Theorems · Theorem · category theory
CategoryTheory.epi_of_nonzero_to_simple
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
[CategoryTheory.Limits.HasEqualizers C] {X Y : C} [CategoryTheory.Simple Y] {f : X ⟶ Y}
[CategoryTheory.Limits.HasImage f], f ≠ 0 → CategoryTheory.Epi fA nonzero morphism f to a simple object is an epimorphism
(assuming f has an image, and C has equalizers).
- Defined in
- Mathlib.CategoryTheory.Simple
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 32 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites13
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.IsIsoproof · cited by 1,156
- CategoryTheory.Epistatement and proof · cited by 688
- CategoryTheory.Limits.HasImagestatement and proof · cited by 107
- CategoryTheory.Limits.image.ιproof · cited by 104
- CategoryTheory.Limits.HasEqualizersstatement and proof · cited by 60
- CategoryTheory.Limits.factorThruImageproof · cited by 55
- CategoryTheory.Simplestatement and proof · cited by 39
- CategoryTheory.Limits.image.facproof · cited by 27
- CategoryTheory.isIso_of_mono_of_nonzeroproof · cited by 5
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