Theorems · Theorem · category theory
CategoryTheory.Injective.factors
∀ {C : Type u₁} {inst : CategoryTheory.Category.{v₁, u₁} C} {J : C} [self : CategoryTheory.Injective J] {X Y : C}
(g : X ⟶ J) (f : X ⟶ Y) [CategoryTheory.Mono f], ∃ h, CategoryTheory.CategoryStruct.comp f h = gAn object J is injective iff every morphism into J can be obtained by extending a monomorphism.
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 3 from the axioms · uses no axioms
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.
- CategoryTheory.Categorystatement and proof · cited by 32,673
- Quiver.Homstatement · cited by 32,603
- CategoryTheory.CategoryStruct.compstatement · cited by 17,999
- CategoryTheory.Monostatement · cited by 893
- CategoryTheory.Injectivestatement and proof · cited by 70
Cited by9
Results whose statement or proof uses this declaration.
- CategoryTheory.Injective.factorThruproof · cited by 12
- CategoryTheory.Injective.comp_factorThruproof · cited by 10
- CategoryTheory.Injective.of_isoproof · cited by 3
- CategoryTheory.Adjunction.map_injectiveproof · cited by 2
- Module.injective_module_of_injective_objectproof · cited by 2
- CategoryTheory.Retract.injectiveproof · cited by 1
- CategoryTheory.Adjunction.injective_of_map_injectiveproof · cited by 1
- LightCondensed.internallyProjective_free_natUnionInftyproof · cited by 0
- CategoryTheory.Functor.injective_of_map_injectiveproof · cited by 0