Theorems · Definition · category theory

CategoryTheory.Limits.colimitFiberwiseColimitIso

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {F : CategoryTheory.Functor C CategoryTheory.Cat} →
      {H : Type u₂} →
        [inst_1 : CategoryTheory.Category.{v₂, u₂} H] →
          (G : CategoryTheory.Functor (CategoryTheory.Grothendieck F) H) →
            [inst_2 :
                ∀ {X Y : C} (f : X ⟶ Y),
                  CategoryTheory.Limits.HasColimit
                    ((F.map f).toFunctor.comp ((CategoryTheory.Grothendieck.ι F Y).comp G))] →
              [inst_3 : CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.fiberwiseColimit G)] →
                CategoryTheory.Limits.colimit (CategoryTheory.Limits.fiberwiseColimit G) ≅
                  CategoryTheory.Limits.colimit G

For every functor G on the Grothendieck construction Grothendieck F, if G has a colimit and every fiber of G has a colimit, then taking this colimit is isomorphic to first taking the fiberwise colimit and then the colimit of the resulting functor.

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Grothendieck

Cited by

7 results in Mathlib

Foundations

Depth 55 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasColimitCategoryTheory.Limits.HasColimit