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chore(library/hott): make precategory use the isomorphic structure

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Jakob von Raumer 2014-12-02 19:43:03 -05:00 committed by Leonardo de Moura
parent 63afac301c
commit 5923392395
1 changed files with 4 additions and 4 deletions
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8
library/hott/algebra/precategory/morphism.lean
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@ -46,10 +46,6 @@ namespace morphism
theorem id_is_iso [instance] : is_iso (ID a) := theorem id_is_iso [instance] : is_iso (ID a) :=
is_iso.mk !id_compose !id_compose is_iso.mk !id_compose !id_compose
-- In a precategory, equal objects are isomorphic
definition iso_of_path (p : a ≈ b) : Σ (f : hom a b), is_iso f :=
path.rec_on p ⟨ id , id_is_iso ⟩
theorem inverse_is_iso [instance] (f : a ⟶ b) [H : is_iso f] : is_iso (f⁻¹) := theorem inverse_is_iso [instance] (f : a ⟶ b) [H : is_iso f] : is_iso (f⁻¹) :=
is_iso.mk !compose_inverse !inverse_compose is_iso.mk !compose_inverse !inverse_compose
@ -150,6 +146,10 @@ namespace morphism
--is_equivalence.mk (is_reflexive.mk refl) (is_symmetric.mk symm) (is_transitive.mk trans) --is_equivalence.mk (is_reflexive.mk refl) (is_symmetric.mk symm) (is_transitive.mk trans)
end isomorphic end isomorphic
-- In a precategory, equal objects are isomorphic
definition iso_of_path (p : a ≈ b) : isomorphic a b :=
path.rec_on p (isomorphic.mk id)
inductive is_mono [class] (f : a ⟶ b) : Type := inductive is_mono [class] (f : a ⟶ b) : Type :=
mk : (∀c (g h : hom c a), f ∘ g ≈ f ∘ h → g ≈ h) → is_mono f mk : (∀c (g h : hom c a), f ∘ g ≈ f ∘ h → g ≈ h) → is_mono f
inductive is_epi [class] (f : a ⟶ b) : Type := inductive is_epi [class] (f : a ⟶ b) : Type :=