nLab
commutativity of limits and colimits

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.

This page lists some of these.

Filtered colimits commute with finite limits

For CC a small filtered category, the functor colim C:[C,Set]Setcolim_C : [C,Set] \to Set commutes with finite limits.

More in detail, let

then the canonical morphism

colim Clim DFlim Dcolim CF colim_C lim_D F \stackrel{\simeq}{\to} lim_D colim_C F

is an isomorphism.

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

Sifted colimits commute with finite products

Similarly to the example of filtered limits, for CC a small sifted category, the functor colim C:[C,Set]Setcolim_C : [C,Set] \to Set commutes with finite products. 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 CC×CC \to C \times C is a final functor.

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

More precisely, if GG is a finite group, CC is a small cofiltered category and F:CGSetF : C \to G-Set is a functor, the canonical map

(limF)/Glim jF(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

If AA is a set, CC is a connected category, and F:C×ASetF : C\times A\to Set is a functor, the canonical map

aAlim cCF(c,a)lim cC aAF(c,a) \coprod_{a\in A} \lim_{c\in C} F(c,a)\to \lim_{c\in C} \coprod_{a\in A}F(c,a)

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

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 AH\mathbf{H}^A\to\mathbf{H} induces an equivalence of (∞,1)-topoi H AH /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.

Certain colimits are stable by base change

Let CC be a category with pullbacks and colimits of shape DD.

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

(colim DF)× ZY colim DF Y Z \array{ (colim_D F) \times_Z Y &\to& colim_D F \\ \downarrow && \downarrow \\ Y &\to & Z }

the canonical morphism

colim dD(F(d)× ZY)(colim DF)× ZY 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

but not in for instance

  • C=C = Ab.

Remark

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

Revised on January 12, 2014 10:59:27 by Marc Hoyois (24.148.85.118)