{factorization structure for sinks} if every sink $\{f_i \colon X_i \to Y\}_{i\in I}$ can be factored as $f_i = m e_i$, where $m\in M$ and the sink $\{e_i \colon X_i \to Z\}_{i\in I}$ is orthogonal to $M$. Note there is no restriction on the sinks involved to be [[small category|small]].

Some authors write $\mathbf{E}$ for the collection of sinks orthogonal to $M$, and say that $(\mathbf{E},M)$ is a factorization structure for sinks, or that the category $C$ is an ``$(\mathbf{E},M)$-category''. However, since $\mathbf{E}$ is uniquely determined by $(E,M)$, and $E$ is precisely the 1-ary sinks in $\mathbf{E}$, there is little harm in saying that $(E,M)$ is a factorization structure for sinks.

The dual notion is a \textbf{factorization structure for cosinks} (``sources'').

\hypertarget{on_size_and_foundations}{}\subsubsection*{{On size and foundations}}\label{on_size_and_foundations}

Observe that if $C$ is [[large category|large]], then the collection $\mathbf{E}$ contains [[proper classes]] as elements. Therefore, in some [[foundations]] such as [[ZF]], it is not definable as a single thing. In [[NBG]] one may treat it as a ``hyperclass'' defined by a first-order formula, in the same way that one treats classes in ZF. If we use a [[Grothendieck universe]] to define smallness, of course, there is no problem.

\hypertarget{constructions}{}\subsection*{{Constructions}}\label{constructions}

Many well-known factorization systems, and ways to construct factorization systems, extend to factorization structures for sinks and/or cosinks.

\begin{itemize}%
\item \href{/nlab/show/M-complete+category#ConstructingOFS}{This theorem} implies that if a category is [[M-complete category|M-complete]] for a class $M$ of morphisms which consists of monomorphisms and is closed under composition and pullback, then $M$ is the right class in a factorization structure for sinks.


\item If $p\colon A\to B$ is a [[Grothendieck fibration]], then factorization structures for sinks can be lifted from $B$ to $A$. Dually, if $p$ is an opfibration, we can lift factorization structures for cosinks. For the ``mismatched'' types of lifting, we require more: $p$ must be a [[topological concrete category|topological functor]].



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

\begin{itemize}%
\item In [[Set]], the factorization system (epi, mono) extends to both a factorization structure for sinks and one for cosinks. The epi-sinks are those that are jointly epimorphic, and the mono-sinks are those that are jointly monomorphic.


\item By lifting (epi,mono) to [[Top]] (or any other topological category over $Set$), we obtain two factorization structures for sinks: (jointly surjective, subspace inclusions) and (final topologies, injections). Dually, we have two factorization structures for cosinks: (surjections, initial topologies) and (quotient maps, jointly injective).



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

\hypertarget{_consists_of_monics}{}\subsubsection*{{$M$ consists of monics}}\label{_consists_of_monics}

Recall that a factorization system $(E,M)$ is called \emph{proper} if $E\subseteq Epi$ and $M\subseteq Mono$. In the case of a factorization structure for sinks, the second of these is automatic.

\begin{utheorem}
If $(E,M)$ is a factorization structure for sinks, then $M$ consists of monomorphisms.

\end{utheorem}
\begin{proof}
Let $m\colon A\to B$ be in $M$, and suppose that $m r = m s$ for some $r,s\colon X\to A$, but $r\neq s$. Consider the sink $\{m r: X \to B \}_{f\in Mor(C)}$ consisting of one copy of $m r$ ($= m s$) for each arrow of the ambient category $C$, and factor this sink as an $E$-sink $\{e_f\colon X \to Y\}_{f\in Mor(C)}$ followed by an $M$-morphism $n\colon Y\to B$.

Now there are at least $2^{|Mor(C)|}$ different sinks $\{g_f\colon X \to A\}_{f\in Mor(C)}$ such that $m g_f = n e_f$, since we may take each $g_f$ to be either $r$ or $s$. Therefore, by orthogonality, there are at least $2^{|Mor(C)|}$ different morphisms $Y\to A$, a contradiction to [[Cantor's theorem]], since there can be at most $Mor(C)$.

\end{proof}
This proof is of course quite reminiscent of Freyd's theorem that any [[complete small category]] is a [[preorder]]. In fact, Freyd's theorem is a consequence of this one (or at least of its proof). For given a complete small category, there is a factorization structure acting at least on \emph{small} sinks, where $M$ is the class of all morphisms and $E$ the class of families of injections into coproducts. (Any complete small category is also cocomplete, by the [[adjoint functor theorem]].) Therefore, all morphisms in a complete small category are monic, including the unique maps to the [[terminal object]]; hence the category is a preorder.

Since Freyd's theorem can fail in [[constructive mathematics]], we should expect the use of [[excluded middle]] to also be essential in proving the above property of factorization structures.

[[!redirects factorization structures]] [[!redirects factorization structure for sinks]] [[!redirects factorization structure for sources]] [[!redirects factorization structure for cosinks]] [[!redirects factorization structures for sinks]] [[!redirects factorization structures for sources]] [[!redirects factorization structures for cosinks]]



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