Theorems · Theorem · category theory

CategoryTheory.isIso_of_hom_simple

∀ {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},
  f ≠ 0 → CategoryTheory.IsIso f

The part of Schur's lemma that holds in any preadditive category with kernels: that a nonzero morphism between simple objects is an isomorphism.

Defined in: Mathlib.CategoryTheory.Preadditive.Schur
Cited by: 1 results in Mathlib
Foundations: Depth 36 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Preadditive CategoryTheory.Limits.HasKernels CategoryTheory.Simple CategoryTheory.Simple

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites7

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.Preadditivestatement and proof · cited by 3,309
CategoryTheory.IsIsostatement · cited by 1,156
CategoryTheory.Limits.HasKernelsstatement and proof · cited by 67
CategoryTheory.Simplestatement and proof · cited by 39
CategoryTheory.isIso_of_mono_of_nonzeroproof · cited by 5

Cited by1

Results whose statement or proof uses this declaration.

CategoryTheory.isIso_iff_nonzeroproof · cited by 3