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 refers to the situation when we have a diagram of the form
and the canonical morphism
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, in its easiest possible formulation, is stated for $X$ as above (1), and requires the canonical map
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.