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Reserved Notation "x = y :> T"
(at level 70, y at next level, no associativity).
Reserved Notation "x = y" (at level 70, no associativity).
Reserved Notation "x + y" (at level 50, left associativity).
Reserved Notation "n .+1" (at level 2, left associativity, format "n .+1").
Reserved Notation "n .+2" (at level 2, left associativity, format "n .+2").
Reserved Notation "g 'o' f" (at level 40, left associativity).
Reserved Notation "! x" (at level 3, format "'!' x").
Declare Scope path_scope.
Delimit Scope type_scope with type.
Delimit Scope trunc_scope with trunc.
Global Open Scope trunc_scope.
Global Open Scope path_scope.
Global Open Scope type_scope.
Global Set Universe Polymorphism.
Notation "A -> B" := (forall (_ : A), B) : type_scope.
Create HintDb typeclass_instances discriminated.
Definition Relation (A : Type) := A -> A -> Type.
Class Symmetric {A} (R : Relation A) :=
symmetry : forall x y, R x y -> R y x.
Notation Type0 := Set.
Cumulative Inductive paths {A : Type} (a : A) : A -> Type :=
idpath : paths a a.
Arguments idpath {A a} , [A] a.
Notation "x = y :> A" := (@paths A x y) : type_scope.
Notation "x = y" := (x = y :>_) : type_scope.
Definition inverse {A : Type} {x y : A} (p : x = y) : y = x
:= match p with idpath => idpath end.
Global Instance symmetric_paths {A} : Symmetric (@paths A) | 0 := @inverse A.
Notation "1" := idpath : path_scope.
Class Contr_internal (A : Type) := Build_Contr {
center : A ;
contr : (forall y : A, center = y)
}.
Inductive trunc_index : Type :=
| minus_two : trunc_index
| trunc_S : trunc_index -> trunc_index.
Bind Scope trunc_scope with trunc_index.
Notation "n .+1" := (trunc_S n) : trunc_scope.
Notation "n .+2" := (n.+1.+1)%trunc : trunc_scope.
Fixpoint IsTrunc_internal (n : trunc_index) (A : Type) : Type :=
match n with
| minus_two => Contr_internal A
| n'.+1 => forall (x y : A), IsTrunc_internal n' (x = y)
end.
Class IsTrunc (n : trunc_index) (A : Type) : Type :=
Trunc_is_trunc : IsTrunc_internal n A.
Notation Contr := (IsTrunc minus_two).
Notation IsHSet := (IsTrunc minus_two.+2).
Inductive Empty : Type0 := .
Inductive Unit : Type0 := tt : Unit.
Module Export Core.
Set Implicit Arguments.
Delimit Scope morphism_scope with morphism.
Delimit Scope category_scope with category.
Delimit Scope object_scope with object.
Record PreCategory :=
Build_PreCategory' {
object :> Type;
morphism : object -> object -> Type;
identity : forall x, morphism x x;
compose : forall s d d',
morphism d d'
-> morphism s d
-> morphism s d'
where "f 'o' g" := (compose f g);
associativity : forall x1 x2 x3 x4
(m1 : morphism x1 x2)
(m2 : morphism x2 x3)
(m3 : morphism x3 x4),
(m3 o m2) o m1 = m3 o (m2 o m1);
associativity_sym : forall x1 x2 x3 x4
(m1 : morphism x1 x2)
(m2 : morphism x2 x3)
(m3 : morphism x3 x4),
m3 o (m2 o m1) = (m3 o m2) o m1;
left_identity : forall a b (f : morphism a b), identity b o f = f;
right_identity : forall a b (f : morphism a b), f o identity a = f;
identity_identity : forall x, identity x o identity x = identity x;
trunc_morphism : forall s d, IsHSet (morphism s d)
}.
Bind Scope category_scope with PreCategory.
Arguments identity {!C%_category} / x%_object : rename.
Arguments compose {!C%_category} / {s d d'}%_object (m1 m2)%_morphism : rename.
Infix "o" := compose : morphism_scope.
End Core.
Delimit Scope functor_scope with functor.
Local Open Scope morphism_scope.
Section Functor.
Variables C D : PreCategory.
Record Functor :=
{
object_of :> C -> D;
morphism_of : forall s d, morphism C s d
-> morphism D (object_of s) (object_of d);
composition_of : forall s d d'
(m1 : morphism C s d) (m2: morphism C d d'),
morphism_of _ _ (m2 o m1)
= (morphism_of _ _ m2) o (morphism_of _ _ m1);
identity_of : forall x, morphism_of _ _ (identity x)
= identity (object_of x)
}.
End Functor.
Generalizable Variables A B m n f.
Fixpoint trunc_index_inc (k : trunc_index) (n : nat)
: trunc_index
:= match n with
| O => k
| S m => (trunc_index_inc k m).+1
end.
Definition nat_to_trunc_index (n : nat) : trunc_index
:= (trunc_index_inc minus_two n).+2.
Definition int_to_trunc_index (v : Decimal.int) : option trunc_index
:= match v with
| Decimal.Pos d => Some (nat_to_trunc_index (Nat.of_uint d))
| Decimal.Neg d => match Nat.of_uint d with
| 2%nat => Some minus_two
| 1%nat => Some (minus_two.+1)
| 0 => Some (minus_two.+2)
| _ => None
end
end.
Definition num_int_to_trunc_index v : option trunc_index :=
match Nat.of_num_int v with
| Some n => Some (nat_to_trunc_index n)
| None => None
end.
Fixpoint trunc_index_to_little_uint n acc :=
match n with
| minus_two => acc
| minus_two.+1 => acc
| minus_two.+2 => acc
| trunc_S n => trunc_index_to_little_uint n (Decimal.Little.succ acc)
end.
Definition trunc_index_to_int n :=
match n with
| minus_two => Decimal.Neg (Nat.to_uint 2)
| minus_two.+1 => Decimal.Neg (Nat.to_uint 1)
| n => Decimal.Pos (Decimal.rev (trunc_index_to_little_uint n Decimal.zero))
end.
Definition trunc_index_to_num_int n :=
Number.IntDecimal (trunc_index_to_int n).
Global Instance istrunc_succ `{IsTrunc n A}
: IsTrunc n.+1 A | 1000.
Admitted.
Global Instance contr_unit : Contr Unit | 0 := let x := {|
center := tt;
contr := fun t : Unit => match t with tt => 1 end
|} in x.
Definition groupoid_category X `{IsTrunc (trunc_S (trunc_S (trunc_S minus_two))) X} : PreCategory.
Admitted.
Definition discrete_category X `{IsHSet X} := groupoid_category X.
Section indiscrete_category.
Variable X : Type.
Definition indiscrete_category : PreCategory
:= @Build_PreCategory' X
(fun _ _ => Unit)
(fun _ => tt)
(fun _ _ _ _ _ => tt)
(fun _ _ _ _ _ _ _ => idpath)
(fun _ _ _ _ _ _ _ => idpath)
(fun _ _ f => match f with tt => idpath end)
(fun _ _ f => match f with tt => idpath end)
(fun _ => idpath)
_.
End indiscrete_category.
Local Open Scope nat_scope.
Fixpoint CardinalityRepresentative (n : nat) : Type0 :=
match n with
| 0 => Empty
| 1 => Unit
| S n' => (CardinalityRepresentative n' + Unit)%type
end.
Coercion CardinalityRepresentative : nat >-> Sortclass.
Global Instance trunc_cardinality_representative (n : nat)
: IsHSet (CardinalityRepresentative n).
Admitted.
Definition nat_category (n : nat) :=
match n with
| 0 => indiscrete_category 0
| 1 => indiscrete_category 1
| S (S n') => discrete_category (S (S n'))
end.
Notation "1" := (nat_category 1) : category_scope.
Notation terminal_category := (nat_category 1) (only parsing).
Class IsTerminalCategory (C : PreCategory)
`{Contr (object C)}
`{forall s d, Contr (morphism C s d)} := {}.
Global Instance: IsTerminalCategory 1 | 0 := {}.
Generalizable All Variables.
Section functors.
Variable C : PreCategory.
Definition from_terminal `{@IsTerminalCategory one Hone Hone'} (c : C)
: Functor one C
:= Build_Functor
one C
(fun _ => c)
(fun _ _ _ => identity c)
(fun _ _ _ _ _ => symmetry _ _ (@identity_identity _ _))
(fun _ => idpath).
End functors.
Local Notation "! x" := (@from_terminal _ terminal_category _ _ _ x) : functor_scope.
Definition CC_Functor' (C : PreCategory) (D : PreCategory) := Functor C D.
Coercion cc_functor_from_terminal' (C : PreCategory) (x : C) : CC_Functor' _ C
:= (!x)%functor.
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