Theorems · Definition · category theory

CategoryTheory.Grothendieck.fiberwiseColimitMapCompEquivalence

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {F G : CategoryTheory.Functor C CategoryTheory.Cat} →
      (α : F ⟶ G) →
        [∀ (X : C), (α.app X).toFunctor.Final] →
          (H : CategoryTheory.Functor (CategoryTheory.Grothendieck G) (Type u₂)) →
            CategoryTheory.Limits.fiberwiseColimit ((CategoryTheory.Grothendieck.map α).comp H) ≅
              CategoryTheory.Limits.fiberwiseColimit H

A natural transformation α : F ⟶ G between functors F G : C ⥤ Cat which is final on each fiber (α.app X) induces an equivalence of fiberwise colimits of map α ⋙ H and H for each functor H : Grothendieck G ⥤ Type.

Defined in

Mathlib.CategoryTheory.Limits.Final

Cited by

0 results in Mathlib

Foundations

Depth 52 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Functor.Final