Theorems · Theorem · category theory

CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.exists_pushouts

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {G : C} [CategoryTheory.Abelian C],
  CategoryTheory.IsSeparator G →
    ∀ {X Y : C} (p : X ⟶ Y) [CategoryTheory.Mono p],
      ¬CategoryTheory.IsIso p →
        ∃ X' i p',
          ∃ (_ : (CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms G).pushouts i) (_ :
            ¬CategoryTheory.IsIso i) (_ : CategoryTheory.Mono p'), CategoryTheory.CategoryStruct.comp i p' = p

If p : X ⟶ Y is a monomorphism that is not an isomorphism, there exists a subobject X' of Y containing X (but different from X) such that the inclusion X ⟶ X' is a pushout of a monomorphism in the family generatingMonomorphisms G.

Defined in

Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Cited by

1 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.Mono