Theorems · Theorem · category theory
CategoryTheory.isIso_of_mono_of_nonzero
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C] {X Y : C}
[CategoryTheory.Simple Y] {f : X ⟶ Y} [CategoryTheory.Mono f], f ≠ 0 → CategoryTheory.IsIso fA nonzero monomorphism to a simple object is an isomorphism.
- Defined in
- Mathlib.CategoryTheory.Simple
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 5 from the axioms · uses no axioms
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.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
- CategoryTheory.IsIsostatement · cited by 1,156
- CategoryTheory.Monostatement and proof · cited by 893
- CategoryTheory.Simplestatement and proof · cited by 39
- CategoryTheory.Simple.mono_isIso_iff_nonzeroproof · cited by 6
Cited by5
Results whose statement or proof uses this declaration.
- CategoryTheory.Simple.of_isoproof · cited by 3
- CategoryTheory.kernel_zero_of_nonzero_from_simpleproof · cited by 1
- CategoryTheory.isIso_of_hom_simpleproof · cited by 1
- CategoryTheory.mono_to_simple_zero_of_not_isoproof · cited by 0
- CategoryTheory.epi_of_nonzero_to_simpleproof · cited by 0