nLab
commutativity of limits and colimits

Idea

In general, limits and colimits do not commute.

It is therefore of interest to list the special conditions under which certain limits do commute with certain colimits.

This page lists some of these.

Filtered colimits commute with finite limits

For $C$ a small filtered category, the functor ${\mathrm{colim}}_C:[C,\mathrm{Set}]\to \mathrm{Set}$ commutes with finite limits.

More in detail, let

• $C$ be a small filtered category

• $D$ be a finite category;

• $F:C\times D^{\mathrm{op}}\to \mathrm{Set}$ a functor;

then the canonical morphism

$${\mathrm{colim}}_C{\lim }_{D}F\stackrel{\simeq }{\to }{\lim }_D{\mathrm{colim}}_{C}F$$colim_C lim_D F \stackrel{\simeq}{\to} lim_D colim_C F

is an isomorphism.

In fact, $C$ is a filtered category if and only if this is true for all finite $D$ and all functors $F:C\times D^{\mathrm{op}}\to \mathrm{Set}$.

Sifted colimits commute with finite products

Similarly to the example of filtered limits, for $C$ a small sifted category, the functor ${\mathrm{colim}}_C:[C,\mathrm{Set}]\to \mathrm{Set}$ commutes with finite limits. In fact, this is usually taken to be the definition of a sifted category, and then a theorem of Gabriel and Ulmer characterizes sifted categories as those for which the diagonal functor $C\to C\times C$ is a final functor.

Taking orbits under the action of a finite group commutes with cofiltered limits

More precisely, if $G$ is a finite group, $C$ is a small cofiltered category and $F:C\to G-\mathrm{Set}$ is a functor, the canonical map

$$(\lim F)/G\to \underset{j\in F}{\lim }(F(j)/G)$$(\lim F)/G \to \lim_{j \in F} (F(j)/G)

is an isomorphism. This fact is mentioned by André Joyal in Foncteurs analytiques et espèces de structures; a proof can be found here.

Certain colimits are stable by base change

Let $C$ be a category with pullbacks and colimits of shape $D$.

We say that colimits of shape $D$ are stable by base change or stable under pullback if for every functor $F:D\to C$ and for all pullback diagrams of the form

$$\begin{array}{ccc}({\mathrm{colim}}_{D}F){\times }_{Z}Y& \to & {\mathrm{colim}}_{D}F\\ \downarrow & & \downarrow \\ Y& \to & Z\end{array}$$\array{ (colim_D F) \times_Z Y &\to& colim_D F \\ \downarrow && \downarrow \\ Y &\to & Z }

the canonical morphism

$${\mathrm{colim}}_{d\in D}(F(d){\times }_{Z}Y)\stackrel{\simeq }{\to }({\mathrm{colim}}_{D}F){\times }_{Z}Y$$colim_{d \in D} (F(d) \times_Z Y) \stackrel{\simeq}{\to} (colim_D F) \times_Z Y

is an isomorphism.

All colimits are stable under base change in for instance

• $C=$ Set;
• hence for $C=$ a presheaf category $[S^{\mathrm{op}},\mathrm{Set}]$ (since colimits in such $C$ are computed objectwise in $\mathrm{Set}$), see limits and colimits by example);
• more generally, any topos;

but not in for instance

• $C=$ Ab.

Remark

In topos theory and (∞,1)-topos theory one says that colimits are universal if they are preserved under pullback.