# nLab permuting limits and colimits

Contents

### Context

#### Limits and colimits

limits and colimits

# Contents

## Idea

In category theory one may ask 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”.

There are (at least) two different types of such “permutation” of limits and colimits:

## Commutativity of limits and colimits

Commutativity of limits and colimits refers to the situation when we have a diagram of the form

(1)$X \,\colon\, I\times J\to C$

and the canonical morphism

(2)$\underset {i\in I} {colim} \; \underset {j\in J} {lim} \; X_{i,j} \longrightarrow \underset {j\in J} {lim} \; \underset {i\in I} {colim} \; X_{i,j}$

is an isomorphism (or an equivalence in the case of ∞-categories).

This situation is discussed at commutativity of limits and colimits

## Distributivity of limits over colimits

Distributivity of limits over colimits, in its easiest possible formulation, is stated for $X$ as above (1), and requires the canonical map

(3)$\underset {i\in I^J} {colim} \; \underset {j\in J} {lim} \; X_{i(j),j} \longrightarrow \underset {j\in J} {lim} \; \underset {i\in I} {colim} X_{i,j}$

to be an isomorphism (or an equivalence in the case of ∞-categories), where now – on the left of (3)$i$ is an object of the functor category $I^J$, and $i(j)$ denotes the value of this functor at $j\in J$.

More generally, one may allow the indexing category $I$ to depend on $j\in J$, in that the functor $I\times J\to J$ is replaced by some arbitrary Grothendieck fibration in categories and $I^J$ by the category of its sections.

This situation is discussed at distributivity of limits over colimits.

Last revised on December 18, 2022 at 18:28:13. See the history of this page for a list of all contributions to it.