commutativity of limits and colimits in nLab

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Contents

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 CC and DD be (usually small) categories, and EE a category that has both CC-colimits and DD-limits. Then for any functor F:C×D→EF \colon C \times D \to E, there is a canonical morphism

(1)colim Clim DF→lim Dcolim CF. colim_C lim_D F \to lim_D colim_C F.

We say that CC-colimits commute with DD-limits in EE if this is an isomorphism for all such FF. This is equivalent to both of the statements:

• The functor colim C:[C,E]→Ecolim_C : [C,E] \to E preserves (i.e. “commutes with”) DD-limits.
• The functor lim D:[D,E]→Elim_D : [D,E] \to E preserves CC-colimits.

Examples

Preservation by functor categories and localizations

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

Also, if CC-colimits commute with DD-limits in EE, and if E′E' is a reflective subcategory of EE with a reflector LL that preserves DD-limits, then CC-colimits also commute with DD-limits in E′E'. This follows because the functor colim C:[C,E′]→E′colim_C : [C,E'] \to E' factors as the composite [C,E′]↪[C,E]→colim CE→LE′[C,E'] \hookrightarrow [C,E] \xrightarrow{colim_C} E \xrightarrow{L} E' in which all three functors preserve DD-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, CC is a filtered category if and only if CC-colimits commute with finite limits in SetSet. 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→C×CC \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 GG is a finite group, CC is a small cofiltered category and F:C→GSetF : C \to G Set is a functor, the canonical map

(limF)/G→lim j∈F(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.

Coproducts commute with connected limits

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

Classes of limits and sound doctrines

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

• 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]