feat(hott/algebra) show that functors preserve inverses and isos
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@ -87,6 +87,30 @@ namespace functor
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: functor.mk F H₁ id₁ comp₁ = functor.mk F H₂ id₂ comp₂ :=
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functor_eq (λc, idp) (λa b f, !id_leftright ⬝ !pH)
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protected definition preserve_iso (F : C ⇒ D) {a b : C} (f : hom a b) [H : is_iso f] :
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is_iso (F f) :=
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begin
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fapply @is_iso.mk, apply (F (f⁻¹)),
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repeat (apply concat ; apply inverse ; apply (respect_comp F) ;
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apply concat ; apply (ap (λ x, to_fun_hom F x)) ;
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[apply left_inverse | apply right_inverse] ;
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apply (respect_id F) ),
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end
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attribute preserve_iso [instance]
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protected definition respect_inv (F : C ⇒ D) {a b : C} (f : hom a b) [H : is_iso f] :
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F (f⁻¹) = (F f)⁻¹ :=
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begin
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fapply @left_inverse_eq_right_inverse, apply (F f),
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apply concat, apply inverse, apply (respect_comp F),
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apply concat, apply (ap (λ x, to_fun_hom F x)),
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apply left_inverse, apply respect_id,
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apply concat, apply inverse, apply (respect_comp F),
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apply concat, apply (ap (λ x, to_fun_hom F x)),
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apply right_inverse, apply respect_id,
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end
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protected definition assoc (H : C ⇒ D) (G : B ⇒ C) (F : A ⇒ B) :
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H ∘f (G ∘f F) = (H ∘f G) ∘f F :=
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!functor_mk_eq_constant (λa b f, idp)
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