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Module FirstComment.
Set Implicit Arguments.
Generalizable All Variables.
Set Asymmetric Patterns.
Set Universe Polymorphism.
Reserved Notation "x # y" (at level 40, left associativity).
Reserved Notation "x #m y" (at level 40, left associativity).
Delimit Scope object_scope with object.
Delimit Scope morphism_scope with morphism.
Delimit Scope category_scope with category.
Record Category (obj : Type) :=
{
Object :> _ := obj;
Morphism : obj -> obj -> Type;
Identity : forall x, Morphism x x;
Compose : forall s d d', Morphism d d' -> Morphism s d -> Morphism s d'
}.
Bind Scope object_scope with Object.
Bind Scope morphism_scope with Morphism.
Arguments Object {obj%type} C%category / : rename.
Arguments Morphism {obj%type} !C%category s d : rename. (* , simpl nomatch. *)
Arguments Identity {obj%type} [!C%category] x%object : rename.
Arguments Compose {obj%type} [!C%category s%object d%object d'%object] m1%morphism m2%morphism : rename.
Bind Scope category_scope with Category.
Record Functor
`(C : @Category objC)
`(D : @Category objD)
:= {
ObjectOf :> objC -> objD;
MorphismOf : forall s d, C.(Morphism) s d -> D.(Morphism) (ObjectOf s) (ObjectOf d)
}.
Record NaturalTransformation
`(C : @Category objC)
`(D : @Category objD)
(F G : Functor C D)
:= {
ComponentsOf :> forall c, D.(Morphism) (F c) (G c)
}.
Definition ProductCategory
`(C : @Category objC)
`(D : @Category objD)
: @Category (objC * objD)%type.
refine (@Build_Category _
(fun s d => (C.(Morphism) (fst s) (fst d) * D.(Morphism) (snd s) (snd d))%type)
(fun o => (Identity (fst o), Identity (snd o)))
(fun s d d' m2 m1 => (Compose (fst m2) (fst m1), Compose (snd m2) (snd m1)))).
Defined.
Infix "*" := ProductCategory : category_scope.
Record IsomorphismOf `{C : @Category objC} {s d} (m : C.(Morphism) s d) :=
{
IsomorphismOf_Morphism :> C.(Morphism) s d := m;
Inverse : C.(Morphism) d s
}.
Record NaturalIsomorphism
`(C : @Category objC)
`(D : @Category objD)
(F G : Functor C D)
:= {
NaturalIsomorphism_Transformation :> NaturalTransformation F G;
NaturalIsomorphism_Isomorphism : forall x : objC, IsomorphismOf (NaturalIsomorphism_Transformation x)
}.
Section PreMonoidalCategory.
Context `(C : @Category objC).
Variable TensorProduct : Functor (C * C) C.
Let src {C : @Category objC} {s d} (_ : Morphism C s d) := s.
Let dst {C : @Category objC} {s d} (_ : Morphism C s d) := d.
Local Notation "A # B" := (TensorProduct (A, B)).
Local Notation "A #m B" := (TensorProduct.(MorphismOf) ((@src _ _ _ A, @src _ _ _ B)) ((@dst _ _ _ A, @dst _ _ _ B)) (A, B)%morphism).
Let TriMonoidalProductL_ObjectOf (abc : C * C * C) : C :=
(fst (fst abc) # snd (fst abc)) # snd abc.
Let TriMonoidalProductR_ObjectOf (abc : C * C * C) : C :=
fst (fst abc) # (snd (fst abc) # snd abc).
Let TriMonoidalProductL_MorphismOf (s d : C * C * C) (m : Morphism (C * C * C) s d) :
Morphism C (TriMonoidalProductL_ObjectOf s) (TriMonoidalProductL_ObjectOf d).
Admitted.
Let TriMonoidalProductR_MorphismOf (s d : C * C * C) (m : Morphism (C * C * C) s d) :
Morphism C (TriMonoidalProductR_ObjectOf s) (TriMonoidalProductR_ObjectOf d).
Admitted.
Definition TriMonoidalProductL : Functor (C * C * C) C.
refine (Build_Functor (C * C * C) C
TriMonoidalProductL_ObjectOf
TriMonoidalProductL_MorphismOf).
Defined.
Definition TriMonoidalProductR : Functor (C * C * C) C.
refine (Build_Functor (C * C * C) C
TriMonoidalProductR_ObjectOf
TriMonoidalProductR_MorphismOf).
Defined.
Variable Associator : NaturalIsomorphism TriMonoidalProductL TriMonoidalProductR.
Section AssociatorCoherenceCondition.
Variables a b c d : C.
(* going from top-left *)
Let AssociatorCoherenceConditionT0 : Morphism C (((a # b) # c) # d) ((a # (b # c)) # d)
:= Associator (a, b, c) #m Identity (C := C) d.
Let AssociatorCoherenceConditionT1 : Morphism C ((a # (b # c)) # d) (a # ((b # c) # d))
:= Associator (a, b # c, d).
Let AssociatorCoherenceConditionT2 : Morphism C (a # ((b # c) # d)) (a # (b # (c # d)))
:= Identity (C := C) a #m Associator (b, c, d).
Let AssociatorCoherenceConditionB0 : Morphism C (((a # b) # c) # d) ((a # b) # (c # d))
:= Associator (a # b, c, d).
Let AssociatorCoherenceConditionB1 : Morphism C ((a # b) # (c # d)) (a # (b # (c # d)))
:= Associator (a, b, c # d).
Definition AssociatorCoherenceCondition' :=
Compose AssociatorCoherenceConditionT2 (Compose AssociatorCoherenceConditionT1 AssociatorCoherenceConditionT0)
= Compose AssociatorCoherenceConditionB1 AssociatorCoherenceConditionB0.
End AssociatorCoherenceCondition.
Definition AssociatorCoherenceCondition := Eval unfold AssociatorCoherenceCondition' in
forall a b c d : C, AssociatorCoherenceCondition' a b c d.
End PreMonoidalCategory.
Section MonoidalCategory.
Variable objC : Type.
Let AssociatorCoherenceCondition' := Eval unfold AssociatorCoherenceCondition in @AssociatorCoherenceCondition.
Record MonoidalCategory :=
{
MonoidalUnderlyingCategory :> @Category objC;
TensorProduct : Functor (MonoidalUnderlyingCategory * MonoidalUnderlyingCategory) MonoidalUnderlyingCategory;
IdentityObject : objC;
Associator : NaturalIsomorphism (TriMonoidalProductL TensorProduct) (TriMonoidalProductR TensorProduct);
AssociatorCoherent : AssociatorCoherenceCondition' Associator
}.
End MonoidalCategory.
Section EnrichedCategory.
Context `(M : @MonoidalCategory objM).
Let x : M := IdentityObject M.
(* Anomaly: apply_coercion_args: mismatch between arguments and coercion. Please report. *)
End EnrichedCategory.
End FirstComment.
Module SecondComment.
Set Implicit Arguments.
Set Universe Polymorphism.
Generalizable All Variables.
Record prod (A B : Type) := pair { fst : A; snd : B }.
Arguments fst {A B} _.
Arguments snd {A B} _.
Infix "*" := prod : type_scope.
Notation " ( x , y ) " := (@pair _ _ x y).
Record Category (obj : Type) :=
Build_Category {
Object :> _ := obj;
Morphism : obj -> obj -> Type;
Identity : forall x, Morphism x x;
Compose : forall s d d', Morphism d d' -> Morphism s d -> Morphism s d'
}.
Arguments Identity {obj%type} [!C] x : rename.
Arguments Compose {obj%type} [!C s d d'] m1 m2 : rename.
Record > Category' :=
{
LSObject : Type;
LSUnderlyingCategory :> @Category LSObject
}.
Section Functor.
Context `(C : @Category objC).
Context `(D : @Category objD).
Record Functor :=
{
ObjectOf :> objC -> objD;
MorphismOf : forall s d, C.(Morphism) s d -> D.(Morphism) (ObjectOf s) (ObjectOf d)
}.
End Functor.
Arguments MorphismOf {objC%type} [C] {objD%type} [D] F [s d] m : rename, simpl nomatch.
Section FunctorComposition.
Context `(C : @Category objC).
Context `(D : @Category objD).
Context `(E : @Category objE).
Definition ComposeFunctors (G : Functor D E) (F : Functor C D) : Functor C E.
Admitted.
End FunctorComposition.
Section IdentityFunctor.
Context `(C : @Category objC).
Definition IdentityFunctor : Functor C C.
refine {| ObjectOf := (fun x => x);
MorphismOf := (fun _ _ x => x)
|}.
Defined.
End IdentityFunctor.
Section ProductCategory.
Context `(C : @Category objC).
Context `(D : @Category objD).
Definition ProductCategory : @Category (objC * objD)%type.
refine (@Build_Category _
(fun s d => (C.(Morphism) (fst s) (fst d) * D.(Morphism) (snd s) (snd d))%type)
(fun o => (Identity (fst o), Identity (snd o)))
(fun s d d' m2 m1 => (Compose (fst m2) (fst m1), Compose (snd m2) (snd m1)))).
Defined.
End ProductCategory.
Definition OppositeCategory `(C : @Category objC) : Category objC :=
@Build_Category objC
(fun s d => Morphism C d s)
(Identity (C := C))
(fun _ _ _ m1 m2 => Compose m2 m1).
Parameter FunctorCategory : forall `(C : @Category objC) `(D : @Category objD), @Category (Functor C D).
Parameter TerminalCategory : Category unit.
Section ComputableCategory.
Variable I : Type.
Variable Index2Object : I -> Type.
Variable Index2Cat : forall i : I, @Category (@Index2Object i).
Local Coercion Index2Cat : I >-> Category.
Definition ComputableCategory : @Category I.
refine (@Build_Category _
(fun C D : I => Functor C D)
(fun o : I => IdentityFunctor o)
(fun C D E : I => ComposeFunctors (C := C) (D := D) (E := E))).
Defined.
End ComputableCategory.
Section SmallCat.
Definition LocallySmallCat := ComputableCategory _ LSUnderlyingCategory.
End SmallCat.
Section CommaCategory.
Context `(A : @Category objA).
Context `(B : @Category objB).
Context `(C : @Category objC).
Variable S : Functor A C.
Variable T : Functor B C.
Record CommaCategory_Object := { CommaCategory_Object_Member :> { ab : objA * objB & C.(Morphism) (S (fst ab)) (T (snd ab)) } }.
Let SortPolymorphic_Helper (A T : Type) (Build_T : A -> T) := A.
Definition CommaCategory_ObjectT := Eval hnf in SortPolymorphic_Helper Build_CommaCategory_Object.
Definition Build_CommaCategory_Object' (mem : CommaCategory_ObjectT) := Build_CommaCategory_Object mem.
Global Coercion Build_CommaCategory_Object' : CommaCategory_ObjectT >-> CommaCategory_Object.
Definition CommaCategory : @Category CommaCategory_Object.
Admitted.
End CommaCategory.
Definition SliceCategory_Functor `(C : @Category objC) (a : C) : Functor TerminalCategory C
:= {| ObjectOf := (fun _ => a);
MorphismOf := (fun _ _ _ => Identity a)
|}.
Definition SliceCategoryOver
: forall (objC : Type) (C : Category objC) (a : C),
Category
(CommaCategory_Object (IdentityFunctor C)
(SliceCategory_Functor C a)).
admit.
Defined.
Section CommaCategoryProjectionFunctor.
Context `(A : Category objA).
Context `(B : Category objB).
Context `(C : Category objC).
Variable S : (OppositeCategory (FunctorCategory A C)).
Variable T : (FunctorCategory B C).
Definition CommaCategoryProjection : Functor (CommaCategory S T) (ProductCategory A B).
Admitted.
Definition CommaCategoryProjectionFunctor_ObjectOf
: @SliceCategoryOver _ LocallySmallCat (ProductCategory A B)
:=
existT _
((CommaCategory S T) : Category', tt)
(CommaCategoryProjection) :
CommaCategory_ObjectT (IdentityFunctor _)
(SliceCategory_Functor LocallySmallCat
(ProductCategory A B)).
(* Anomaly: apply_coercion_args: mismatch between arguments and coercion. Please report. *)
(* Toplevel input, characters 110-142:
Error:
In environment
objA : Type
A : Category objA
objB : Type
B : Category objB
objC : Type
C : Category objC
S : OppositeCategory (FunctorCategory A C)
T : FunctorCategory B C
The term "ProductCategory A B:Category (objA * objB)" has type
"Category (objA * objB)" while it is expected to have type
"Object LocallySmallCat".
*)
End CommaCategoryProjectionFunctor.
End SecondComment.
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