Library UniMath.CategoryTheory.Monads.Kleisli
**********************************************************
Contents:
Written by: Joseph Helfer, Matthew Weaver, 2017
- "Kleisli" definition of monad Kleisli
- equivalence of this definition and the "monoidal" definition weq_Kleisli_Monad
Require Import UniMath.Foundations.PartD.
Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.Monads.Monads.
Require Import UniMath.CategoryTheory.Monads.RelativeMonads.
Local Open Scope cat.
Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.Monads.Monads.
Require Import UniMath.CategoryTheory.Monads.RelativeMonads.
Local Open Scope cat.
Section Kleisli_def.
Definition Kleisli (C : precategory_data) : UU := RelMonad (functor_identity C).
Definition RelMonad_from_Kleisli {C : precategory_data} (T : Kleisli C) := (T : RelMonad (functor_identity C)).
Coercion RelMonad_from_Kleisli : Kleisli >-> RelMonad.
End Kleisli_def.
Definition Kleisli (C : precategory_data) : UU := RelMonad (functor_identity C).
Definition RelMonad_from_Kleisli {C : precategory_data} (T : Kleisli C) := (T : RelMonad (functor_identity C)).
Coercion RelMonad_from_Kleisli : Kleisli >-> RelMonad.
End Kleisli_def.
Section monad_types_equiv.
Definition Monad_to_Kleisli {C : precategory} : Monad C → Kleisli C :=
λ T, (functor_on_objects T ,, (pr1 (η T) ,, @bind C T))
,, @Monad_law2 C T ,, (@η_bind C T ,, @bind_bind C T).
Definition Kleisli_to_functor {C : precategory} (T: Kleisli C) : C ⟶ C.
Show proof.
Definition Kleisli_to_μ {C : precategory} (T: Kleisli C) :
Kleisli_to_functor T ∙ Kleisli_to_functor T ⟹ Kleisli_to_functor T.
Show proof.
Definition Kleisli_to_η {C : precategory} (T: Kleisli C) :
functor_identity C ⟹ Kleisli_to_functor T.
Show proof.
Definition Kleisli_to_Monad {C : precategory} (T : Kleisli C) : Monad C.
Show proof.
Proposition Kleisli_to_Monad_to_Kleisli {C : precategory} (hs : has_homsets C) (T : Kleisli C) :
Monad_to_Kleisli (Kleisli_to_Monad T) = T.
Show proof.
Lemma Monad_to_Kleisli_to_Monad_raw_data {C : precategory} (T : Monad C) :
Monad_to_raw_data (Kleisli_to_Monad (Monad_to_Kleisli T)) = Monad_to_raw_data T.
Show proof.
Definition Monad_to_Kleisli_to_Monad {C : precategory} (hs : has_homsets C) (T : Monad C) :
Kleisli_to_Monad (Monad_to_Kleisli T) = T.
Show proof.
Definition isweq_Monad_to_Kleisli {C : precategory} (hs: has_homsets C) :
isweq Monad_to_Kleisli :=
isweq_iso _ _ (Monad_to_Kleisli_to_Monad hs) (Kleisli_to_Monad_to_Kleisli hs).
Definition weq_Kleisli_Monad {C : precategory} (hs : has_homsets C) :
(Monad C) ≃ (Kleisli C) := _,, (isweq_Monad_to_Kleisli hs).
End monad_types_equiv.
Definition Monad_to_Kleisli {C : precategory} : Monad C → Kleisli C :=
λ T, (functor_on_objects T ,, (pr1 (η T) ,, @bind C T))
,, @Monad_law2 C T ,, (@η_bind C T ,, @bind_bind C T).
Definition Kleisli_to_functor {C : precategory} (T: Kleisli C) : C ⟶ C.
Show proof.
use mk_functor.
- use mk_functor_data.
+ exact T.
+ apply r_lift.
- apply is_functor_r_lift.
- use mk_functor_data.
+ exact T.
+ apply r_lift.
- apply is_functor_r_lift.
Definition Kleisli_to_μ {C : precategory} (T: Kleisli C) :
Kleisli_to_functor T ∙ Kleisli_to_functor T ⟹ Kleisli_to_functor T.
Show proof.
use tpair.
- exact (λ (x : C), r_bind T (identity (T x))).
- intros x x' f; simpl.
unfold r_lift.
now rewrite (r_bind_r_bind T), <- assoc, (r_eta_r_bind T (T x')), id_right, (r_bind_r_bind T), id_left.
- exact (λ (x : C), r_bind T (identity (T x))).
- intros x x' f; simpl.
unfold r_lift.
now rewrite (r_bind_r_bind T), <- assoc, (r_eta_r_bind T (T x')), id_right, (r_bind_r_bind T), id_left.
Definition Kleisli_to_η {C : precategory} (T: Kleisli C) :
functor_identity C ⟹ Kleisli_to_functor T.
Show proof.
use tpair.
- exact (r_eta T).
- intros x x' f; simpl.
unfold r_lift.
now rewrite (r_eta_r_bind T x).
- exact (r_eta T).
- intros x x' f; simpl.
unfold r_lift.
now rewrite (r_eta_r_bind T x).
Definition Kleisli_to_Monad {C : precategory} (T : Kleisli C) : Monad C.
Show proof.
use (((Kleisli_to_functor T,, Kleisli_to_μ T) ,, Kleisli_to_η T) ,, _).
do 2 try apply tpair; intros; simpl.
- apply (r_eta_r_bind T).
- unfold r_lift. now rewrite (r_bind_r_bind T), <- assoc, (r_eta_r_bind T (T c)), id_right, (r_bind_r_eta T).
- unfold r_lift. now rewrite !(r_bind_r_bind T), id_left, <- assoc, (r_eta_r_bind T (T c)), id_right.
do 2 try apply tpair; intros; simpl.
- apply (r_eta_r_bind T).
- unfold r_lift. now rewrite (r_bind_r_bind T), <- assoc, (r_eta_r_bind T (T c)), id_right, (r_bind_r_eta T).
- unfold r_lift. now rewrite !(r_bind_r_bind T), id_left, <- assoc, (r_eta_r_bind T (T c)), id_right.
Proposition Kleisli_to_Monad_to_Kleisli {C : precategory} (hs : has_homsets C) (T : Kleisli C) :
Monad_to_Kleisli (Kleisli_to_Monad T) = T.
Show proof.
apply subtypeEquality'.
- apply (maponpaths (λ p, tpair _ _ p )); simpl.
apply dirprod_paths.
* apply idpath.
* repeat (apply funextsec; unfold homot; intro).
simpl; unfold Monads.bind; simpl; unfold r_lift.
now rewrite (r_bind_r_bind T), <- assoc, (r_eta_r_bind T (T x0)), id_right.
- do 2 try apply isapropdirprod;
do 5 try (apply impred; intro);
apply hs.
- apply (maponpaths (λ p, tpair _ _ p )); simpl.
apply dirprod_paths.
* apply idpath.
* repeat (apply funextsec; unfold homot; intro).
simpl; unfold Monads.bind; simpl; unfold r_lift.
now rewrite (r_bind_r_bind T), <- assoc, (r_eta_r_bind T (T x0)), id_right.
- do 2 try apply isapropdirprod;
do 5 try (apply impred; intro);
apply hs.
Lemma Monad_to_Kleisli_to_Monad_raw_data {C : precategory} (T : Monad C) :
Monad_to_raw_data (Kleisli_to_Monad (Monad_to_Kleisli T)) = Monad_to_raw_data T.
Show proof.
apply (maponpaths (λ p, tpair _ _ p )); simpl.
apply dirprod_paths.
+ apply dirprod_paths;
repeat (apply funextsec; unfold homot; intro);
simpl.
* unfold r_lift, r_bind, r_eta; simpl. unfold Monads.bind.
rewrite (functor_comp T), <- assoc.
change (# T x1 · (# T (η T x0) · μ T x0) = #T x1).
now rewrite (@Monad_law2 C T x0), id_right.
* unfold Monads.bind, r_bind; simpl.
now rewrite (functor_id T), id_left.
+ apply idpath.
apply dirprod_paths.
+ apply dirprod_paths;
repeat (apply funextsec; unfold homot; intro);
simpl.
* unfold r_lift, r_bind, r_eta; simpl. unfold Monads.bind.
rewrite (functor_comp T), <- assoc.
change (# T x1 · (# T (η T x0) · μ T x0) = #T x1).
now rewrite (@Monad_law2 C T x0), id_right.
* unfold Monads.bind, r_bind; simpl.
now rewrite (functor_id T), id_left.
+ apply idpath.
Definition Monad_to_Kleisli_to_Monad {C : precategory} (hs : has_homsets C) (T : Monad C) :
Kleisli_to_Monad (Monad_to_Kleisli T) = T.
Show proof.
apply (Monad_eq_raw_data hs).
apply Monad_to_Kleisli_to_Monad_raw_data.
apply Monad_to_Kleisli_to_Monad_raw_data.
Definition isweq_Monad_to_Kleisli {C : precategory} (hs: has_homsets C) :
isweq Monad_to_Kleisli :=
isweq_iso _ _ (Monad_to_Kleisli_to_Monad hs) (Kleisli_to_Monad_to_Kleisli hs).
Definition weq_Kleisli_Monad {C : precategory} (hs : has_homsets C) :
(Monad C) ≃ (Kleisli C) := _,, (isweq_Monad_to_Kleisli hs).
End monad_types_equiv.