Theorems · Theorem · category theory

CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.exists_transfiniteCompositionOfShape

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {G : C} [inst_1 : CategoryTheory.Abelian C],
  CategoryTheory.IsSeparator G →
    ∀ {X : C} [CategoryTheory.IsGrothendieckAbelian.{w, v, u} C] {A : C} (f : A ⟶ X) [CategoryTheory.Mono f],
      ∃ J x x_1 x_2,
        ∃ (x_3 : WellFoundedLT J),
          Nonempty
            ((CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms G).pushouts.TransfiniteCompositionOfShape J
              f)

Let C be a Grothendieck abelian category. Assume that G : C is a generator of C. Then, any morphism in C is a transfinite composition of pushouts of monomorphisms in the family generatingMonomorphisms G which consists of the inclusions of the subobjects of G.

Defined in

Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Cited by

1 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.IsGrothendieckAbelianCategoryTheory.Mono