Theorems · Theorem · category theory

CategoryTheory.NormalMonoCategory.epi_of_zero_cancel

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
  [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasKernels C]
  [CategoryTheory.IsNormalMonoCategory C] [CategoryTheory.Limits.HasZeroObject C] {X Y : C} (f : X ⟶ Y),
  (∀ (Z : C) (g : Y ⟶ Z), CategoryTheory.CategoryStruct.comp f g = 0 → g = 0) → CategoryTheory.Epi f

If f ≫ g = 0 implies g = 0 for all g, then g is a monomorphism.

Defined in

Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers

Cited by

0 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasFiniteProductsCategoryTheory.Limits.HasKernelsCategoryTheory.IsNormalMonoCategoryCategoryTheory.Limits.HasZeroObject