feat(category.constructions.hset): prove that the category of sets is complete
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
f82d1bd566
commit
51edd0ad1d
7 changed files with 97 additions and 29 deletions
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@ -37,7 +37,7 @@ namespace category
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: is_hprop (f = g) :=
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_
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theorem cone_hom_eq (f f' : cone_hom x y) (q : cone_hom.f f = cone_hom.f f') : f = f' :=
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theorem cone_hom_eq {f f' : cone_hom x y} (q : cone_hom.f f = cone_hom.f f') : f = f' :=
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begin
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induction f, induction f', esimp at *, induction q, apply ap (cone_hom.mk f),
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apply @is_hprop.elim, apply pi.is_trunc_pi, intro x, apply is_trunc_eq, -- type class fails
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@ -6,7 +6,7 @@ Authors: Floris van Doorn, Jakob von Raumer
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Category of hsets
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-/
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import ..category types.equiv ..functor types.lift
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import ..category types.equiv ..functor types.lift ..limits
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open eq category equiv iso is_equiv is_trunc function sigma
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@ -46,16 +46,6 @@ namespace category
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local attribute is_equiv_iso_of_equiv [instance]
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-- TODO: move
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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)⁻¹ᶠ
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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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@ -83,7 +73,7 @@ namespace category
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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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assert H₂ : _, from (iso_of_eq_eq_compose A B)⁻¹,
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let H₂ := (iso_of_eq_eq_compose A B)⁻¹ in
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begin
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rewrite H₂ at H₁,
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assumption
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@ -104,4 +94,47 @@ namespace category
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(λa b, lift_functor)
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(λa, eq_of_homotopy (λx, by induction x; reflexivity))
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(λa b c g f, eq_of_homotopy (λx, by induction x; reflexivity))
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open pi sigma.ops
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local attribute Category.to.precategory [unfold 1]
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local attribute category.to_precategory [unfold 2]
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definition is_complete_cone.{u v w} [constructor]
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(I : Precategory.{v w}) (F : I ⇒ set.{max u v w}) : cone_obj F :=
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begin
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fapply cone_obj.mk,
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{ fapply trunctype.mk,
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{ exact Σ(s : Π(i : I), trunctype.carrier (F i)),
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Π{i j : I} (f : i ⟶ j), F f (s i) = (s j)},
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{ exact abstract begin apply is_trunc_sigma, intro s,
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apply is_trunc_pi, intro i,
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apply is_trunc_pi, intro j,
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apply is_trunc_pi, intro f,
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apply is_trunc_eq end end}},
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{ fapply nat_trans.mk,
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{ intro i x, esimp at x, exact x.1 i},
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{ intro i j f, esimp, apply eq_of_homotopy, intro x, esimp at x, induction x with s p,
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esimp, apply p}}
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end
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definition is_complete_set.{u v w} : is_complete.{(max u v w)+1 (max u v w) v w} set :=
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begin
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intro I F, fapply has_terminal_object.mk,
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{ exact is_complete_cone.{u v w} I F},
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{ intro c, esimp at *, induction c with X η, induction η with η p, esimp at *,
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fapply is_contr.mk,
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{ fapply cone_hom.mk,
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{ intro x, esimp at *, fapply sigma.mk,
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{ intro i, exact η i x},
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{ intro i j f, exact ap10 (p f) x}},
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{ intro i, reflexivity}},
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{ esimp, intro h, induction h with f q, apply cone_hom_eq, esimp at *,
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apply eq_of_homotopy, intro x, fapply sigma_eq: esimp,
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{ apply eq_of_homotopy, intro i, exact (ap10 (q i) x)⁻¹},
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{ apply is_hprop.elimo,
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apply is_trunc_pi, intro i,
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apply is_trunc_pi, intro j,
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apply is_trunc_pi, intro f,
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apply is_trunc_eq}}}
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end
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end category
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@ -48,7 +48,7 @@ namespace iso
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split_epi.mk !right_inverse
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definition is_iso_id [instance] [priority 500] (a : ob) : is_iso (ID a) :=
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is_iso.mk !id_comp !id_comp
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is_iso.mk !id_id !id_id
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definition is_iso_inverse [instance] [priority 200] (f : a ⟶ b) [H : is_iso f] : is_iso f⁻¹ :=
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is_iso.mk !right_inverse !left_inverse
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@ -89,13 +89,13 @@ namespace iso
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by cases p;apply inverse_unique
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definition retraction_id (a : ob) : retraction_of (ID a) = id :=
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retraction_eq !id_comp
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retraction_eq !id_id
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definition section_id (a : ob) : section_of (ID a) = id :=
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section_eq !id_comp
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section_eq !id_id
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definition id_inverse (a : ob) [H : is_iso (ID a)] : (ID a)⁻¹ = id :=
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inverse_eq_left !id_comp
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inverse_eq_left !id_id
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definition split_mono_comp [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
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[Hf : split_mono f] [Hg : split_mono g] : split_mono (g ∘ f) :=
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@ -202,7 +202,7 @@ namespace iso
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definition iso_of_eq_con (p : a = b) (q : b = c)
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: iso_of_eq (p ⬝ q) = iso.trans (iso_of_eq p) (iso_of_eq q) :=
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eq.rec_on q (eq.rec_on p (iso_eq !id_comp⁻¹))
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eq.rec_on q (eq.rec_on p (iso_eq !id_id⁻¹))
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section
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open funext
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@ -6,8 +6,8 @@ Authors: Floris van Doorn
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Limits in a category
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-/
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import .constructions.cone .groupoid .constructions.discrete .constructions.product
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.constructions.finite_cats
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import .constructions.cone .constructions.discrete .constructions.product
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.constructions.finite_cats .category
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open is_trunc functor nat_trans eq
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@ -33,6 +33,10 @@ namespace category
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definition terminal_iso_terminal {c c' : ob} (H : is_terminal c) (K : is_terminal c') : c ≅ c' :=
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iso.MK !terminal_morphism !terminal_morphism !hom_terminal_eq !hom_terminal_eq
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local attribute is_terminal [reducible]
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theorem is_hprop_is_terminal [instance] : is_hprop (is_terminal c) :=
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_
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omit C
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structure has_terminal_object [class] (D : Precategory) :=
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@ -42,16 +46,39 @@ namespace category
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abbreviation terminal_object [constructor] := @has_terminal_object.d
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attribute has_terminal_object.is_term [instance]
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variable {D}
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definition terminal_object_iso_terminal_object (H₁ H₂ : has_terminal_object D)
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: @terminal_object D H₁ ≅ @terminal_object D H₂ :=
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terminal_iso_terminal (@has_terminal_object.is_term D H₁) (@has_terminal_object.is_term D H₂)
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theorem is_hprop_has_terminal_object [instance] (D : Category)
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: is_hprop (has_terminal_object D) :=
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begin
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apply is_hprop.mk, intro t₁ t₂, induction t₁ with d₁ H₁, induction t₂ with d₂ H₂,
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assert p : d₁ = d₂,
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{ apply eq_of_iso, apply terminal_iso_terminal H₁ H₂},
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induction p, exact ap _ !is_hprop.elim
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end
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variable (D)
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definition has_limits_of_shape [class] := Π(F : I ⇒ D), has_terminal_object (cone F)
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/-
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The next definitions states that a category is complete with respect to diagrams
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in a certain universe. "is_complete.{o₁ h₁ o₂ h₂}" means that D is complete
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with respect to diagrams of type Precategory.{o₂ h₂}
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-/
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definition is_complete.{o₁ h₁ o₂ h₂} [class] (D : Precategory.{o₁ h₁}) :=
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Π(I : Precategory.{o₂ h₂}), has_limits_of_shape D I
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definition has_limits_of_shape_of_is_complete [instance] [H : is_complete D] (I : Precategory)
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: has_limits_of_shape D I := H I
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variables {I D}
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definition has_terminal_object_of_has_limits_of_shape [instance] [H : has_limits_of_shape D I]
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(F : I ⇒ D) : has_terminal_object (cone F) :=
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H F
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definition has_terminal_object_cone [H : has_limits_of_shape D I]
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(F : I ⇒ D) : has_terminal_object (cone F) := H F
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local attribute has_terminal_object_cone [instance]
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variables (F : I ⇒ D) [H : has_limits_of_shape D I] {i j : I}
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include H
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@ -65,9 +65,7 @@ namespace category
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variables {a b c d : ob} {h : c ⟶ d} {g : hom b c} {f f' : hom a b} {i : a ⟶ a}
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include C
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definition id [reducible] := ID a
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definition id_comp (a : ob) : ID a ∘ ID a = ID a := !id_left
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definition id [reducible] [unfold 2] := ID a
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definition id_leftright (f : hom a b) : id ∘ f ∘ id = f := !id_left ⬝ !id_right
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definition comp_id_eq_id_comp (f : hom a b) : f ∘ id = id ∘ f := !id_right ⬝ !id_left⁻¹
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@ -42,7 +42,7 @@ namespace functor
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(λd d' g, F (id, g))
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(λd, !respect_id)
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(λd₁ d₂ d₃ g' g, calc
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F (id, g' ∘ g) = F (id ∘ id, g' ∘ g) : by rewrite id_comp
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F (id, g' ∘ g) = F (id ∘ id, g' ∘ g) : by rewrite id_id
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... = F ((id,g') ∘ (id, g)) : by esimp
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... = F (id,g') ∘ F (id, g) : by rewrite respect_comp)
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@ -75,7 +75,7 @@ namespace functor
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apply @nat_trans_eq,
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intro d, calc
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natural_map (Fhom F (f' ∘ f)) d = F (f' ∘ f, id) : by rewrite functor_curry_hom_def
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... = F (f' ∘ f, id ∘ id) : by rewrite id_comp
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... = F (f' ∘ f, id ∘ id) : by rewrite id_id
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... = F ((f',id) ∘ (f, id)) : by esimp
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... = F (f',id) ∘ F (f, id) : by rewrite [respect_comp F]
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... = natural_map ((Fhom F f') ∘ (Fhom F f)) d : by esimp
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@ -100,7 +100,7 @@ namespace functor
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Ghom G (ID p) = to_fun_hom (to_fun_ob G p.1) id ∘ natural_map (to_fun_hom G id) p.2 : by esimp
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... = id ∘ natural_map (to_fun_hom G id) p.2 : by rewrite respect_id
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... = id ∘ natural_map nat_trans.id p.2 : by rewrite respect_id
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... = id : id_comp
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... = id : id_id
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theorem functor_uncurry_comp ⦃p p' p'' : C ×c D⦄ (f' : p' ⟶ p'') (f : p ⟶ p')
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: Ghom G (f' ∘ f) = Ghom G f' ∘ Ghom G f :=
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@ -412,6 +412,16 @@ namespace sigma
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definition equiv_subtype [H : Πa, is_hprop (B a)] (u v : {a | B a}) : (u.1 = v.1) ≃ (u = v) :=
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equiv.mk !subtype_eq _
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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)⁻¹ᶠ
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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)]
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(u v : Σa, B a) : is_equiv (subtype_eq_inv u v) :=
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_
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/- truncatedness -/
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theorem is_trunc_sigma (B : A → Type) (n : trunc_index)
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[HA : is_trunc n A] [HB : Πa, is_trunc n (B a)] : is_trunc n (Σa, B a) :=
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