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\begin{document}

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\section*{commutativity of limits and colimits}

\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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{preservation_by_functor_categories_and_localizations}{Preservation by functor categories and localizations}\dotfill \pageref*{preservation_by_functor_categories_and_localizations} \linebreak
\noindent\hyperlink{FilteredColimitsCommuteWithFiniteLimits}{Filtered colimits commute with finite limits}\dotfill \pageref*{FilteredColimitsCommuteWithFiniteLimits} \linebreak
\noindent\hyperlink{SiftedColimitsCommuteWithFiniteProducts}{Sifted colimits commute with finite products}\dotfill \pageref*{SiftedColimitsCommuteWithFiniteProducts} \linebreak
\noindent\hyperlink{taking_orbits_under_the_action_of_a_finite_group_commutes_with_cofiltered_limits}{Taking orbits under the action of a finite group commutes with cofiltered limits}\dotfill \pageref*{taking_orbits_under_the_action_of_a_finite_group_commutes_with_cofiltered_limits} \linebreak
\noindent\hyperlink{coproducts_commute_with_connected_limits}{Coproducts commute with connected limits}\dotfill \pageref*{coproducts_commute_with_connected_limits} \linebreak
\noindent\hyperlink{classes_of_limits_and_sound_doctrines}{Classes of limits and sound doctrines}\dotfill \pageref*{classes_of_limits_and_sound_doctrines} \linebreak
\noindent\hyperlink{ColimitsStableByBaseChange}{Relation to stability under base change}\dotfill \pageref*{ColimitsStableByBaseChange} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

In general, [[limit]]s and [[colimit]]s do not commute. It is therefore of interest to list the special conditions under which certain limits do commute with certain colimits.

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

Let $C$ and $D$ be (usually [[small category|small]]) categories, and $E$ a category that has both $C$-colimits and $D$-limits. Then for any functor $F \colon C \times D \to E$, there is a canonical morphism

\begin{equation}
colim_C lim_D F 
  \to
   lim_D colim_C F.
\label{FormulaForCommutativityOfLimitsOverColimits}\end{equation}
We say that \textbf{$C$-colimits commute with $D$-limits in $E$} if this is an [[isomorphism]] for all such $F$. This is equivalent to both of the statements:

\begin{itemize}%
\item The functor $colim_C : [C,E] \to E$ [[preserved limit|preserves]] (i.e. ``commutes with'') $D$-limits.
\item The functor $lim_D : [D,E] \to E$ preserves $C$-colimits.

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

\hypertarget{preservation_by_functor_categories_and_localizations}{}\subsubsection*{{Preservation by functor categories and localizations}}\label{preservation_by_functor_categories_and_localizations}

If $C$-colimits commute with $D$-limits in $E$, then the same is true in any [[functor category]] $[J,E]$, since limits and colimits in the latter are both pointwise in $E$.

Also, if $C$-colimits commute with $D$-limits in $E$, and if $E'$ is a [[reflective subcategory]] of $E$ with a reflector $L$ that preserves $D$-limits, then $C$-colimits also commute with $D$-limits in $E'$. This follows because the functor $colim_C : [C,E'] \to E'$ factors as the composite $[C,E'] \hookrightarrow [C,E] \xrightarrow{colim_C} E \xrightarrow{L} E'$ in which all three functors preserve $D$-limits.

\hypertarget{FilteredColimitsCommuteWithFiniteLimits}{}\subsubsection*{{Filtered colimits commute with finite limits}}\label{FilteredColimitsCommuteWithFiniteLimits}

In $Set$, [[filtered category|filtered]] colimits commute with finite limits. In fact, $C$ is a [[filtered category]] \emph{if and only if} $C$-colimits commute with finite limits in $Set$. More generally, filtered colimits commute with [[L-finite category|L-finite]] limits.

By the above remarks, it follows that filtered colimits commute with finite limits in any [[Grothendieck topos]].

\hypertarget{SiftedColimitsCommuteWithFiniteProducts}{}\subsubsection*{{Sifted colimits commute with finite products}}\label{SiftedColimitsCommuteWithFiniteProducts}

Again in [[Set]] (and hence also in any [[topos]]), [[sifted category|sifted colimits]] commute with [[finite products]]. In fact, this is usually taken to be the definition of a [[sifted category]], and then a theorem of \href{sifted+colimit#GabrielUlmer71}{Gabriel-Ulmer 71} characterizes sifted categories as those for which the [[diagonal functor]] $C \to C \times C$ is a [[final functor]].

As a special case, [[categories with finite products are cosifted]].

For more on this see at \emph{[[distributivity of products and colimits]]}.

\hypertarget{taking_orbits_under_the_action_of_a_finite_group_commutes_with_cofiltered_limits}{}\subsubsection*{{Taking orbits under the action of a finite group commutes with cofiltered limits}}\label{taking_orbits_under_the_action_of_a_finite_group_commutes_with_cofiltered_limits}

This means that if $G$ is a finite group, $C$ is a small cofiltered category and $F : C \to G Set$ is a functor, the canonical map

\begin{displaymath}
(\lim F)/G \to \lim_{j \in F} (F(j)/G)
\end{displaymath}
is an isomorphism. This fact is mentioned by Andr\'e{} Joyal in \emph{Foncteurs analytiques et esp\`e{}ces de structures}; a proof can be found \href{http://www.matem.unam.mx/~omar/notes/fingrpcomm.html}{here}.

\hypertarget{coproducts_commute_with_connected_limits}{}\subsubsection*{{Coproducts commute with connected limits}}\label{coproducts_commute_with_connected_limits}

\begin{example}
\label{CoproductsCommutingWithConnectedLimits}\hypertarget{CoproductsCommutingWithConnectedLimits}{}
Let $A$ be a [[set]], $C$ a [[connected category]], and $F \colon C\times A \longrightarrow Set$ a [[functor]]. Then the canonical morphism

\begin{displaymath}
\underset{a\in A}{\coprod} \underset{\longleftarrow}{\lim}_{c\in C} F(c,a)
    \longrightarrow 
   \underset{\longleftarrow}{\lim}_{c\in C} \coprod_{a\in A}F(c,a)
\end{displaymath}
is an [[isomorphism]]. This remains true if [[Set]] is replaced by any [[Grothendieck topos]].

\end{example}
More generally, if $\mathbf{H}$ is an [[(∞,1)-topos]], $A$ is an [[n-groupoid]], and $C$ is a small [[(∞,1)-category]] whose [[classifying space]] is [[n-connected]], then $C$-limits commute with $A$-colimits in $\mathbf{H}$. This follows from the fact that the colimit functor $\mathbf{H}^A\to\mathbf{H}$ induces an equivalence of (∞,1)-topoi $\mathbf{H}^A\simeq \mathbf{H}_{/A}$. For example, if $C$ is a [[cofiltered (∞,1)-category]] or even a [[cosifted (∞,1)-category]], then the classifying space of $C$ is weakly contractible and hence $C$-limits commute with $A$-colimits in $\mathbf{H}$ for any [[∞-groupoid]] $A$.

\hypertarget{classes_of_limits_and_sound_doctrines}{}\subsubsection*{{Classes of limits and sound doctrines}}\label{classes_of_limits_and_sound_doctrines}

In general, for any class of limits $\Phi$, one may consider the class of all colimits that commute with $\Phi$-limits and dually. These classes of limits and colimits share many of the properties of the above examples, especially when $\Phi$ is a [[sound doctrine]].

\hypertarget{ColimitsStableByBaseChange}{}\subsection*{{Relation to stability under base change}}\label{ColimitsStableByBaseChange}

 Stability of a colimit under pullback looks informally like a ``commutativity'' condition between colimits and pullbacks, but it is not actually in general an instance of the general notion of commutativity of limits and colimits, though it is an instance of [[distributivity of limits over colimits]]. See also [[pullback-stable colimit]] for more.

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

\begin{itemize}%
\item [[distributivity of limits over colimits]]
\item [[pullback-stable colimit]]

\end{itemize}
[[!redirects commutativity of limits and colimits]] [[!redirects commutativity of limits with colimits]] [[!redirects commutativity of a limit with a colimit]] [[!redirects commutativity of limits over colimits]] [[!redirects commutativity of a limit over a colimit]]

[[!redirects commutativity of colimits and limits]] [[!redirects commutativity of colimits with limits]] [[!redirects commutativity of a colimit with a limit]]

[[!redirects limits commuting with colimits]]



\end{document}