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\section*{ternary factorization system}

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

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

[[!include factorization systems - contents]]

\hypertarget{ternary_factorisation_systems}{}\section*{{Ternary factorisation systems}}\label{ternary_factorisation_systems}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{the_sixth_class_of_maps}{The sixth class of maps}\dotfill \pageref*{the_sixth_class_of_maps} \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}

Just as an (orthogonal/unique) [[orthogonal factorization system|factorization system]] $(E,M)$ on a [[category]] $C$ gives a way to factor every [[morphism]] of $C$ as an $E$-map followed by an $M$-map, a \emph{ternary (orthogonal) factorization system} $(E,F,M)$ gives a way to factor every map of $C$ as an $E$-map followed by an $F$-map followed by an $M$-map.

This is a special case of a notion of [[k-ary factorization system]].

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

It turns out that a convenient way to state the definition is in terms of a pair of ordinary (orthogonal) factorization systems. We define a \textbf{ternary factorization system} on $C$ to consist of a pair $(L_1,R_1)$ and $(L_2,R_2)$ of ordinary orthogonal factorization systems such that $L_1 \subseteq L_2$ (or equivalently $R_2 \subseteq R_1$).

The three classes of map $(E,F,M)$ are then defined by $E=L_1$, $F = L_2\cap R_1$, and $M=R_2$. This is justified by:

\begin{uprop}
Given a ternary factorization system as above, any morphism $f:A\to B$ factors as

\begin{displaymath}
A \overset{L_1}{\to} im_2(f) \overset{L_2 \cap R_1}{\to} im_1(f) \overset{R_2}{\to} B
\end{displaymath}
in an essentially unique way.

\end{uprop}
\begin{proof}
Consider the two ternary factorizations of $f$ obtained by

\begin{enumerate}%
\item First factoring $f$ into an $L_1$-map followed by an $R_1$-map, then factoring the $R_1$-part into an $L_2$-map followed by an $R_2$-map; and
\item First factoring $f$ into an $L_2$-map followed by an $R_2$-map, then factoring the $L_2$-part into an $L_1$-map followed by an $R_1$-map.

\end{enumerate}
Note that both start with an $L_1$ map and end with an $R_2$ map. By a straightforward exercise in orthogonality, we can get comparison maps in both directions between these two factorizations which make them isomorphic. Therefore, since the first produces a middle map which is in $L_2$ and the second produces a middle map which is in $R_1$, this middle map must in fact be in $L_2\cap R_1$. Finally, any other such ternary factorization of $f$ induces an $(L_1,R_1)$ and $(L_2,R_2)$ factorization by composing pairwise, and uniqueness of these two implies uniqueness of the ternary factorization.

More explicitly, we factor $f$ as

$\backslash$begin\{center\}$\backslash$begin\{tikzcd\} \& C\_1 $\backslash$arr,``\{$\backslash$lambda\_2\}'' $\backslash$ardrr,``\{r\_1\}''' \& C\_2 $\backslash$ardr,``\{$\backslash$rho\_2\}'' $\backslash$ A $\backslash$arur,``\{$\backslash$ell\_1\}'' $\backslash$ardr,``\{$\backslash$lambda\_1\}''' $\backslash$ardrr,``\{$\backslash$ell\_2\}'' \&\&\& B$\backslash$ \& D\_1 $\backslash$arr,``\{$\backslash$rho\_1\}''' \& D\_2 $\backslash$arur,``\{r\_2\}''' $\backslash$end\{tikzcd\}$\backslash$end\{center\}

with $\lambda_i, \ell_i\in L_i, \rho_j,r_j\in R_j$. Then since $R_2\subseteq R_1$, we have $r_2 \rho_1 \in R_1$, so that $(\ell_1,r_1)$ and $(\lambda_1,r_2\rho_1)$ are both $(L_1,R_1)$-factorizations of $f$ and thus we have a unique compatible isomorphism $C_1\cong D_1$. Similarly, $(\lambda_2 \ell_1, \rho_2)$ and $(\ell_2,r_2)$ are both $(L_2,R_2)$-factorizations, so we have a unique compatible isomorphism $C_2\cong D_2$. This gives a diagram

$\backslash$begin\{center\}$\backslash$begin\{tikzcd\} \& C\_1 $\backslash$arr,``\{$\backslash$lambda\_2\}'' $\backslash$ardd,``$\backslash$cong'' \& C\_2 $\backslash$ardr,``\{$\backslash$rho\_2\}'' $\backslash$ardd,``$\backslash$cong''' $\backslash$ A $\backslash$arur,``\{$\backslash$ell\_1\}'' $\backslash$ardr,``\{$\backslash$lambda\_1\}''' \&\&\& B$\backslash$ \& D\_1 $\backslash$arr,``\{$\backslash$rho\_1\}''' \& D\_2 $\backslash$arur,``\{r\_2\}''' $\backslash$end\{tikzcd\}$\backslash$end\{center\}

with two commutative triangles, and the middle square also commutes since both sides are lifts in a lifting problem of $\ell_1$ against $r_2$. Finally, since $\lambda_2\in L_2$ is isomorphic to $\rho_1\in R_1$ in the arrow category, both are in fact in $L_2\cap R_1$.

\end{proof}
Conversely, just as for a binary factorization system, the extra requirement of orthogonality can be deduced from uniqueness of the factorizations, a unique and \emph{functorial} ternary factorization implies that it ``splits'' into a pair of binary factorization systems, i.e. a ternary factorization system as defined here. This is remarked on \href{http://golem.ph.utexas.edu/category/2010/07/ternary_factorization_systems.html#c034162}{here}.

One can also characterize the notion in terms of a ternary factorization with a ``ternary orthogonality'' property; see the paper of Pultr and Tholen referenced below.

\hypertarget{the_sixth_class_of_maps}{}\subsection*{{The sixth class of maps}}\label{the_sixth_class_of_maps}

In addition to $L_1$, $R_1$, $L_2$, $R_2$, and $L_2\cap R_1$, a ternary factorization system also determines a sixth important class of morphisms, namely those whose $(L_2\cap R_1)$-part is an isomorphism, or equivalently those that can be factored as an $L_1$-map followed by an $R_2$-map. We therefore call this class $R_2 L_1$.

\begin{uprop}
In a ternary factorization system, $L_1 = L_2 \cap R_2L_1$ and $R_2 = R_1 \cap R_2L_1$.

\end{uprop}
\begin{proof}
In both cases $\subseteq$ is obvious. Conversely, if $f \in L_2 \cap R_2 L_1$, say $f = m e$ for $m\in R_2$ and $e\in L_1$, then orthogonality in the square

\begin{displaymath}
\itexarray{a & \overset{e}{\to} & c\\
  ^f \downarrow && \downarrow ^m\\
  b & \underset{id}{\to} & b}
\end{displaymath}
exhibits $f$ as a retract of $e$ in $Arr(C)$, whence $f\in L_1$ since $L_1$ is closed under retracts.

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

\begin{itemize}%
\item In [[Top]], let $L_1=$ quotient maps, $R_1=$ injective continuous maps, $L_2=$ surjective continuous functions, and $R_2=$ subspace embeddings. Here $L_2\cap R_1=$ bijective continuous maps, and the two intermediate objects in the ternary factorization of a continuous map are obtained by imposing the coarsest and the finest compatible topologies on its set-theoretic image.


\item More generally, if a category has both ([[epimorphism|epi]], [[strong monomorphism|strong mono]]) and ([[strong epimorphism|strong epi]], [[monomorphism|mono]]) factorizations, then since strong epis are epi, we have a ternary factorization. Here $L_2\cap R_1$ is the class of monic epics, sometimes called [[bimorphisms]]. The maps in $R_2 L_1$ are sometimes called [[strict morphisms]].


\item On [[Cat]] there is a 2-categorical version of a ternary factorization system, determined by the [[factorization system on a 2-category|2-categorical factorization systems]] ([[essentially surjective functor|eso]]+[[full functor|full]], [[faithful functor|faithful]]) and (eso, [[fully faithful functor|full and faithful]]). Here $L_2\cap R_1$ is the class of eso+faithful functors, while $R_2 L_1$ is the class of full functors. This factorization system plays an important role in the study of [[stuff, structure, property]].

Restricted to [[groupoids]] this is the [[1-image]]-[[2-image]] factorization, the 3-stage [[Postnikov system]] of groupoids.


\item On [[Topos]] there is also a 2-categorical ternary factorization system composed of the binary 2-categorical factorization systems ([[hyperconnected geometric morphism|hyperconnected]], [[localic geometric morphism|localic]]) and ([[geometric surjection|surjection]], [[geometric embedding|inclusion]]). Here the maps in $L_2\cap R_1$ have no name other than ``localic surjections,'' and those in $R_2 L_1$ have no established name (although they are briefly mentioned in A4.6.10 of the [[Elephant]]).


\item Suppose that $C$ has a binary factorization system $(E,M)$ and that $p\colon A\to C$ is an [[ambifibration]] relative to $(E,M)$: i.e. every arrow in $E$ has an opcartesian lift and every arrow in $M$ has a [[cartesian morphism|cartesian]] lift. (In particular, $p$ could be a [[bifibration]].) Then there is a ternary factorization system on $A$ for which $L_1$ is the class of opcartesian arrows over $E$, $R_2$ is the class of cartesian arrows over $M$, and $L_2\cap R_1$ is the class of vertical arrows (those lying over identities). See \href{http://golem.ph.utexas.edu/category/2010/07/ternary_factorization_systems.html#c034162}{this comment}.

For instance, the above factorization system on $Top$ is induced in this way via the [[forgetful functor]] $Top\to Set$ from the (epi,mono) factorization system on [[Set]].


\item A similar example is given by a [[span]] $A \overset{p}{\leftarrow} E \overset{q}{\to} B$ of categories where $p$ is a [[fibration]] whose cartesian morphisms are $q$-vertical and $q$ is an [[opfibration]] whose opcartesian morphisms are $p$-vertical (that is, the span $(p,q)$ is both a left and a right fibration in the sense of Street). Then the two factorization systems on $E$ given by the $q$-opcartesian and $q$-vertical morphisms on the one hand, and the $p$-vertical and $p$-cartesian morphisms on the other, satisfy the $L_1 \subseteq L_2$ condition above, so that every morphism in $E$ factors as a $q$-opcartesian morphism followed by a morphism that is both $p$- and $q$-vertical, followed by a $p$-cartesian morphism.

Such a span is a [[two-sided fibration]] if $L_1R_2 \subseteq R_2L_1$, that is if the three-way factorization of the composite of a $p$-cartesian morphism followed by a $q$-opcartesian one has its middle term an isomorphism.



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

\begin{itemize}%
\item The notion of [[model category]] involves a pair of [[weak factorization systems]] called (acyclic cofibration, fibration) and (cofibration, acyclic fibration) which are compatible in the same sense as above. However, non-uniqueness of these factorizations means that the resulting ``ternary factorization'' of a morphism is not unique. The class corresponding to $R_2 L_1$ is important, however: it is precisely the class of [[weak equivalences]].


\item The notion of [[k-ary factorization system]] is a generalization to factorizations into $k$ morphisms.


\item Just as [[strict factorization system]]s can be identified with [[distributive laws]] in the [[bicategory]] of [[spans]], so ``strict'' ternary (and k-ary) factorization systems can be identified with \href{http://arxiv.org/abs/0710.1120}{iterated distributive law}s in $Span$.



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

\begin{itemize}%
\item A. Pultr and W. Tholen, \emph{Free Quillen Factorization Systems}. Georgian Math. J.9 (2002), No. 4, 807-820


\item \href{http://golem.ph.utexas.edu/category/2010/07/ternary_factorization_systems.html}{Cafe discussion}



\end{itemize}
[[!redirects ternary factorization system]] [[!redirects ternary factorization systems]] [[!redirects ternary factorisation system]] [[!redirects ternary factorisation systems]] [[!redirects double factorization system]] [[!redirects double factorization systems]] [[!redirects double factorisation system]] [[!redirects double factorisation systems]] [[!redirects 3-way factorization system]] [[!redirects 3-way factorization systems]] [[!redirects 3-way factorisation system]] [[!redirects 3-way factorisation systems]] [[!redirects 3-step factorization system]] [[!redirects 3-step factorization systems]] [[!redirects 3-step factorisation system]] [[!redirects 3-step factorisation systems]] [[!redirects 3-stage factorization system]] [[!redirects 3-stage factorization systems]] [[!redirects 3-stage factorisation system]] [[!redirects 3-stage factorisation systems]] [[!redirects 3-fold factorization system]] [[!redirects 3-fold factorization systems]] [[!redirects 3-fold factorisation system]] [[!redirects 3-fold factorisation systems]]

[[!redirects Quillen factorization system]] [[!redirects Quillen factorization systems]] [[!redirects Quillen factorisation system]] [[!redirects Quillen factorisation systems]]



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