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\begin{document}

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\section*{filtered limit}

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

\hypertarget{limits_and_colimits}{}\paragraph*{{Limits and colimits}}\label{limits_and_colimits}

[[!include infinity-limits - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\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{commutation_with_small_limits}{Commutation with $\kappa$-small limits}\dotfill \pageref*{commutation_with_small_limits} \linebreak
\noindent\hyperlink{flat_functors_and_points_of_presheaf_toposes}{Flat functors and points of presheaf toposes}\dotfill \pageref*{flat_functors_and_points_of_presheaf_toposes} \linebreak
\noindent\hyperlink{description_in_set_grp_top_and_alike}{Description in Set, Grp, Top and alike}\dotfill \pageref*{description_in_set_grp_top_and_alike} \linebreak
\noindent\hyperlink{more}{More}\dotfill \pageref*{more} \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}

Most of the earliest instances of [[limits]] and [[colimits]] used in mathematics were for [[diagrams]] indexed by the [[partially ordered set]] of [[natural numbers]], which we now call \emph{[[sequential limit|sequential]] (co)limits}. Many of the nice features of these (co)limits apply more generally to \emph{[[codirected limits]]} and \emph{[[directed colimits]]}, where the indexing category is a (co)-[[directed set]] (much as [[sequences]] in [[topology]] generalise fruitfully to [[nets]]). A \emph{filtered (co)limit} is a further generalisation of this, essentially removing the requirement that the indexing category be a [[poset]] while preserving the directedness aspect in a [[categorification|categorified]] way.

Another very important class of early limits and colimits involved situations that generalised [[intersections]] and [[unions]]. If one is looking at a [[family of subsets]] of some [[set]], then one can close it up under finite intersections and/or unions (if they are not already included) and use it to index diagrams. For instance, the family of [[continuous functions]] defined on [[open neighbourhoods]] of some point in a [[topological space]] will have this property. It was noticed that these limits and colimits behaved very nicely and a closer look showed that it was the \emph{(co)filtering} nature of the indexing category that was the key. This also leads us to filtered (co)limits.

So, a \emph{filtered colimit} is a [[colimit]] over a [[diagram]] from a [[filtered category]], and a \emph{cofiltered limit} (sometimes called a filtered limit) is a [[limit]] over a [[diagram]] from a [[cofiltered category]]. Taken in a suitable category such as [[Set]], \textbf{a colimit being filtered is equivalent to its commuting with [[finite limits]]}.

More generally, for $\kappa$ a [[regular cardinal]], a \emph{$\kappa$-filtered colimit} is one over a $\kappa$-filtered category (and dually), and when taken with values in [[Set]] these are precisely the colimits that commute with $\kappa$-[[small limit]]s.

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

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{filtered colimit} or \textbf{finitely filtered colimit} is a [[colimit]] of a [[functor]] $F\colon D \to C$ where $D$ is a [[filtered category]].

For $\kappa$ a [[regular cardinal]] a \textbf{$\kappa$-filtered colimit} is one over a $\kappa$-filtered category.

Similarly, a \textbf{cofiltered limit} is a [[limit]] of a functor $F\colon D \to C$ where $D$ is a [[cofiltered category]], or equivalently of a [[contravariant functor]] $F\colon D \to C$ (that is a functor $F\colon D^{op} \to C$) where $D$ is a filtered category.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
A cofiltered limit may also be called a \textbf{filtered limit} (although this can be unclear); the respective terms \textbf{filtered [[direct limit]]} and \textbf{filtered [[inverse limit]]} are also popular.

\end{remark}
A [[functor]] that preserves all finitely filtered colimits is called a \emph{[[finitary functor]]} .

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

\hypertarget{commutation_with_small_limits}{}\subsubsection*{{Commutation with $\kappa$-small limits}}\label{commutation_with_small_limits}

The following is the crucial property of filtered colimits: that they commute with [[finite limits]].

\begin{remark}
\label{}\hypertarget{}{}
For $C$ and $D$ two [[diagram]] categories and

\begin{displaymath}
F : C \times D \to Set
\end{displaymath}
a diagram, there is a canonical morphism

\begin{displaymath}
\lambda
   :
  {\lim_\to}_C {\lim_\leftarrow}_D F
   \to 
  {\lim_\leftarrow}_D {\lim_\to}_C  F
\end{displaymath}
from the [[colimit]] over $C$ of the [[limit]] over $D$ to the limit over $D$ of the colimit over $C$ of $F$:

$\lambda$ is given by a [[cone]], whose components

\begin{displaymath}
\lambda_d : 
    {\lim_\to}_C {\lim_\leftarrow}_D F
     \to
    {\lim_\to}_C  F(-,d)
\end{displaymath}
are in turn given by a [[cocone]] with components

\begin{displaymath}
(\lambda_d)_c : 
    {\lim_\leftarrow}_D F(c,-)
     \to
    {\lim_\to}_C  F(-,d)
   \,.
\end{displaymath}
This finally take to have as components

\begin{displaymath}
{\lim_\leftarrow}_D F(c,d)
     \to   
       F(c,d)
     \to
    {\lim_\to}_C  F(c,d)
  \,.
\end{displaymath}
One checks that this indeed implies that all the components are natural and gives the existence of the original morphism.

Notice that in general $\lambda$ is \emph{not} an [[isomorphism]].

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
We say the [[limit]] ${\lim_\leftarrow}_D F(-,-)$ \textbf{commutes} with the colimit ${\lim_\to}_C F(-,-)$ if the morphism $\lambda$ above is an [[isomorphism]]

\begin{displaymath}
{\lim_\to}_C {\lim_\leftarrow}_D F
   \stackrel{\simeq}{\to}
  {\lim_\leftarrow}_D {\lim_\to}_C  F
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{FilteredColimitsCommuterWithFiniteLimits}\hypertarget{FilteredColimitsCommuterWithFiniteLimits}{}
In [[Set]], filtered colimits commute with [[finite limits]].

In fact, [[filtered categories]] $C$ are precisely those shapes of [[diagram]] categories such that colimits over them commute with all finite limits in Sets.

More generally, for $\kappa$ a [[regular cardinal]], then $\kappa$-filtered colimits commute with $\kappa$-small limits.

\end{prop}
A detailed components proof of the first part is in \hyperlink{Borceux}{Borceux, theorem I2.13.4} or (\hyperlink{BJTS14}{BJTS 14}).

For more on this see also [[limits and colimits by example]].

\begin{warning}
\label{}\hypertarget{}{}
It is not true that filtered colimits and finite limits commute in \emph{any} category $C$ which has them. A simple example is where $C$ is the [[poset]] of [[closed subspaces]] of the [[one-point compactification]] $\mathbb{N} \cup \{\infty\}$ of the [[discrete space]] of [[natural numbers]]. If $A = \{\infty\}$ and $B_i$ ranges over finite subsets of $\mathbb{N}$, then $A \times colim_i B_i = \{\infty\} \cap (\mathbb{N} \cup \{\infty\}) = \{\infty\}$, but $colim_i A \times B_i = colim_i \{\infty\} \cap B_i = colim_i \emptyset = \emptyset$.

\end{warning}
According to 1.5 and 1.21 in [[LPAC]], a category has $\kappa$-[[directed colimits]] precisely if it has $\kappa$-filtered ones, and a functor preserves $\kappa$-directed colimits iff it preserves $\kappa$-filtered ones. A proof of this result, following Adamek \& Rosicky, may be found [[theorem:directed colimits imply filtered colimits|here]].

The fact that directed colimits suffice to obtain all filtered ones may be regarded as a convenience similar to the fact that all [[colimits]] can be constructed from [[coproducts]] and [[coequalizers]]. Of course, a [[duality|dual]] result holds for [[codirected limits]].

\hypertarget{flat_functors_and_points_of_presheaf_toposes}{}\subsubsection*{{Flat functors and points of presheaf toposes}}\label{flat_functors_and_points_of_presheaf_toposes}

Let $C$ be a small category. A functor $F: C \to Set$ is \textbf{[[flat functor|flat]]} if it is a filtered colimit of [[representable functor]]s.

Equivalently, a [[module]] $F: C \to Set$ is flat if and only if the [[tensor product of functors|tensor product]]

\begin{displaymath}
- \otimes_C F: Set^{C^{op}} \to Set
\end{displaymath}
is [[left exact functor|left exact]]. One may prove as a corollary that if $C$ is [[finitely complete category|finitely complete]], $F$ is flat if and only if it is [[left exact functor|left exact]] (preserves finite limits). Since this tensor product is automatically a left adjoint, we have the following basic result:

\begin{prop}
\label{}\hypertarget{}{}
For $C$ a [[small category]], the category of [[point of a topos|topos points]] of the [[presheaf topos]] $Set^{C^{op}}$ (i.e., [[geometric morphism]]s $Set \to Set^{C^{op}}$ and natural transformations between them) is equivalent to the category of flat modules on $C$.

\end{prop}
\hypertarget{description_in_set_grp_top_and_alike}{}\subsubsection*{{Description in Set, Grp, Top and alike}}\label{description_in_set_grp_top_and_alike}

Elements in filtered colimits in [[Set]] and [[Grp]] are given as classes of equivalences, so called [[germs]]. Filtered limits in [[Set]] and [[Top]] are given as families of compatible elements, so called [[threads]].

\hypertarget{more}{}\subsubsection*{{More}}\label{more}

(More was/is to be written here, including an application to [[geometric realization]], relation to [[Diaconescu's theorem]], and perhaps more.)

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

\begin{itemize}%
\item [[filtered category]], [[compact object]]


\item [[sifted colimit]], [[sifted (∞,1)-colimit]]


\item [[filtered (∞,1)-colimit]], [[filtered homotopy colimit]]



\end{itemize}
Filtered colimits are also important in the theory of [[locally presentable category|locally presentable]] and [[accessible category|accessible]] categories. See also [[pro-object]] and [[ind-object]].

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

Section 2.13 in part I of

\begin{itemize}%
\item [[Francis Borceux]], \emph{[[Handbook of Categorical Algebra]]}

\end{itemize}
Also

\begin{itemize}%
\item Marie Bjerrum, [[Peter Johnstone]], [[Tom Leinster]], William F. Sawin, \emph{Notes on commutation of limits and colimits} (\href{http://arxiv.org/abs/1409.7860}{arXiv:1409.7860})

\end{itemize}
[[!redirects filtered limit]] [[!redirects filtered limits]] [[!redirects cofiltered limit]] [[!redirects cofiltered limits]] [[!redirects filtered colimit]] [[!redirects filtered colimits]] [[!redirects filtered inverse limit]] [[!redirects filtered inverse limits]] [[!redirects cofiltered inverse limit]] [[!redirects cofiltered inverse limits]] [[!redirects filtered direct limit]] [[!redirects filtered direct limits]]



\end{document}