nLab permuting limits and colimits

Contents

Context

Limits and colimits

limits and colimits

Contents

Idea

One can wonder when an expression of the form $lim colim$ can be replaced by an expression of the form $colim lim$, 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 $X\colon I\times J\to C$ and the canonical morphism $colim_{i\in I} lim_{j\in J} X_{i,j} \to lim_{j\in J} colim_{i\in I} X_{i,j}$ 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 $X$ as above, and requires the canonical map $colim_{i\in I^J} lim_{j\in J} X_{i(j),j} \to lim_{j\in J} colim_{i\in I} X_{i,j}$ to be an isomorphism (or an equivalence in the case of ∞-categories). Note that here $i$ is an element of the functor category $I^J$, and $i(j)$ denotes its value at $j\in J$.

More generally, one may allow the indexing category $I$ to depend on $j\in J$, i.e., the functor $I\times J\to J$ is replaced by some arbitrary Grothendieck fibration in categories. Then $I^J$ 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.