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\begin{document}

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\section*{k-ary factorization system}

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

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

[[!include factorization systems - contents]]

\hypertarget{ary_factorisation_systems}{}\section*{{$k$-ary factorisation systems}}\label{ary_factorisation_systems}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{infinitary_factorisation_systems}{Infinitary factorisation systems}\dotfill \pageref*{infinitary_factorisation_systems} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A $k$-ary factorization system is a generalization of (binary) [[orthogonal factorization systems]] and [[ternary factorization systems]] to factorizations into a string of $k$ morphisms.

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

For $k \gt 0$ a [[natural number]] and $C$ a [[category]] (or $\infty$-[[infinity-category|category]]), a \textbf{$k$-ary factorisation system} on $C$ consists of $(k - 1)$ factorisation systems $(E_i, M_i)$ (for $0 \lt i \lt k$) on $C$, such that

\begin{itemize}%
\item $M_i \subseteq M_{i + 1}$ whenever this is meaningful (equivalently, $E_i \subseteq E_{i - 1}$).

\end{itemize}
We can extend this to include two other factorisation systems, one for $i = 0$ and one for $i = k$:

\begin{itemize}%
\item $M_0$ consists of only [[isomorphisms]]/[[equivalences]] (equivalently, $E_0$ consists of all morphisms), and
\item $M_k$ consists of all morphisms (equivalently, $E_k$ consists of only isomorphisms/equivalences).

\end{itemize}
Given a $k$-ary factorisation system, the ([[coimage|co]])[[image]] of $(E_i,M_i)$ is the \textbf{$i$-(co)image} of the entire $k$-ary factorisation system.

Note that every (higher) category has a unique $1$-ary factorisation system, since no structure at all is required. We also say that a [[groupoid]] (or $\infty$-[[infinity-groupoid|groupoid]]) has a (necessarily unique) $0$-ary factorisation system; this makes sense since we have $M_0 = M_k$ (and $E_0 = E_k$) in that case. A [[discrete category]] has a (necessarily unique) $(-1)$-ary factorisation system.

A $k$-ary factorisation system may also be called a \textbf{$k$-step factorisation system} or a \textbf{$(k+1)$-stage factorisation system}. You can see why if you count the basic morphisms (steps) and objects (stages) that $k - 1$ overlapping factorisation systems produce from a morphism.

\hypertarget{infinitary_factorisation_systems}{}\subsection*{{Infinitary factorisation systems}}\label{infinitary_factorisation_systems}

Here is an incomplete attempt at a general definition:

Fix any [[ordinal number]] (or [[opposite poset|opposite]] thereof, or any [[poset]], really) $\alpha$. Then an \textbf{$\alpha$-stage factorisation system} (in an ambient $\infty$-category $C$) consists of an $\alpha$-indexed family of factorisation systems $(E_i, M_i)$ in $C$ such that:

\begin{itemize}%
\item $M_i \subseteq M_j$ whenever $i \leq j$ (equivalently, $E_i \supseteq E_j$ whenever $i \leq j$),
\item each morphism $f\colon X \to Y$ is both the [[inverse limit]] $\underset{i \to \infty}\lim \im_i f$ in the [[slice category]] $C/Y$ and the [[direct limit]] $\underset{i \to -\infty}\colim \coim_i f$ in the [[coslice category]] $X/C$, and
\item for each $f\colon X \to Y$, $\id_Y$ is $\underset{i \to -\infty}\colim \im_i f$ and $\id_X$ is $\underset{i \to \infty}\lim \coim_i f$.

\end{itemize}
This seems to be correct whenever $\alpha$ really is either an ordinal or the opposite thereof, as well as some other posets such as $\omega^op + \omega$ (which is the poset of [[integers]]), but it seems to be missing something for (for example) $\omega + \omega^op$. Notice that, when $\alpha$ is \emph{both} an ordinal and the opposite thereof, we recover the above definition of an $(alpha-1)$-ary factorisation system.

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

\begin{itemize}%
\item For $k = 3$ one speaks of a \emph{[[ternary factorization system]]}. See there for more examples


\item In an [[(∞,1)-topos]] the [[(epi, mono) factorization system]] in a [[topos]] splits up to an $\infty$-ary factorization system consisting of the [[(n-epi, n-mono) factorization systems]] (the [[n-image]]-factorization) for all $n \in \mathbb{N}$. This is called the \emph{[[Postnikov system]]}.



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

\begin{itemize}%
\item \href{http://golem.ph.utexas.edu/category/2010/07/ternary_factorization_systems.html}{Cafe discussion} mainy on the ternary version


\item \href{https://nforum.ncatlab.org/discussion/1629}{Forum discussion} including the k-ary case, even when k is infinite



\end{itemize}
[[!redirects k-ary factorization system]] [[!redirects k-ary factorization systems]] [[!redirects k-ary factorisation system]] [[!redirects k-ary factorisation systems]] [[!redirects n-ary factorization system]] [[!redirects n-ary factorization systems]] [[!redirects n-ary factorisation system]] [[!redirects n-ary factorisation systems]] [[!redirects alpha-ary factorization system]] [[!redirects alpha-ary factorization systems]] [[!redirects alpha-ary factorisation system]] [[!redirects alpha-ary factorisation systems]]

[[!redirects k-fold factorization system]] [[!redirects k-fold factorization systems]] [[!redirects k-fold factorisation system]] [[!redirects k-fold factorisation systems]] [[!redirects n-fold factorization system]] [[!redirects n-fold factorization systems]] [[!redirects n-fold factorisation system]] [[!redirects n-fold factorisation systems]] [[!redirects alpha-fold factorization system]] [[!redirects alpha-fold factorization systems]] [[!redirects alpha-fold factorisation system]] [[!redirects alpha-fold factorisation systems]]

[[!redirects k-step factorization system]] [[!redirects k-step factorization systems]] [[!redirects k-step factorisation system]] [[!redirects k-step factorisation systems]] [[!redirects n-step factorization system]] [[!redirects n-step factorization systems]] [[!redirects n-step factorisation system]] [[!redirects n-step factorisation systems]] [[!redirects alpha-step factorization system]] [[!redirects alpha-step factorization systems]] [[!redirects alpha-step factorisation system]] [[!redirects alpha-step factorisation systems]]

[[!redirects k-stage factorization system]] [[!redirects k-stage factorization systems]] [[!redirects k-stage factorisation system]] [[!redirects k-stage factorisation systems]] [[!redirects n-stage factorization system]] [[!redirects n-stage factorization systems]] [[!redirects n-stage factorisation system]] [[!redirects n-stage factorisation systems]] [[!redirects alpha-stage factorization system]] [[!redirects alpha-stage factorization systems]] [[!redirects alpha-stage factorisation system]] [[!redirects alpha-stage factorisation systems]]

[[!redirects infinitary factorization system]] [[!redirects infinitary factorization systems]]



\end{document}