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\begin{document}

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\section*{filtered category}

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

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

[[!include category theory - contents]]

\hypertarget{filtered_categories}{}\section*{{Filtered categories}}\label{filtered_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{ordinary_filteredness}{Ordinary filteredness}\dotfill \pageref*{ordinary_filteredness} \linebreak
\noindent\hyperlink{higher_filteredness}{Higher filteredness}\dotfill \pageref*{higher_filteredness} \linebreak
\noindent\hyperlink{generalized_filteredness}{Generalized filteredness}\dotfill \pageref*{generalized_filteredness} \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 filtered category is a [[categorification]] of the concept of [[directed set]]. In addition to having an upper bound (but not necessarily a [[coproduct]]) for every pair of objects, there must also be an upper bound (but not necessarily a [[coequaliser]]) for every pair of parallel morphisms.

A [[diagram]] $F:D\to C$ where $D$ is a filtered category is called a \textbf{filtered diagram}. A colimit of a filtered diagram is called a \textbf{[[filtered colimit]]}.

A category whose opposite is filtered is called \textbf{cofiltered}.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{ordinary_filteredness}{}\subsubsection*{{Ordinary filteredness}}\label{ordinary_filteredness}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{(finitely) filtered category} is a [[category]] $C$ in which every finite [[diagram]] has a [[cocone]].

\end{defn}
That is, for any finite category $D$ and any [[functor]] $F:D\to C$, there exists an object $c\in C$ and a [[natural transformation]] $F\to \Delta c$ where $\Delta c:D\to C$ is the constant diagram at $c$. If $D^+$ is the result of freely adjoining a terminal object to a category $D$, then the condition is the same as that any functor $F: D \to C$ with finite domain admits an extension $\tilde{F}: D^+ \to C$.

Equivalently, filtered categories can be characterized as those categories where, for every finite diagram $J$, the diagonal functor $\Delta : C \to C^J$ is [[final functor|final]]. This point of view can be generalized to other kinds of categories whose colimits are well-behaved with respect to a type of limit, such as [[sifted colimit|sifted]] categories.

This can be rephrased in more elementary terms by saying that:

\begin{itemize}%
\item There exists an object of $C$ (the case when $D=\emptyset$)
\item For any two objects $c_1,c_2\in C$, there exists an object $c_3\in C$ and morphisms $c_1\to c_3$ and $c_2\to c_3$.
\item For any two [[parallel morphisms]] $f,g:c_1\to c_2$ in $C$, there exists a morphism $h:c_2\to c_3$ such that $h f = h g$.

\end{itemize}
Just as all finite [[colimit|colimits]] can be constructed from initial objects, binary coproducts, and coequalizers, so a cocone on any finite diagram can be constructed from these three.

In [[constructive mathematics]], the elementary rephrasing above is equivalent to every [[Bishop-finite]] diagram admitting a cocone.

\hypertarget{higher_filteredness}{}\subsubsection*{{Higher filteredness}}\label{higher_filteredness}

More generally, if $\kappa$ is an infinite [[regular cardinal]] (or an [[arity class]]), then a \textbf{$\kappa$-filtered category} is one such that any diagram $D\to C$ has a cocone where $D$ has $\lt \kappa$ arrows, or equivalently that any functor $F: D \to C$ whose domain has fewer than $\kappa$ morphisms admits an extension $\tilde{F}: D^+ \to C$. The usual filtered categories are then the case $\kappa = \omega$, i.e., where the $D$ have fewer than $\omega$ morphisms (in other words are finite). (We could also say in this case ``$\aleph_0$-filtered'', but $\omega$-filtered is more usual in the literature.)

Note that a [[preorder]] is $\kappa$-filtered as a category just when it is $\kappa$-[[direction|directed]] as a preorder.

\hypertarget{generalized_filteredness}{}\subsubsection*{{Generalized filteredness}}\label{generalized_filteredness}

Even more generally, if $\mathcal{J}$ is a class of small categories, a category $C$ is called \textbf{$\mathcal{J}$-filtered} if $C$-colimits commute with $\mathcal{J}$-limits in [[Set]]. When $\mathcal{J}$ is the class of all $\kappa$-small categories for an infinite regular cardinal $\kappa$, then $\mathcal{J}$-filteredness is the same as $\kappa$-filteredness as defined above. See \hyperlink{ABLR}{ABLR}.

If $\mathcal{J}$ is the class consisting of the [[terminal category]] and the [[empty category]] --- which is to say, the class of $\kappa$-small categories when $\kappa$ is the finite regular cardinal $2$ --- then being $\mathcal{J}$-filtered in this sense is equivalent to being [[connected category|connected]]. Note that this is not what the explicit definition given above for infinite regular cardinals would specialize to by simply setting $\kappa=2$ (that would be simply [[inhabited set|inhabitation]]).

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

\begin{itemize}%
\item A filtered [[preorder]] is the same as a [[direction|directed]] one: a \textbf{filtered [[(0,1)-category]]}.


\item Every category with a [[terminal object]] is filtered.


\item Every category which has finite [[colimit]]s is filtered.


\item A [[product]] of filtered categories is filtered.



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

\begin{itemize}%
\item [[sifted category]], [[sifted (∞,1)-category]]


\item [[directed set]], \textbf{filtered category}, [[filtered (∞,1)-category]]


\item [[direct category]]


\item [[filtered colimit]]



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

\begin{itemize}%
\item [[Jiří Adámek]], [[Francis Borceux]], [[Stephen Lack]], and [[Jiří Rosický]], \emph{A classification of accessible categories}, Journal of Pure and Applied Algebra 175:7-30, 2002, (\href{http://maths.mq.edu.au/~slack/papers/acc.html}{web page with PS fulltext}).

\end{itemize}
[[!redirects filtered categories]] [[!redirects cofiltered category]] [[!redirects cofiltered categories]] [[!redirects filtrant category]] [[!redirects filtrant categories]] [[!redirects filtered diagram]]



\end{document}