Theorems · Theorem · category theory

CategoryTheory.NormalEpiCategory.mono_of_cancel_zero

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

If g ≫ f = 0 implies g = 0 for all g, then f 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.HasFiniteCoproductsCategoryTheory.Limits.HasCokernelsCategoryTheory.IsNormalEpiCategoryCategoryTheory.Limits.HasZeroObject