Theorems · Theorem · category theory

CategoryTheory.MorphismProperty.relative.of_exists

∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  {F : CategoryTheory.Functor C D} {X Y : D} {P : CategoryTheory.MorphismProperty C} [F.Faithful] [F.Full]
  [P.RespectsIso] {f : X ⟶ Y},
  (∀ ⦃a : C⦄ (g : F.obj a ⟶ Y), ∃ b fst snd, ∃ (_ : CategoryTheory.IsPullback fst (F.map snd) f g), P snd) →
    CategoryTheory.MorphismProperty.relative F P f

Given a morphism property P which respects isomorphisms, then to show that a morphism f : X ⟶ Y satisfies P.relative it suffices to show that: * The morphism is representable. * For any morphism g : F.obj a ⟶ G, the property P holds for some represented pullback of f by g.

Defined in

Mathlib.CategoryTheory.MorphismProperty.Representable

Cited by

2 results in Mathlib

Foundations

Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.FaithfulCategoryTheory.Functor.FullCategoryTheory.MorphismProperty.RespectsIso