Theorems · Theorem · category theory
CategoryTheory.Functor.mono_map_iff_mono
∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
(F : CategoryTheory.Functor C D) {X Y : C} (f : Y ⟶ X) [hF₁ : F.PreservesMonomorphisms]
[hF₂ : F.ReflectsMonomorphisms], CategoryTheory.Mono (F.map f) ↔ CategoryTheory.Mono f- Defined in
- Mathlib.CategoryTheory.Functor.EpiMono
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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.Functor.objstatement · cited by 19,642
- CategoryTheory.Functorstatement and proof · cited by 16,252
- CategoryTheory.Functor.mapstatement · cited by 8,698
- CategoryTheory.Monostatement and proof · cited by 893
- CategoryTheory.Functor.PreservesMonomorphismsstatement and proof · cited by 41
- CategoryTheory.Functor.mono_of_mono_mapproof · cited by 24
- CategoryTheory.Functor.ReflectsMonomorphismsstatement and proof · cited by 11
Cited by3
Results whose statement or proof uses this declaration.
- SimplexCategory.mono_iff_injectiveproof · cited by 6
- CategoryTheory.ConcreteCategory.mono_iff_injective_of_preservesPullbackproof · cited by 4
- AlgebraicGeometry.IsClosedImmersion.iff_isFinite_and_monoproof · cited by 2