mathlib documentation


Filtered colimits commute with finite limits. #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

We show that for a functor F : J × K ⥤ Type v, when J is finite and K is filtered, the universal morphism colimit_limit_to_limit_colimit F comparing the colimit (over K) of the limits (over J) with the limit of the colimits is an isomorphism.

(In fact, to prove that it is injective only requires that J has finitely many objects.)

References #

Injectivity doesn't need that we have finitely many morphisms in J, only that there are finitely many objects.

This follows this proof from

  • Borceux, Handbook of categorical algebra 1, Theorem 2.13.4 although with different names.