Theorems · Definition · category theory
CategoryTheory.retraction
{C : Type u₁} →
[inst : CategoryTheory.Category.{v₁, u₁} C] → {X Y : C} → (f : Y ⟶ X) → [hf : CategoryTheory.IsSplitMono f] → X ⟶ YThe chosen retraction of a split monomorphism.
- Defined in
- Mathlib.CategoryTheory.EpiMono
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 4 from the axioms · uses Classical.choice
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- Nonempty.someproof · cited by 340
- CategoryTheory.IsSplitMonostatement and proof · cited by 33
- CategoryTheory.SplitMono.retractionproof · cited by 14
- CategoryTheory.IsSplitMono.exists_splitMonoproof · cited by 4
Cited by18
Results whose statement or proof uses this declaration.
- CategoryTheory.IsSplitMono.idstatement · cited by 6
- CategoryTheory.Limits.binaryBiconeOfIsSplitMonoOfCokernelproof · cited by 5
- CategoryTheory.Limits.coneOfIsSplitMonostatement · cited by 3
- DerivedCategory.to_singleFunctor_obj_eq_zero_of_injectiveproof · cited by 1
- CategoryTheory.isIso_of_epi_of_isSplitMonoproof · cited by 1
- CategoryTheory.Monad.isSplitMono_iff_isIso_unitproof · cited by 1
- CategoryTheory.retraction.congr_simpstatement and proof · cited by 1
- CategoryTheory.RegularMono.ofIsSplitMonoproof · cited by 0
- CategoryTheory.Limits.binaryBiconeOfIsSplitMonoOfCokernel_fststatement · cited by 0
- CategoryTheory.Limits.binaryBiconeOfIsSplitMonoOfCokernel_inrstatement · cited by 0
- CategoryTheory.IsIso.of_mono_retractionstatement and proof · cited by 0
- CategoryTheory.Adjunction.full_R_of_isSplitMono_counit_appproof · cited by 0