nLab
permuting limits and colimits

Contents

Contents

Idea

One can wonder when an expression of the form limcolimlim colim can be replaced by an expression of the form colimlimcolim lim, i.e., under which circumstances limits and colimits can be permuted.

Commutativity of limits and colimits

Commutativity? of limits and colimits refers to the situation

when we have a diagram X:I×JCX\colon I\times J\to C and the canonical morphism colim iIlim jJX i,jlim jJcolim iIX i,jcolim_{i\in I} lim_{j\in J} X_{i,j} \to lim_{j\in J} colim_{i\in I} X_{i,j} is an isomorphism (or an equivalence in the case of ∞-categories).

The article commutativity of limits and colimits explores the situations for which such a commutativity property holds.

Distributivity of limits over colimits

Distributivity? of limits over colimits, in its

easiest possible formulation, is stated for XX as above, and requires the canonical map colim iI Jlim jJX i(j),jlim jJcolim iIX i,jcolim_{i\in I^J} lim_{j\in J} X_{i(j),j} \to lim_{j\in J} colim_{i\in I} X_{i,j} to be an isomorphism (or an equivalence in the case of ∞-categories). Note that here ii is an element of the functor category I JI^J, and i(j)i(j) denotes its value at jJj\in J.

More generally, one may allow the indexing category II to depend on jJj\in J, i.e., the functor I×JJI\times J\to J is replaced by some arbitrary Grothendieck fibration in categories. Then I JI^J is replaced by the category of sections of this Grothendieck fibration.

The article distributivity of limits over colimits explores the situations for which such a distributivity property holds.

Last revised on December 25, 2015 at 17:06:27. See the history of this page for a list of all contributions to it.