This file contains typeclasses for abstract categories and type constructor classes (functors, monads, etc.). These are not the specialized definitions used by Tealeaves but can be used for general abstract nonsense.
The implementation here closely resembles that of math-classes, see https://github.com/coq-community/math-classes but this development focuses more on monad-related abstractions such as modules and comonads.
From Tealeaves Require Export

Declare Scope category_scope.
Delimit Scope category_scope with cat.
Open Scope category_scope.


Operational typeclasses

Class Arrows (Obj : Type) : Type :=
  homset : Obj -> Obj -> Type.

Class Identities Obj `{Arrows Obj} :=
  catid : forall (x : Obj), homset x x.

Class Composition Obj `{Arrows Obj} :=
  comp : forall (x y z : Obj), homset y z -> homset x y -> homset x z.


Module Notations.
  Infix "⟶" := (homset) (at level 90, right associativity) : category_scope.
  Infix "⊙":= (comp _ _ _) (at level 40, left associativity) : category_scope.
  Notation "F ⇒ G" := (forall a : _, homset (F a) (G a)) (at level 50) : category_scope.
End Notations.

Import Notations.

Category typeclass

Section category.

    (Obj : Type)
    `{Arrows Obj}
    `{! Identities Obj}
    `{! Composition Obj}.

  Class Category :=
    { cat_assoc {w x y z} (c : y z) (b : x y) (a : w x) :
        c (b a) = (c b) a;
      cat_id_r {x y} (a : x y):
        a catid x = a;
      cat_id_l {x y} (a : x y) :
        catid y a = a;

End category.


Section functor.

    `{Category C}
    `{Category D}
    (F : C -> D).

  Open Scope category_scope.

  Class Fmap : Type :=
    fmap: forall {a b : C} (f : a b), F a F b.

  #[global] Arguments fmap {Fmap a b} (_)%cat.

  (* don't register category fields as coercions to avoid loops *)
  Class Functor `(Fmap) : Prop :=
    { func_src : Category C;
      func_tgt : Category D;
      fmap_id : forall a : C,
          fmap (catid a) = catid (F a);
      fmap_fmap : forall a b c (f : a b) (g : b c),
          fmap g fmap f = fmap (g f);

End functor.

Natural transformations

Section natural_transformation.

    `{Category C}
    `{Category D}
    `{! Functor (F : C -> D) Ffmap}
    `{! Functor (G : C -> D) Gfmap}.

  Class Natural (η : forall x, F x G x) :=
    naturality : forall {x y} (f : x y),
      fmap G f η x = η y fmap F f.

End natural_transformation.

Monoidal structure on endofunctors

Endofunctors are closed under composition and identities. This gives monoidal structure on the category of endofunctors but we do not need to formalize that far.
Section endofunctor_id.

    `{Category C}.

  #[global] Instance Fmap_one : Fmap (fun x => x) :=
    (fun (a b : C) (f : a b) => f).

  Definition fmap_id_one : forall (a : C),
      fmap (fun x => x) (catid a) = catid a := ltac:(reflexivity).

  Definition fmap_fmap_one a b c (f : a b) (g : b c) :
    fmap (fun x => x) g fmap (fun x => x) f = fmap (fun x => x) (g f) := ltac:(reflexivity).

  #[global] Program Instance Functor_one : Functor (fun x => x) _ :=
    {| fmap_id := fmap_id_one;
       fmap_fmap := fmap_fmap_one;

End endofunctor_id.

Section endofunctor_composition.

    `{Category C}
    `{Category D}
    `{Category E}
    `{! Functor (F : C -> D) fmap_F}
    `{! Functor (G : D -> E) fmap_G}.

  #[global] Instance Fmap_compose : Fmap (G F) :=
    fun a b f => fmap G (fmap F f).

  Lemma fmap_id_compose : `(fmap (G F) (catid a) = catid (G (F a))).
    intros ?; unfold fmap, Fmap_compose.
    now rewrite (fmap_id F), (fmap_id G).

  Lemma fmap_fmap_compose (a b c : C) (f : a b) (g : b c) :
    fmap (G F) g fmap (G F) f = fmap (G F) (g f).
    unfold fmap, Fmap_compose, compose.
    now rewrite (fmap_fmap G), (fmap_fmap F).

  #[global] Instance Functor_compose : Functor (G F) _ :=
    {| fmap_id := fmap_id_compose;
       fmap_fmap := fmap_fmap_compose;

End endofunctor_composition.

Monads and modules

Section monad_operations.

    `{Category C}
    (T : C -> C).

  Class Join := join : T T T.

  Class Return := ret : (fun x => x) T.

End monad_operations.

Section monad.

    `{Category C}
    (T : C -> C)
    `{! Functor T fmap_T}
    `{! Return T} `{! Join T}.

  Class Monad :=
    { mon_functor :> Functor T fmap_T;
      mon_ret_natural :> Natural (ret T);
      mon_join_natural :> Natural (join T);
      mon_join_fmap_ret :
        `(join T a (fmap T (ret T a)) = catid (T a));
      mon_join_ret :
        `(join T a (ret T (T a)) = catid (T a));
      mon_join_join :
        `(join T a (join T (T a)) = join T a (fmap T (join T a)));

End monad.

Section monad_homomorphism.

    `{Category C}
    `{! Functor (T : C -> C) Tfmap}
    `{! Functor (U : C -> C) Ufmap}
    `{! Return T} `{! Join T}
    `{! Return U} `{! Join U}
    `{! Monad T} `{! Monad U}.

  Class Monad_Hom (mhom : forall a, T a U a) :=
    { mhom_domain : Monad T;
      mhom_codomain : Monad U;
      mhom_natural : Natural mhom;
      mhom_ret :
        `(mhom a (ret T a) = ret U a);
      mhom_join :
        `(mhom a (join T a) = join U a (mhom (U a) (fmap T (mhom a))));

End monad_homomorphism.

Right modules of endofunctors

Section RightModule.

    {C D : Type}
    (F : C -> D)
    (T : C -> C)
    `{Monad C T}
    `{Category D}
    `{! Functor F Ffmap }.

  Class RightAction := right_action : F T F.

  Class RightModule `{RightAction} :=
    { rmod_monad :> Monad T;
      rmod_object :> Functor F Ffmap;
      rmod_natural : Natural (right_action);
      rmod_ret :
        `(right_action x fmap F (ret T x) = catid (F x));
      rmod_join :
        `(right_action x right_action (T x) = right_action x fmap F (join T x));

End RightModule.

The bind operation

Section bind.

    {C D : Type}
    (F : C -> D)
    (T : C -> C)
    `{RightModule C D F T}.

  Class Bind := bind : forall {a b : C} (f : a T b), F a F b.

  #[global] Arguments bind {Bind a b} (_)%cat.

  Definition compose_kleisli {a b c : C} :
    (b T c) -> (a T b) -> a T c :=
    fun g f => join T c (fmap T g f).

End bind.

Module KleisliNotation.
  Notation "g ⋆ f" := (compose_kleisli _ g f) (at level 60) : category_scope.
End KleisliNotation.

Import KleisliNotation.

Section module_bind.

    `{RightModule C D F T}
    `{! Category C}
    `{! Category D}.

  #[global] Instance Bind_Module : Bind F T :=
    fun {a b} {f : a T b} => right_action F T b fmap F f.

    {a b c : C}.

  Lemma bind_ret_fmap : forall (f : a b),
      bind F T (ret T b f) = fmap F f.
    intros. unfold bind, Bind_Module.
    rewrite <- (fmap_fmap F).
    rewrite (cat_assoc D).
    rewrite (rmod_ret F T).
    rewrite (cat_id_l D).

  Lemma bind_functorial : forall (f : a T b) (g : b T c),
      bind F T (g f) = (bind F T g) (bind F T f).
    intros f g. unfold bind, Bind_Module.
    unfold compose_kleisli.
    rewrite <- (fmap_fmap F).
    rewrite <- (fmap_fmap F).
    repeat rewrite (cat_assoc D).
    rewrite <- (rmod_join F T).
    change (fmap F (fmap T g)) with (fmap (F T) g).
    repeat rewrite <- (cat_assoc D).
    rewrite (cat_assoc D (right_action F T (T c)) _ _).
    rewrite <- (naturality (Natural := rmod_natural F T) g).
    now repeat rewrite <- (cat_assoc D).

  Lemma bind_fmap : forall (g : b T c) (f : a b),
      bind F T g (fmap F f) = bind F T (g f).
    unfold bind, Bind_Module.
    rewrite <- (fmap_fmap F).
    now rewrite <- (cat_assoc D).

  Lemma bind_ret_l :
    bind F T (ret T a) = catid (F a).
    intros. unfold bind, Bind_Module.
    now rewrite (rmod_ret F T).

End module_bind.