Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
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 a small filtered category, the functor commutes with finite limits.
More in detail, let
then the canonical morphism
is an isomorphism.
In fact, is a filtered category if and only if this is true for all finite and all functors .
Sifted colimits commute with finite products
Similarly to the example of filtered limits, for a small sifted category, the functor 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 is a final functor.
For more on this see at distributivity of products and colimits.
Taking orbits under the action of a finite group commutes with cofiltered limits
More precisely, if is a finite group, is a small cofiltered category and is a functor, the canonical map
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
Let be a set, a connected category, and a functor. Then the canonical morphism
is an isomorphism. This remains true if Set is replaced by any Grothendieck topos.
More generally, if is an (∞,1)-topos, is an n-groupoid, and is a small (∞,1)-category whose classifying space is n-connected, then -limits commute with -colimits in . This follows from the fact that the colimit functor induces an equivalence of (∞,1)-topoi . For example, if is a cofiltered (∞,1)-category or even a cosifted (∞,1)-category, then the classifying space of is weakly contractible and hence -limits commute with -colimits in for any ∞-groupoid .
Classes of limits and sound doctrines
In general, for any class of limits , one may consider the class of all colimits that commute with -limits and dually. These classes of limits and colimits share many of the properties of the above examples, especially when is a sound doctrine.
Certain colimits are stable by base change
Let be a category with pullbacks and colimits of shape .
We say that colimits of shape are stable by base change or stable under pullback if for every functor and for all pullback diagrams of the form
the canonical morphism
is an isomorphism.
All colimits are stable under base change in for instance
but not in for instance
In topos theory and (∞,1)-topos theory one says that colimits are universal if they are preserved under pullback.