144 lines
5 KiB
Text
144 lines
5 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: algebra.category.constructions
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Authors: Floris van Doorn
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-/
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import .basic algebra.precategory.constructions types.equiv types.trunc
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--open eq eq.ops equiv category.ops iso category is_trunc
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open eq category equiv iso is_equiv category.ops is_trunc iso.iso function sigma
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namespace category
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namespace set
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local attribute is_equiv_subtype_eq [instance]
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definition iso_of_equiv {A B : Precategory_hset} (f : A ≃ B) : A ≅ B :=
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iso.MK (to_fun f)
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(equiv.to_inv f)
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(eq_of_homotopy (sect (to_fun f)))
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(eq_of_homotopy (retr (to_fun f)))
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definition equiv_of_iso {A B : Precategory_hset} (f : A ≅ B) : A ≃ B :=
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equiv.MK (to_hom f)
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(iso.to_inv f)
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(ap10 (right_inverse (to_hom f)))
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(ap10 (left_inverse (to_hom f)))
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definition is_equiv_iso_of_equiv (A B : Precategory_hset) : is_equiv (@iso_of_equiv A B) :=
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adjointify _ (λf, equiv_of_iso f)
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(λf, iso.eq_mk idp)
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(λf, equiv.eq_mk idp)
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local attribute is_equiv_iso_of_equiv [instance]
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open sigma.ops
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definition subtype_eq_inv {A : Type} {B : A → Type} [H : Πa, is_hprop (B a)] (u v : Σa, B a)
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: u = v → u.1 = v.1 :=
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(subtype_eq u v)⁻¹ᵉ
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local attribute subtype_eq_inv [reducible]
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definition is_equiv_subtype_eq_inv {A : Type} {B : A → Type} [H : Πa, is_hprop (B a)] (u v : Σa, B a)
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: is_equiv (subtype_eq_inv u v) :=
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_
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definition iso_of_eq_eq_compose (A B : hset) : @iso_of_eq _ _ A B =
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@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
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@ap _ _ (to_fun (trunctype.sigma_char 0)) A B :=
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eq_of_homotopy (λp, eq.rec_on p idp)
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definition equiv_equiv_iso (A B : Precategory_hset) : (A ≃ B) ≃ (A ≅ B) :=
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equiv.MK (λf, iso_of_equiv f)
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(λf, equiv.MK (to_hom f)
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(iso.to_inv f)
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(ap10 (right_inverse (to_hom f)))
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(ap10 (left_inverse (to_hom f))))
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(λf, iso.eq_mk idp)
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(λf, equiv.eq_mk idp)
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definition equiv_eq_iso (A B : Precategory_hset) : (A ≃ B) = (A ≅ B) :=
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ua !equiv_equiv_iso
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definition is_univalent_hset (A B : Precategory_hset) : is_equiv (@iso_of_eq _ _ A B) :=
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have H : is_equiv (@iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
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@ap _ _ (to_fun (trunctype.sigma_char 0)) A B), from
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@is_equiv_compose _ _ _ _ _
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(@is_equiv_compose _ _ _ _ _
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(@is_equiv_compose _ _ _ _ _
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_
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(@is_equiv_subtype_eq_inv _ _ _ _ _))
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!univalence)
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!is_equiv_iso_of_equiv,
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(iso_of_eq_eq_compose A B)⁻¹ ▹ H
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end set
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definition category_hset [reducible] [instance] : category hset :=
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category.mk' hset precategory_hset set.is_univalent_hset
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definition Category_hset [reducible] : Category :=
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Category.mk hset category_hset
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namespace ops
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abbreviation set := Category_hset
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end ops
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section functor
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open functor nat_trans
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variables {C : Precategory} {D : Category} {F G : D ^c C}
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definition eq_of_iso_functor_ob (η : F ≅ G) (c : C) : F c = G c :=
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by apply eq_of_iso; apply componentwise_iso; exact η
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definition eq_of_iso_functor (η : F ≅ G) : F = G :=
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begin
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fapply functor_eq,
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{exact (eq_of_iso_functor_ob η)},
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{intros (c, c', f), --unfold eq_of_iso_functor_ob, --TODO: report: this fails
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apply concat,
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{apply (ap (λx, to_hom x ∘ to_fun_hom F f ∘ _)), apply (retr iso_of_eq)},
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apply concat,
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{apply (ap (λx, _ ∘ to_fun_hom F f ∘ (to_hom x)⁻¹)), apply (retr iso_of_eq)},
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apply inverse, apply naturality_iso}
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end
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definition iso_of_eq_eq_of_iso_functor (η : F ≅ G) : iso_of_eq (eq_of_iso_functor η) = η :=
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begin
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apply iso.eq_mk,
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apply nat_trans_eq_mk,
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intro c,
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rewrite natural_map_hom_of_eq, esimp {eq_of_iso_functor},
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rewrite ap010_functor_eq, esimp {hom_of_eq,eq_of_iso_functor_ob},
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rewrite (retr iso_of_eq),
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end
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definition eq_of_iso_functor_iso_of_eq (p : F = G) : eq_of_iso_functor (iso_of_eq p) = p :=
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begin
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apply functor_eq2,
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intro c,
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esimp {eq_of_iso_functor},
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rewrite ap010_functor_eq,
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esimp {eq_of_iso_functor_ob},
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rewrite componentwise_iso_iso_of_eq,
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rewrite (sect iso_of_eq)
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end
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definition is_univalent_functor (D : Category) (C : Precategory) : is_univalent (D ^c C) :=
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λF G, adjointify _ eq_of_iso_functor
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iso_of_eq_eq_of_iso_functor
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eq_of_iso_functor_iso_of_eq
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end functor
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definition Category_functor_of_precategory (D : Category) (C : Precategory) : Category :=
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category.MK (D ^c C) (is_univalent_functor D C)
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definition Category_functor (D : Category) (C : Category) : Category :=
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Category_functor_of_precategory D C
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namespace ops
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infixr `^c2`:35 := Category_functor
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end ops
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end category
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