Theorems · Theorem · category theory
CategoryTheory.Simple.mono_isIso_iff_nonzero
∀ {C : Type u} {inst : CategoryTheory.Category.{v, u} C} {inst_1 : CategoryTheory.Limits.HasZeroMorphisms C} {X : C}
[self : CategoryTheory.Simple X] {Y : C} (f : Y ⟶ X) [CategoryTheory.Mono f], CategoryTheory.IsIso f ↔ f ≠ 0- Defined in
- Mathlib.CategoryTheory.Simple
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 4 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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 · cited by 32,603
- CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
- CategoryTheory.IsIsostatement · cited by 1,156
- CategoryTheory.Monostatement · cited by 893
- CategoryTheory.Simplestatement and proof · cited by 39
Cited by6
Results whose statement or proof uses this declaration.
- CategoryTheory.isIso_of_mono_of_nonzeroproof · cited by 5
- CategoryTheory.Simple.of_isoproof · cited by 3
- CategoryTheory.id_nonzeroproof · cited by 3
- CategoryTheory.Functor.simple_of_simple_objproof · cited by 2
- CategoryTheory.zero_not_simpleproof · cited by 0
- CategoryTheory.indecomposable_of_simpleproof · cited by 0