Theorems · Theorem · category theory
CategoryTheory.Iso.hom_inv_id_assoc
∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] {X Y : C} (self : X ≅ Y) {Z : C} (h : X ⟶ Z),
CategoryTheory.CategoryStruct.comp self.hom (CategoryTheory.CategoryStruct.comp self.inv h) = hComposition of the two directions of an isomorphism is the identity on the source.
- Defined in
- Mathlib.CategoryTheory.Iso
- Cited by
- 187 results in Mathlib
- Foundations
- Depth 6 from the axioms, rests on 17 definitions · uses Quot.sound
- Assumes
- CategoryTheory.Category
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.Iso.homstatement and proof · cited by 7,684
- CategoryTheory.Iso.invstatement and proof · cited by 6,514
- CategoryTheory.Category.assocproof · cited by 6,433
- CategoryTheory.Isostatement and proof · cited by 3,963
- CategoryTheory.Category.id_compproof · cited by 1,998
- CategoryTheory.Iso.hom_inv_idproof · cited by 264
Cited by187
Results whose statement or proof uses this declaration.
- CategoryTheory.Iso.inv_comp_eqproof · cited by 50
- CategoryTheory.BraidedCategory.braiding_tensor_right_homproof · cited by 14
- CategoryTheory.ShortComplex.homologyπ_naturalityproof · cited by 10
- CategoryTheory.MonoidalCategory.associator_naturality_leftproof · cited by 10
- CategoryTheory.ShortComplex.homologyι_naturalityproof · cited by 7
- SSet.horn.faceSingletonComplIso_inv_ιproof · cited by 7
- CategoryTheory.braiding_tensorUnit_rightproof · cited by 7
- CategoryTheory.Limits.zero_of_source_iso_zeroproof · cited by 7
- CategoryTheory.CartesianMonoidalCategory.associator_hom_snd_sndproof · cited by 6
- AlgebraicGeometry.Proj.SpecMap_awayMap_awayιproof · cited by 5
- Bimod.AssociatorBimod.hom_inv_idproof · cited by 5
- CategoryTheory.MonoidalCategory.associator_conjugationproof · cited by 5