Theorems · Theorem · category theory

CategoryTheory.isIso_iff_nonzero

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
  [CategoryTheory.Limits.HasKernels C] {X Y : C} [CategoryTheory.Simple X] [CategoryTheory.Simple Y] (f : X ⟶ Y),
  CategoryTheory.IsIso f ↔ f ≠ 0

As a corollary of Schur's lemma for preadditive categories, any morphism between simple objects is (exclusively) either an isomorphism or zero.

Defined in

Mathlib.CategoryTheory.Preadditive.Schur

Cited by

3 results in Mathlib

Foundations

Depth 37 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasKernelsCategoryTheory.SimpleCategoryTheory.Simple