Theorems · Definition · category theory
CategoryTheory.Injective.factorThru
{C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] →
{J X Y : C} → [CategoryTheory.Injective J] → (X ⟶ J) → (f : X ⟶ Y) → [CategoryTheory.Mono f] → Y ⟶ JLet J be injective and g a morphism into J, then g can be factored through any monomorphism.
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses Classical.choice
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 and proof · cited by 32,603
- CategoryTheory.Monostatement and proof · cited by 893
- CategoryTheory.Injectivestatement and proof · cited by 70
- CategoryTheory.Injective.factorsproof · cited by 8
Cited by15
Results whose statement or proof uses this declaration.
- CategoryTheory.Injective.comp_factorThrustatement · cited by 10
- CategoryTheory.ShortComplex.Exact.comp_descToInjectiveproof · cited by 6
- CategoryTheory.ShortComplex.Exact.descToInjectiveproof · cited by 4
- CategoryTheory.Abelian.epiWithInjectiveKernel_iffproof · cited by 3
- CategoryTheory.Injective.hasLiftingProperty_of_isZeroproof · cited by 2
- CochainComplex.isSplitMono_from_singleFunctor_obj_of_injectiveproof · cited by 1
- FDRep.simple_iff_end_is_rank_oneproof · cited by 1
- CategoryTheory.Injective.comp_factorThru_assocstatement and proof · cited by 1
- CategoryTheory.ShortComplex.ShortExact.splittingOfInjectiveproof · cited by 1
- CategoryTheory.InjectiveResolution.descFZeroproof · cited by 1