Theorems · Definition · category theory

SheafOfModules.GeneratingSections.ofEpi

{C : Type u'} →
  [inst : CategoryTheory.Category.{v', u'} C] →
    {J : CategoryTheory.GrothendieckTopology C} →
      {R : CategoryTheory.Sheaf J RingCat} →
        [inst_1 : CategoryTheory.HasWeakSheafify J AddCommGrpCat] →
          [inst_2 : J.WEqualsLocallyBijective AddCommGrpCat] →
            {M N : SheafOfModules R} →
              M.GeneratingSections → (p : M ⟶ N) → [CategoryTheory.Epi p] → N.GeneratingSections

If M ⟶ N is an epimorphism and that M is generated by some sections, then N is generated by the images of these sections.

Defined in

Mathlib.Algebra.Category.ModuleCat.Sheaf.Generators

Cited by

8 results in Mathlib

Foundations

Depth 110 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.HasWeakSheafifyCategoryTheory.GrothendieckTopology.WEqualsLocallyBijectiveCategoryTheory.Epi