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Library CT.Monad

Require Import CT.Category.
Require Import CT.Functor.
Require Import CT.NaturalTransformation.
Require Import CT.Instance.Functor.ComposeFunctor.
Require Import CT.Instance.Functor.Endofunctor.
Require Import CT.Instance.Functor.IdentityFunctor.
Require Export CT.Instance.NaturalTransformation.HCNaturalTransformation.
Require Export CT.Instance.NaturalTransformation.IdentityNaturalTransformation.
Require Export CT.Instance.NaturalTransformation.VCNaturalTransformation.

Monad.

A monad is an endofunctor on

$C$ together with two natural transformations,

$\eta$ and

$\mu$ which satisfy two coherence conditions.

Record Monad {C : Category} :=
  { T : Endofunctor C;

    eta : NaturalTransformation (@IdentityFunctor C) T;

    mu : NaturalTransformation (ComposeFunctor T T) T;


    monad_coherence_assoc :
      nt_components (VCNaturalTransformation (HCNaturalTransformation mu (IdentityNaturalTransformation T)) mu) =
      nt_components (VCNaturalTransformation (HCNaturalTransformation (IdentityNaturalTransformation T) mu) mu);

    monad_coherence_id_1 :
      nt_components (VCNaturalTransformation (HCNaturalTransformation eta (IdentityNaturalTransformation T)) mu) =
      nt_components (IdentityNaturalTransformation T);

    monad_coherence_id_2 :
      nt_components (VCNaturalTransformation (HCNaturalTransformation (IdentityNaturalTransformation T) eta) mu) =
      nt_components (IdentityNaturalTransformation T);
  }.

Definition bind {C : Category} (m : @Monad C) x :=
  nt_components (mu m) (F_ob (T m) x).