Theorems · Theorem · category theory
CategoryTheory.IsSplitMono.id
∀ {C : Type u₁} [inst : CategoryTheory.Category.{v₁, u₁} C] {X Y : C} (f : Y ⟶ X) [hf : CategoryTheory.IsSplitMono f],
CategoryTheory.CategoryStruct.comp f (CategoryTheory.retraction f) = CategoryTheory.CategoryStruct.id Y- Defined in
- Mathlib.CategoryTheory.EpiMono
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 5 from the axioms · uses Classical.choice
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.CategoryStruct.compstatement · cited by 17,999
- CategoryTheory.CategoryStruct.idstatement · cited by 6,235
- Nonempty.someproof · cited by 340
- CategoryTheory.IsSplitMonostatement and proof · cited by 33
- CategoryTheory.retractionstatement · cited by 14
- CategoryTheory.IsSplitMono.exists_splitMonoproof · cited by 4
- CategoryTheory.SplitMono.idproof · cited by 3
Cited by6
Results whose statement or proof uses this declaration.
- DerivedCategory.to_singleFunctor_obj_eq_zero_of_injectiveproof · cited by 1
- CategoryTheory.Limits.IsZero.iff_isSplitMono_eq_zeroproof · cited by 1
- CategoryTheory.isIso_of_epi_of_isSplitMonoproof · cited by 1
- CategoryTheory.Monad.isSplitMono_iff_isIso_unitproof · cited by 1
- CategoryTheory.Adjunction.full_R_of_isSplitMono_counit_appproof · cited by 0
- CategoryTheory.IsSplitMono.id_assocproof · cited by 0