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. (See also at permuting limits and colimits.)

Definition

Let $C$ and $D$ be (usually 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

(1)$$colim_C lim_D F \to lim_D colim_C F.$$

We say that $C$-colimits commute with $D$-limits in $E$ if this is an isomorphism for all such $F$. This is equivalent to both of the statements:

• The functor $colim_C : [C,E] \to E$ preserves (i.e. “commutes with”) $D$-limits.
• The functor $lim_D : [D,E] \to E$ preserves $C$-colimits.

Examples

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.

Filtered colimits commute with finite limits

In Set, filtered colimits commute with finite limits (e.g. MacLane (1971), §IX.2 Thm. 1 (p. 211)).

In fact, $C$ is a filtered category if and only if $C$-colimits commute with finite limits in $Set$. More generally, filtered colimits commute with L-finite limits.

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

Sifted colimits commute with finite products

Again in Set (and hence also in any Grothendieck topos), 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 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 distributivity of products and colimits.

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

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

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

Coproducts commute with connected limits

Proposition

Let $A$ be a set, $C$ a connected category, and $F \colon C\times A \longrightarrow Set$ a functor. Then the canonical morphism

$$\underset{a\in A}{\coprod} \underset{\longleftarrow}{\lim}_{c\in C} F(c,a) \longrightarrow \underset{\longleftarrow}{\lim}_{c\in C} \coprod_{a\in A}F(c,a)$$

is an isomorphism. This remains true if Set is replaced by any Grothendieck topos.

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 $A$.

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.

Relation to stability under base change

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.

Related concepts

• distributivity of limits over colimits, permuting limits and colimits

• pullback-stable colimit

• absolute colimits

References

• Peter Hilton, Commuting limits, Cahiers de Topologie et Géométrie Différentielle Catégoriques 10.1 (1968): 127-138.

• Beno Eckmann, Peter John Hilton, Commuting limits with colimits, Journal of Algebra 11 1 (1969) 116-144.

• Armin Frei, and John L. MacDonald, Limits in categories of relations and limit-colimit commutation, Journal of Pure and Applied Algebra 1 2 (1971) 179-197

• Saunders MacLane, §IX.2 in: Categories for the Working Mathematician, Graduate texts in mathematics, Springer (1971) [doi:10.1007/978-1-4757-4721-8]

• François Foltz, Sur la commutation des limites, Diagrammes 5 (1981) F1-F33

• Marie Bjerrum, Peter Johnstone, Tom Leinster, William F. Sawin, Notes on commutation of limits and colimits, Theory and Applications of Categories 30 (2015), 527-532 [arXiv:1409.7860, tac:3015]