In category theory one may ask when an expression of the form “” can be replaced by an expression of the form “”, 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 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) – is an object of the functor category , and denotes the value of this functor at .
More generally, one may allow the indexing category to depend on , in that the functor is replaced by some arbitrary Grothendieck fibration in categories and 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.