Theorems · Theorem · category theory

CategoryTheory.Limits.hasColimit_of_hasColimit_fiberwiseColimit_of_hasColimit

∀ {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))]
  [CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.fiberwiseColimit G)], CategoryTheory.Limits.HasColimit G

We can infer that a functor G : Grothendieck F ⥤ H, with F : C ⥤ Cat, has a colimit from the fact that each of its fibers has a colimit and that these fiberwise colimits, as a functor C ⥤ H have a colimit.

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Grothendieck

Cited by

4 results in Mathlib

Foundations

Depth 54 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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