Theorems · Theorem · category theory
CategoryTheory.mono_iff_injective
∀ {X Y : Type u} (f : X ⟶ Y), CategoryTheory.Mono f ↔ Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom f)A morphism in Type is a monomorphism if and only if it is injective.
- Defined in
- Mathlib.CategoryTheory.Types.Basic
- Cited by
- 20 results in Mathlib
- Foundations
- Depth 21 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Quiver.Homstatement and proof · cited by 32,603
- CategoryTheory.ConcreteCategory.homstatement and proof · cited by 4,022
- TypeCat.Funstatement · cited by 1,307
- CategoryTheory.Monostatement and proof · cited by 893
Cited by20
Results whose statement or proof uses this declaration.
- CategoryTheory.injective_of_monoproof · cited by 14
- CategoryTheory.ConcreteCategory.mono_of_injectiveproof · cited by 12
- TopCat.mono_iff_injectiveproof · cited by 8
- CategoryTheory.ConcreteCategory.injective_of_mono_of_preservesPullbackproof · cited by 5
- subtype_val_monoproof · cited by 4
- CategoryTheory.ConcreteCategory.mono_iff_injective_of_preservesPullbackproof · cited by 4
- SSet.S.existsUnique_nproof · cited by 2
- CategoryTheory.Limits.Types.mono_of_isPushout_of_isPullbackproof · cited by 1
- CategoryTheory.Limits.Types.coequalizer_preimage_image_eq_of_preimage_eqproof · cited by 1
- CategoryTheory.Limits.Types.pushoutCocone_inl_eq_inr_iff_of_isColimitproof · cited by 1
- CategoryTheory.Limits.Types.isPushout_of_isPullback_of_monoproof · cited by 1
- CategoryTheory.GrothendieckTopology.Point.fiber_map_injective_of_monoproof · cited by 1