Theorems · Definition · category theory

CategoryTheory.Functor.PushoutObjObj.ofIsInitialRight

{C₁ : Type u₁} →
  {C₂ : Type u₂} →
    {C₃ : Type u₃} →
      [inst : CategoryTheory.Category.{v₁, u₁} C₁] →
        [inst_1 : CategoryTheory.Category.{v₂, u₂} C₂] →
          [inst_2 : CategoryTheory.Category.{v₃, u₃} C₃] →
            (F : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C₃)) →
              {X₁ Y₁ : C₁} →
                (f₁ : X₁ ⟶ Y₁) →
                  {X₂ Y₂ : C₂} →
                    (f₂ : X₂ ⟶ Y₂) →
                      [CategoryTheory.Limits.PreservesColimitsOfShape (CategoryTheory.Discrete PEmpty.{1}) (F.obj X₁)] →
                        [CategoryTheory.Limits.PreservesColimitsOfShape (CategoryTheory.Discrete PEmpty.{1})
                              (F.obj Y₁)] →
                          CategoryTheory.Limits.IsInitial X₂ → F.PushoutObjObj f₁ f₂

A Functor.PushoutObjObj structure for a functor F : C₁ ⥤ C₂ ⥤ C₃ and morphisms f₁ : X₁ ⟶ Y₁ and f₂ : X₂ ⟶ Y₂ when X₂ is initial and both F.obj X₁ and F.obj Y₁ preserve the initial object.

Defined in

Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj

Cited by

6 results in Mathlib

Foundations

Depth 38 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.PreservesColimitsOfShapeCategoryTheory.Limits.PreservesColimitsOfShape