nLab permuting limits and colimits

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Idea

In category theory one may ask 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”.

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:I×JC X \,\colon\, I\times J\to C

and the canonical morphism

(2)colimiIlimjJX i,jlimjJcolimiIX i,j \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 XX as above (1), and requires the canonical map

(3)colimiI JlimjJX i(j),jlimjJcolimiIX i,j \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)ii is an object of the functor category I JI^J, and i(j)i(j) denotes the value of this functor at jJj\in J.

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