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Set Implicit Arguments.
Set Universe Polymorphism.
Module success.
Unset Primitive Projections.
Record category :=
{ ob : Type;
hom : ob -> ob -> Type;
comp : forall x y z, hom y z -> hom x y -> hom x z }.
Delimit Scope hom_scope with hom.
Bind Scope hom_scope with hom.
Arguments hom : clear implicits.
Arguments comp _ _ _ _ _%hom _%hom : clear implicits.
Notation "f 'o' g" := (comp _ _ _ _ f g) (at level 40, left associativity) : hom_scope.
Record functor (C D : category) :=
{ ob_of : ob C -> ob D;
hom_of : forall x y, hom C x y -> hom D (ob_of x) (ob_of y) }.
Delimit Scope functor_scope with functor.
Bind Scope functor_scope with functor.
Arguments hom_of _ _ _%functor _ _ _%hom.
Notation "F '_1' m" := (hom_of F _ _ m) (at level 10, no associativity) : hom_scope.
Axiom f_comp : forall C D E, functor D E -> functor C D -> functor C E.
Notation "f 'o' g" := (@f_comp _ _ _ f g) (at level 40, left associativity) : functor_scope.
Check ((_ o _) _1 _)%hom. (* ((?16 o ?17) _1 ?20)%hom
: hom ?15 (ob_of (?16 o ?17) ?18) (ob_of (?16 o ?17) ?19) *)
End success.
Module failure.
Set Primitive Projections.
Record category :=
{ ob : Type;
hom : ob -> ob -> Type;
comp : forall x y z, hom y z -> hom x y -> hom x z }.
Delimit Scope hom_scope with hom.
Bind Scope hom_scope with hom.
Arguments hom : clear implicits.
Arguments comp _ _ _ _ _%hom _%hom : clear implicits.
Notation "f 'o' g" := (comp _ _ _ _ f g) (at level 40, left associativity) : hom_scope.
Record functor (C D : category) :=
{ ob_of : ob C -> ob D;
hom_of : forall x y, hom C x y -> hom D (ob_of x) (ob_of y) }.
Delimit Scope functor_scope with functor.
Bind Scope functor_scope with functor.
Arguments hom_of _ _ _%functor _ _ _%hom.
Notation "F '_1' m" := (hom_of F _ _ m) (at level 10, no associativity) : hom_scope.
Notation "F '_2' m" := (hom_of F%functor _ _ m) (at level 10, no associativity) : hom_scope.
Axiom f_comp : forall C D E, functor D E -> functor C D -> functor C E.
Notation "f 'o' g" := (@f_comp _ _ _ f g) (at level 40, left associativity) : functor_scope.
Check ((_ o _) _2 _)%hom. (* ((?14 o ?15)%functor _1 ?18)%hom
: hom ?13 (ob_of (?14 o ?15)%functor ?16)
(ob_of (?14 o ?15)%functor ?17) *)
Check ((_ o _) _1 _)%hom. (* Toplevel input, characters 7-19:
Error:
The term "(?23 o ?24)%hom" has type "hom ?19 ?20 ?22"
while it is expected to have type "functor ?25 ?26". *)
End failure.
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