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916823d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0040
universe u v w
variable {X : Type uX} [Category.{vX} X]
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C ⥤ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U_Faithful
abbrev StructuredArrowOver (x : X) (C : ConcreteCat (X := X)): Type _ :=
StructuredArrow x C.U
def IsUniversalArrowOver (x : X) {C : ConcreteCat (X := X)} (u : StructuredArrowOver x C) : Prop :=
∀ (v : StructuredArrowOver x C),
∃! (g : u.right ⟶ v.right), u.hom ≫ C.U.map g = v.hom
def IsFreeObjectOver (x : X) {C : ConcreteCat (X := X)} (z : C.C) : Prop :=
∃ (f : StructuredArrowOver x C), f.right = z ∧ IsUniversalArrowOver (x := x) (C := C) f
def HasFreeObject (C : ConcreteCat (X := X)) : Prop :=
∀ (x : X), ∃ (z : C.C), IsFreeObjectOver (x := x) (z := z)
structure SupLatCat where
carrier : Type u
[inst : CompleteSemilatticeSup carrier]
attribute [instance] SupLatCat.inst
instance : CoeSort SupLatCat (Type u) := ⟨SupLatCat.carrier⟩
def of (α : Type u) [CompleteSemilatticeSup α] : SupLatCat := ⟨α⟩
structure Hom (A B : SupLatCat.{u}) where
toFun : A → B
map_sSup' : ∀ s : Set A, toFun (sSup s) = sSup (toFun '' s)
instance (A B : SupLatCat) : CoeFun (Hom A B) (fun _ => A → B) := ⟨Hom.toFun⟩
@[simp] lemma Hom.map_sSup {A B : SupLatCat} (f : Hom A B) (s : Set A) :
f (sSup s) = sSup (f '' s) :=
f.map_sSup' s
@[ext] lemma Hom.ext {A B : SupLatCat} {f g : Hom A B}
(h : ∀ a, f a = g a) : f = g := by
cases f with
| mk fto fmap =>
cases g with
| mk gto gmap =>
have hto : fto = gto := funext (by intro a; exact h a)
cases hto
have : fmap = gmap := by
apply Subsingleton.elim
cases this
rfl
def id (A : SupLatCat) : Hom A A :=
{ toFun := (_root_.id : A → A)
map_sSup' := by
intro s
simp }
def comp {A B C : SupLatCat} (f : Hom A B) (g : Hom B C) : Hom A C :=
{ toFun := fun a => g (f a)
map_sSup' := by
intro s
calc
g (f (sSup s)) = g (sSup (f '' s)) := by
simp
_ = sSup (g '' (f '' s)) := by
simp
_ = sSup ((fun x => g (f x)) '' s) := by
simp [Set.image_image] }
instance : Category SupLatCat where
Hom A B := Hom A B
id A := id A
comp f g := comp f g
id_comp := by intro A B f; ext a; rfl
comp_id := by intro A B f; ext a; rfl
assoc := by intro A B C D f g h; ext a; rfl
def forget : SupLatCat ⥤ Type u :=
{ obj := fun A => A.carrier
map := fun {X Y} (f : X ⟶ Y) => f.toFun
map_id := by intro A; rfl
map_comp := by intro A B C f g; rfl }
instance : forget.Faithful where
map_injective := by
intro X Y f g h
apply Hom.ext
intro x
simpa using congrArg (fun k => k x) h
def SupLatCatConcrete : ConcreteCat (X := Type u) :=
{ C := SupLatCat.{u}
U := (forget) }
theorem SupLat_Has_Free_Object :
HasFreeObject SupLatCatConcrete:= by
sorry
end CAT_statement_S_0040
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