nLab permuting limits and colimits

1. 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,

• distributivity:

2. 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

3. 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.

4. Related concepts

• commutativity of limits and colimits

• distributivity of limits and colimits