Library UniMath.CategoryTheory.Monads.Monads
**********************************************************
Contents:
Written by: Benedikt Ahrens (started March 2015)
Extended by: Anders Mörtberg, 2016
Extended by: Ralph Matthes, 2017 (Section MonadsUsingCoproducts)
Rewrite by: Ralph Matthes, 2023, defining the category of monads through a displayed category over the endofunctors, this gives the previously missing univalence result nearly for free
- Definition of monads (Monad)
- category of monads category_Monad C on C
- Forgetful functor forgetfunctor_Monad from monads to endofunctors on C
- Haskell style bind operation (bind)
- A substitution operator for monads (monadSubst)
- A helper lemma for proving equality of Monads (Monad_eq_raw_data)
- Proof that precategory_Monad C is univalent if C is
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.FunctorCategory.
Require Import UniMath.CategoryTheory.whiskering.
Require Import UniMath.CategoryTheory.Adjunctions.Core.
Require Import UniMath.CategoryTheory.Limits.Terminal.
Require Import UniMath.CategoryTheory.Limits.BinCoproducts.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.CategoryTheory.DisplayedCats.Total.
Require Import UniMath.CategoryTheory.DisplayedCats.Constructions.
Require Import UniMath.CategoryTheory.DisplayedCats.Isos.
Require Import UniMath.CategoryTheory.DisplayedCats.SIP.
Require Import UniMath.CategoryTheory.DisplayedCats.Univalence.
Require Import UniMath.CategoryTheory.Core.Univalence.
Local Open Scope cat.
Section Monad_disp_def.
Context {C : category}.
Definition disp_Monad_data (F : functor C C) : UU :=
(F ∙ F ⟹ F) × (functor_identity C ⟹ F).
Definition disp_μ {F : functor C C} (T : disp_Monad_data F) : F ∙ F ⟹ F := pr1 T.
Definition disp_η {F : functor C C} (T : disp_Monad_data F) : functor_identity C ⟹ F := pr2 T.
Definition disp_Monad_laws {F : functor C C} (T : disp_Monad_data F) : UU :=
(
(∏ c : C, disp_η T (F c) · disp_μ T c = identity (F c))
×
(∏ c : C, #F (disp_η T c) · disp_μ T c = identity (F c)) )
×
(∏ c : C, #F (disp_μ T c) · disp_μ T c = disp_μ T (F c) · disp_μ T c).
Lemma isaprop_disp_Monad_laws {F : functor C C} (T : disp_Monad_data F) : isaprop (disp_Monad_laws T).
Proof.
repeat apply isapropdirprod;
apply impred; intro c; apply C.
Qed.
Definition disp_Monad_Mor_laws {F F' : functor C C} (T : disp_Monad_data F) (T' : disp_Monad_data F') (α : F ⟹ F') : UU :=
(∏ a : C, disp_μ T a · α a = α (F a) · #F' (α a) · disp_μ T' a) ×
(∏ a : C, disp_η T a · α a = disp_η T' a).
Lemma isaprop_disp_Monad_Mor_laws {F F' : functor C C} (T : disp_Monad_data F) (T' : disp_Monad_data F') (α : F ⟹ F')
: isaprop (disp_Monad_Mor_laws T T' α).
Proof.
apply isapropdirprod;
apply impred; intro c; apply C.
Qed.
Lemma monads_category_id_subproof {F : functor C C} (T : disp_Monad_data F) (Tlaws : disp_Monad_laws T) :
disp_Monad_Mor_laws T T (nat_trans_id F).
Proof.
split; intros a; simpl.
- now rewrite id_left, id_right, functor_id, id_left.
- now apply id_right.
Qed.
Lemma monads_category_comp_subproof {F F' F'' : functor C C}
(T : disp_Monad_data F) (Tlaws : disp_Monad_laws T)
(T' : disp_Monad_data F') (T'laws : disp_Monad_laws T')
(T'' : disp_Monad_data F'') (T''laws : disp_Monad_laws T'')
(α : F ⟹ F') (α' : F' ⟹ F'') :
disp_Monad_Mor_laws T T' α → disp_Monad_Mor_laws T' T'' α' → disp_Monad_Mor_laws T T'' (nat_trans_comp _ _ _ α α').
Proof.
intros Hα Hα'.
split; intros; simpl.
- rewrite assoc.
set (H:=pr1 Hα a); simpl in H.
rewrite H; clear H; rewrite <- !assoc.
set (H:=pr1 Hα' a); simpl in H.
rewrite H; clear H.
rewrite functor_comp.
apply maponpaths.
now rewrite !assoc, nat_trans_ax.
- rewrite assoc.
eapply pathscomp0; [apply cancel_postcomposition, (pr2 Hα)|].
apply (pr2 Hα').
Qed.
Definition monads_category_disp : disp_cat [C,C].
Proof.
use disp_cat_from_SIP_data.
- intro F. exact (∑ T : disp_Monad_data F, disp_Monad_laws T).
- intros F F' [T Tlaws] [T' T'laws] α. exact (disp_Monad_Mor_laws T T' α).
- intros F F' [T Tlaws] [T' T'laws] α. apply isaprop_disp_Monad_Mor_laws.
- intros F [T Tlaws]. apply (monads_category_id_subproof _ Tlaws).
- intros F F' F'' [T Tlaws] [T' T'laws] [T'' T''laws] α α'.
apply (monads_category_comp_subproof _ Tlaws _ T'laws _ T''laws).
Defined.
Definition category_Monad : category := total_category monads_category_disp.
Definition Monad : UU := ob category_Monad.
Coercion functor_from_Monad (T : Monad) : functor C C := pr1 T.
Definition μ (T : Monad) : T ∙ T ⟹ T := pr112 T.
Definition η (T : Monad) : functor_identity C ⟹ T := pr212 T.
Lemma Monad_law1 {T : Monad} : ∏ c : C, η T (T c) · μ T c = identity (T c).
Proof.
exact (pr1 (pr122 T)).
Qed.
Lemma Monad_law2 {T : Monad} : ∏ c : C, #T (η T c) · μ T c = identity (T c).
Proof.
exact (pr2 (pr122 T)).
Qed.
Lemma Monad_law3 {T : Monad} : ∏ c : C, #T (μ T c) · μ T c = μ T (T c) · μ T c.
Proof.
exact (pr222 T).
Qed.
Lemma monads_category_disp_eq (F : functor C C) (T T' : monads_category_disp F) : pr1 T = pr1 T' -> T = T'.
Proof.
intro H.
induction T as [T Tlaws].
induction T' as [T' T'laws].
use total2_paths_f; [apply H |].
apply isaprop_disp_Monad_laws.
Qed.
Lemma monads_category_Pisset (F : functor C C) : isaset (∑ T : disp_Monad_data F, disp_Monad_laws T).
Proof.
change isaset with (isofhlevel 2).
apply isofhleveltotal2.
{ apply isasetdirprod; apply [C,C]. }
intro T.
apply isasetaprop.
apply isaprop_disp_Monad_laws.
Qed.
Lemma monads_category_Hstandard {F : functor C C}
(T : disp_Monad_data F) (Tlaws : disp_Monad_laws T)
(T' : disp_Monad_data F) (T'laws : disp_Monad_laws T') :
disp_Monad_Mor_laws T T' (nat_trans_id F) → disp_Monad_Mor_laws T' T (nat_trans_id F) → T,, Tlaws = T',, T'laws.
Proof.
intros H H'.
apply subtypeInjectivity.
{ intro T0. apply isaprop_disp_Monad_laws. }
cbn.
induction T as [μ η].
induction T' as [μ' η'].
apply dirprodeq; cbn.
- apply nat_trans_eq; [apply C|]. intro a.
assert (Hinst := pr1 H a).
cbn in Hinst.
rewrite id_right in Hinst.
rewrite id_left in Hinst.
rewrite functor_id in Hinst.
rewrite id_left in Hinst.
exact Hinst.
- apply nat_trans_eq; [apply C|]. intro a.
assert (Hinst := pr2 H a).
cbn in Hinst.
rewrite id_right in Hinst.
exact Hinst.
Qed.
Definition is_univalent_monads_category_disp : is_univalent_disp monads_category_disp.
Proof.
use is_univalent_disp_from_SIP_data.
- exact monads_category_Pisset.
- intros F [T Tlaws] [T' T'laws]. apply monads_category_Hstandard.
Defined.
End Monad_disp_def.
Arguments category_Monad _ : clear implicits.
Arguments Monad _ : clear implicits.
Definition is_univalent_category_Monad (C : univalent_category) :
is_univalent (category_Monad C).
Proof.
apply SIP.
- apply is_univalent_functor_category. apply C.
- apply monads_category_Pisset.
- intros F [T Tlaws] [T' T'laws]. apply monads_category_Hstandard.
Defined.
Section pointfree.
Context (C : category) (T0: functor C C) (T : disp_Monad_data T0).
Let EndC := [C, C].
Let η := disp_η T.
Let μ := disp_μ T.
Definition Monad_laws_pointfree : UU :=
(
(nat_trans_comp _ _ _ (pre_whisker T0 η) μ = identity(C:=EndC) T0)
×
(nat_trans_comp _ _ _ (post_whisker η T0) μ = identity(C:=EndC) T0) )
×
(nat_trans_comp _ _ _ (post_whisker μ T0) μ = nat_trans_comp _ _ _ (pre_whisker T0 μ) μ).
Lemma pointfree_is_equiv: Monad_laws_pointfree <-> disp_Monad_laws T.
Proof.
split.
- intro H. induction H as [[H1 H2] H3].
split.
+ split.
* intro c. apply (maponpaths pr1) in H1. apply toforallpaths in H1. apply H1.
* intro c. apply (maponpaths pr1) in H2. apply toforallpaths in H2. apply H2.
+ intro c. apply (maponpaths pr1) in H3. apply toforallpaths in H3. apply H3.
- intro H. induction H as [[H1 H2] H3].
split.
+ split.
* apply nat_trans_eq_alt. exact H1.
* apply nat_trans_eq_alt. exact H2.
+ apply nat_trans_eq; try apply homset_property. exact H3.
Qed.
Let T0' := T0 : EndC.
Let η' := η: EndC⟦functor_identity C, T0'⟧.
Let μ' := μ: EndC⟦functor_compose T0' T0', T0'⟧.
Definition Monad_laws_pointfree_in_functor_category : UU :=
(
(#(pre_composition_functor _ _ _ T0') η' · μ' = identity(C:=EndC) T0')
×
(#(post_composition_functor _ _ _ T0') η' · μ' = identity(C:=EndC) T0') )
×
(#(post_composition_functor _ _ _ T0') μ' · μ' = (#(pre_composition_functor _ _ _ T0') μ') · μ').
the last variant of the laws is convertible with the one before
Goal
Monad_laws_pointfree = Monad_laws_pointfree_in_functor_category.
Proof.
apply idpath.
Qed.
End pointfree.
Definition Monad_Mor {C : category} (T T' : Monad C) : UU
:= category_Monad C ⟦T, T'⟧.
Definition Monad_Mor_laws {C : category} {T T' : Monad C} (α : T ⟹ T')
: UU :=
(∏ a : C, μ T a · α a = α (T a) · #T' (α a) · μ T' a) ×
(∏ a : C, η T a · α a = η T' a).
Coercion nat_trans_from_monad_mor {C : category} (T T' : Monad C) (s : Monad_Mor T T')
: T ⟹ T' := pr1 s.
Coercion nat_trans_from_monad_mor_laws {C : category} (T T' : Monad C) (s : Monad_Mor T T')
: Monad_Mor_laws s := pr2 s.
Proposition isaprop_Monad_Mor_laws {C : category} {T T' : Monad C} (α : T ⟹ T')
: isaprop (Monad_Mor_laws α).
Proof.
use isapropdirprod ; use impred ; intro ; apply homset_property.
Qed.
Definition Monad_Mor_η {C : category} {T T' : Monad C} (α : Monad_Mor T T')
: ∏ a : C, η T a · α a = η T' a.
Proof.
exact (pr22 α).
Qed.
Definition Monad_Mor_μ {C : category} {T T' : Monad C} (α : Monad_Mor T T')
: ∏ a : C, μ T a · α a = α (T a) · #T' (α a) · μ T' a.
Proof.
exact (pr12 α).
Qed.
Proposition identity_Monad_Mor_laws
{C : category}
(T : Monad C)
: Monad_Mor_laws (nat_trans_id T).
Proof.
split.
- intro x ; cbn.
rewrite functor_id.
rewrite !id_left, id_right.
apply idpath.
- intro x ; cbn.
rewrite id_right.
apply idpath.
Qed.
Proposition composition_Monad_Mor_laws
{C : category}
{T T' T'' : Monad C}
{α : T ⟹ T'}
{β : T' ⟹ T''}
(Hα : Monad_Mor_laws α)
(Hβ : Monad_Mor_laws β)
: Monad_Mor_laws (nat_trans_comp _ _ _ α β).
Proof.
split.
- intro x ; cbn.
rewrite !assoc.
etrans.
{
apply maponpaths_2.
apply Hα.
}
rewrite !assoc'.
apply maponpaths.
etrans.
{
apply maponpaths.
apply Hβ.
}
rewrite !assoc.
apply maponpaths_2.
etrans.
{
apply maponpaths_2.
apply (nat_trans_ax β).
}
rewrite !assoc'.
rewrite functor_comp.
apply idpath.
- intro x ; cbn.
rewrite !assoc.
etrans.
{
apply maponpaths_2.
apply Hα.
}
apply Hβ.
Qed.
Definition Monad_Mor_equiv {C : category}
{T T' : Monad C} (α β : Monad_Mor T T')
: α = β ≃ (pr1 α = pr1 β).
Proof.
apply subtypeInjectivity; intro a.
apply isaprop_disp_Monad_Mor_laws.
Defined.
Lemma isaset_Monad_Mor {C : category} (T T' : Monad C) : isaset (Monad_Mor T T').
Proof.
apply homset_property.
Qed.
Definition Monad_composition {C : category} {T T' T'' : Monad C}
(α : Monad_Mor T T') (α' : Monad_Mor T' T'')
: Monad_Mor T T'' := α · α'.
Definition forgetfunctor_Monad (C : category) : functor (category_Monad C) [C,C]
:= pr1_category monads_category_disp.
Lemma forgetMonad_faithful (C : category) : faithful (forgetfunctor_Monad C).
Proof.
apply faithful_pr1_category.
intros T T' α.
apply isaprop_disp_Monad_Mor_laws.
Qed.
Monad_laws_pointfree = Monad_laws_pointfree_in_functor_category.
Proof.
apply idpath.
Qed.
End pointfree.
Definition Monad_Mor {C : category} (T T' : Monad C) : UU
:= category_Monad C ⟦T, T'⟧.
Definition Monad_Mor_laws {C : category} {T T' : Monad C} (α : T ⟹ T')
: UU :=
(∏ a : C, μ T a · α a = α (T a) · #T' (α a) · μ T' a) ×
(∏ a : C, η T a · α a = η T' a).
Coercion nat_trans_from_monad_mor {C : category} (T T' : Monad C) (s : Monad_Mor T T')
: T ⟹ T' := pr1 s.
Coercion nat_trans_from_monad_mor_laws {C : category} (T T' : Monad C) (s : Monad_Mor T T')
: Monad_Mor_laws s := pr2 s.
Proposition isaprop_Monad_Mor_laws {C : category} {T T' : Monad C} (α : T ⟹ T')
: isaprop (Monad_Mor_laws α).
Proof.
use isapropdirprod ; use impred ; intro ; apply homset_property.
Qed.
Definition Monad_Mor_η {C : category} {T T' : Monad C} (α : Monad_Mor T T')
: ∏ a : C, η T a · α a = η T' a.
Proof.
exact (pr22 α).
Qed.
Definition Monad_Mor_μ {C : category} {T T' : Monad C} (α : Monad_Mor T T')
: ∏ a : C, μ T a · α a = α (T a) · #T' (α a) · μ T' a.
Proof.
exact (pr12 α).
Qed.
Proposition identity_Monad_Mor_laws
{C : category}
(T : Monad C)
: Monad_Mor_laws (nat_trans_id T).
Proof.
split.
- intro x ; cbn.
rewrite functor_id.
rewrite !id_left, id_right.
apply idpath.
- intro x ; cbn.
rewrite id_right.
apply idpath.
Qed.
Proposition composition_Monad_Mor_laws
{C : category}
{T T' T'' : Monad C}
{α : T ⟹ T'}
{β : T' ⟹ T''}
(Hα : Monad_Mor_laws α)
(Hβ : Monad_Mor_laws β)
: Monad_Mor_laws (nat_trans_comp _ _ _ α β).
Proof.
split.
- intro x ; cbn.
rewrite !assoc.
etrans.
{
apply maponpaths_2.
apply Hα.
}
rewrite !assoc'.
apply maponpaths.
etrans.
{
apply maponpaths.
apply Hβ.
}
rewrite !assoc.
apply maponpaths_2.
etrans.
{
apply maponpaths_2.
apply (nat_trans_ax β).
}
rewrite !assoc'.
rewrite functor_comp.
apply idpath.
- intro x ; cbn.
rewrite !assoc.
etrans.
{
apply maponpaths_2.
apply Hα.
}
apply Hβ.
Qed.
Definition Monad_Mor_equiv {C : category}
{T T' : Monad C} (α β : Monad_Mor T T')
: α = β ≃ (pr1 α = pr1 β).
Proof.
apply subtypeInjectivity; intro a.
apply isaprop_disp_Monad_Mor_laws.
Defined.
Lemma isaset_Monad_Mor {C : category} (T T' : Monad C) : isaset (Monad_Mor T T').
Proof.
apply homset_property.
Qed.
Definition Monad_composition {C : category} {T T' T'' : Monad C}
(α : Monad_Mor T T') (α' : Monad_Mor T' T'')
: Monad_Mor T T'' := α · α'.
Definition forgetfunctor_Monad (C : category) : functor (category_Monad C) [C,C]
:= pr1_category monads_category_disp.
Lemma forgetMonad_faithful (C : category) : faithful (forgetfunctor_Monad C).
Proof.
apply faithful_pr1_category.
intros T T' α.
apply isaprop_disp_Monad_Mor_laws.
Qed.
Definition of bind
Context {C : category} {T : Monad C}.
Definition bind {a b : C} (f : C⟦a,T b⟧) : C⟦T a,T b⟧ := # T f · μ T b.
Lemma η_bind {a b : C} (f : C⟦a,T b⟧) : η T a · bind f = f.
Proof.
unfold bind.
rewrite assoc.
eapply pathscomp0;
[apply cancel_postcomposition, (! (nat_trans_ax (η T) _ _ f))|]; simpl.
rewrite <- assoc.
eapply pathscomp0; [apply maponpaths, Monad_law1|].
apply id_right.
Qed.
Lemma bind_η {a : C} : bind (η T a) = identity (T a).
Proof.
apply Monad_law2.
Qed.
Lemma bind_bind {a b c : C} (f : C⟦a,T b⟧) (g : C⟦b,T c⟧) :
bind f · bind g = bind (f · bind g).
Proof.
unfold bind; rewrite <- assoc.
eapply pathscomp0; [apply maponpaths; rewrite assoc;
apply cancel_postcomposition, (!nat_trans_ax (μ T) _ _ g)|].
rewrite !functor_comp, <- !assoc.
now apply maponpaths, maponpaths, (!Monad_law3 c).
Qed.
End bind.
Section MonadsUsingCoproducts.
Context {C : category} (T : Monad C) (BC : BinCoproducts C).
Local Notation "a ⊕ b" := (BinCoproductObject (BC a b)).
Context {C : category} (T : Monad C) (BC : BinCoproducts C).
Local Notation "a ⊕ b" := (BinCoproductObject (BC a b)).
operation of weakening in a monad
operation of exchange in a monad
Definition mexch (a b c : C): C⟦T (a ⊕ (b ⊕ c)), T (b ⊕ (a ⊕ c))⟧.
Proof.
set (a1 := BinCoproductIn1 (BC _ _) · BinCoproductIn2 (BC _ _): C⟦a, b ⊕ (a ⊕ c)⟧).
set (a21 := BinCoproductIn1 (BC _ _): C⟦b, b ⊕ (a ⊕ c)⟧).
set (a22 := BinCoproductIn2 (BC _ _) · BinCoproductIn2 (BC _ _): C⟦c, b ⊕ (a ⊕ c)⟧).
exact (bind ((BinCoproductArrow _ a1 (BinCoproductArrow _ a21 a22)) · (η T _))).
Defined.
Proof.
set (a1 := BinCoproductIn1 (BC _ _) · BinCoproductIn2 (BC _ _): C⟦a, b ⊕ (a ⊕ c)⟧).
set (a21 := BinCoproductIn1 (BC _ _): C⟦b, b ⊕ (a ⊕ c)⟧).
set (a22 := BinCoproductIn2 (BC _ _) · BinCoproductIn2 (BC _ _): C⟦c, b ⊕ (a ⊕ c)⟧).
exact (bind ((BinCoproductArrow _ a1 (BinCoproductArrow _ a21 a22)) · (η T _))).
Defined.
Section MonadSubst.
Definition monadSubstGen {b:C} (a : C) (e : C⟦b,T a⟧) : C⟦T (b ⊕ a), T a⟧ :=
bind (BinCoproductArrow _ e (η T a)).
Lemma subst_interchange_law_gen (c b a : C) (e : C⟦c,T (b ⊕ a)⟧) (f : C⟦b,T a⟧):
(monadSubstGen _ e) · (monadSubstGen _ f) =
(mexch c b a) · (monadSubstGen _ (f · (mweak c a)))
· (monadSubstGen _ (e · (monadSubstGen _ f))).
Proof.
unfold monadSubstGen, mexch.
do 3 rewrite bind_bind.
apply maponpaths.
apply BinCoproductArrowsEq.
+ do 4 rewrite assoc.
do 2 rewrite BinCoproductIn1Commutes.
rewrite <- assoc.
rewrite bind_bind.
rewrite <- assoc.
rewrite (η_bind(a:=let (pr1, _) := pr1 (BC b (c ⊕ a)) in pr1)).
rewrite <- assoc.
apply pathsinv0.
eapply pathscomp0.
* apply cancel_precomposition.
rewrite assoc.
rewrite BinCoproductIn2Commutes.
rewrite (η_bind(a:=(c ⊕ a))).
apply idpath.
* now rewrite BinCoproductIn1Commutes.
+ rewrite assoc.
rewrite BinCoproductIn2Commutes.
rewrite (η_bind(a:=b ⊕ a)).
do 3 rewrite assoc.
rewrite BinCoproductIn2Commutes.
apply BinCoproductArrowsEq.
* rewrite BinCoproductIn1Commutes.
rewrite <- assoc.
rewrite bind_bind.
do 2 rewrite assoc.
rewrite BinCoproductIn1Commutes.
rewrite <- assoc.
rewrite (η_bind(a:=let (pr1, _) := pr1 (BC b (c ⊕ a)) in pr1)).
rewrite assoc.
rewrite BinCoproductIn1Commutes.
unfold mweak.
rewrite <- assoc.
rewrite bind_bind.
apply pathsinv0.
apply remove_id_right; try now idtac.
rewrite <- bind_η.
apply maponpaths.
rewrite <- assoc.
rewrite (η_bind(a:=let (pr1, _) := pr1 (BC c a) in pr1)).
now rewrite BinCoproductIn2Commutes.
* rewrite BinCoproductIn2Commutes.
rewrite <- assoc.
rewrite bind_bind.
do 2 rewrite assoc.
rewrite BinCoproductIn2Commutes.
do 2 rewrite <- assoc.
rewrite (η_bind(a:=let (pr1, _) := pr1 (BC b (c ⊕ a)) in pr1)).
apply pathsinv0.
eapply pathscomp0.
- apply cancel_precomposition.
rewrite assoc.
rewrite BinCoproductIn2Commutes.
rewrite (η_bind(a:=(c ⊕ a))).
apply idpath.
- now rewrite BinCoproductIn2Commutes.
Qed.
Context (TC : Terminal C).
Local Notation "1" := TC.
Definition monadSubst (a : C) (e : C⟦1,T a⟧) : C⟦T (1 ⊕ a), T a⟧ :=
monadSubstGen a e.
Lemma subst_interchange_law (a : C) (e : C⟦1,T (1 ⊕ a)⟧) (f : C⟦1,T a⟧):
(monadSubst _ e) · (monadSubst _ f) =
(mexch 1 1 a) · (monadSubst _ (f · (mweak 1 a)))
· (monadSubst _ (e · (monadSubst _ f))).
Proof.
apply subst_interchange_law_gen.
Qed.
End MonadSubst.
End MonadsUsingCoproducts.
Definition monadSubstGen {b:C} (a : C) (e : C⟦b,T a⟧) : C⟦T (b ⊕ a), T a⟧ :=
bind (BinCoproductArrow _ e (η T a)).
Lemma subst_interchange_law_gen (c b a : C) (e : C⟦c,T (b ⊕ a)⟧) (f : C⟦b,T a⟧):
(monadSubstGen _ e) · (monadSubstGen _ f) =
(mexch c b a) · (monadSubstGen _ (f · (mweak c a)))
· (monadSubstGen _ (e · (monadSubstGen _ f))).
Proof.
unfold monadSubstGen, mexch.
do 3 rewrite bind_bind.
apply maponpaths.
apply BinCoproductArrowsEq.
+ do 4 rewrite assoc.
do 2 rewrite BinCoproductIn1Commutes.
rewrite <- assoc.
rewrite bind_bind.
rewrite <- assoc.
rewrite (η_bind(a:=let (pr1, _) := pr1 (BC b (c ⊕ a)) in pr1)).
rewrite <- assoc.
apply pathsinv0.
eapply pathscomp0.
* apply cancel_precomposition.
rewrite assoc.
rewrite BinCoproductIn2Commutes.
rewrite (η_bind(a:=(c ⊕ a))).
apply idpath.
* now rewrite BinCoproductIn1Commutes.
+ rewrite assoc.
rewrite BinCoproductIn2Commutes.
rewrite (η_bind(a:=b ⊕ a)).
do 3 rewrite assoc.
rewrite BinCoproductIn2Commutes.
apply BinCoproductArrowsEq.
* rewrite BinCoproductIn1Commutes.
rewrite <- assoc.
rewrite bind_bind.
do 2 rewrite assoc.
rewrite BinCoproductIn1Commutes.
rewrite <- assoc.
rewrite (η_bind(a:=let (pr1, _) := pr1 (BC b (c ⊕ a)) in pr1)).
rewrite assoc.
rewrite BinCoproductIn1Commutes.
unfold mweak.
rewrite <- assoc.
rewrite bind_bind.
apply pathsinv0.
apply remove_id_right; try now idtac.
rewrite <- bind_η.
apply maponpaths.
rewrite <- assoc.
rewrite (η_bind(a:=let (pr1, _) := pr1 (BC c a) in pr1)).
now rewrite BinCoproductIn2Commutes.
* rewrite BinCoproductIn2Commutes.
rewrite <- assoc.
rewrite bind_bind.
do 2 rewrite assoc.
rewrite BinCoproductIn2Commutes.
do 2 rewrite <- assoc.
rewrite (η_bind(a:=let (pr1, _) := pr1 (BC b (c ⊕ a)) in pr1)).
apply pathsinv0.
eapply pathscomp0.
- apply cancel_precomposition.
rewrite assoc.
rewrite BinCoproductIn2Commutes.
rewrite (η_bind(a:=(c ⊕ a))).
apply idpath.
- now rewrite BinCoproductIn2Commutes.
Qed.
Context (TC : Terminal C).
Local Notation "1" := TC.
Definition monadSubst (a : C) (e : C⟦1,T a⟧) : C⟦T (1 ⊕ a), T a⟧ :=
monadSubstGen a e.
Lemma subst_interchange_law (a : C) (e : C⟦1,T (1 ⊕ a)⟧) (f : C⟦1,T a⟧):
(monadSubst _ e) · (monadSubst _ f) =
(mexch 1 1 a) · (monadSubst _ (f · (mweak 1 a)))
· (monadSubst _ (e · (monadSubst _ f))).
Proof.
apply subst_interchange_law_gen.
Qed.
End MonadSubst.
End MonadsUsingCoproducts.
Section Monad'_def.
Definition raw_Monad_data (C : category) : UU :=
∑ F : C -> C, (((∏ a b : ob C, a --> b -> F a --> F b) ×
(∏ a : ob C, F (F a) --> F a)) ×
(∏ a : ob C, a --> F a)).
Coercion functor_data_from_raw_Monad_data {C : category} (T : raw_Monad_data C) :
functor_data C C := make_functor_data (pr1 T) (pr1 (pr1 (pr2 T))).
Definition Monad'_data_laws {C : category} (T : raw_Monad_data C) :=
((is_functor T) ×
(is_nat_trans (functor_composite_data T T) T (pr2 (pr1 (pr2 T))))) ×
(is_nat_trans (functor_identity C) T (pr2 (pr2 T))).
Definition Monad'_data (C : category) := ∑ (T : raw_Monad_data C), Monad'_data_laws T.
Definition Monad'_data_to_Monad_data {C : category} (T : Monad'_data C) : disp_Monad_data (_,, pr1 (pr1 (pr2 T))) :=
((pr2 (pr1 (pr2 (pr1 T))),, (pr2 (pr1 (pr2 T))))),,
(pr2 (pr2 (pr1 T)),, (pr2 (pr2 T))).
Definition Monad' (C : category) := ∑ (T : Monad'_data C),
(disp_Monad_laws (Monad'_data_to_Monad_data T)).
End Monad'_def.
Definition raw_Monad_data (C : category) : UU :=
∑ F : C -> C, (((∏ a b : ob C, a --> b -> F a --> F b) ×
(∏ a : ob C, F (F a) --> F a)) ×
(∏ a : ob C, a --> F a)).
Coercion functor_data_from_raw_Monad_data {C : category} (T : raw_Monad_data C) :
functor_data C C := make_functor_data (pr1 T) (pr1 (pr1 (pr2 T))).
Definition Monad'_data_laws {C : category} (T : raw_Monad_data C) :=
((is_functor T) ×
(is_nat_trans (functor_composite_data T T) T (pr2 (pr1 (pr2 T))))) ×
(is_nat_trans (functor_identity C) T (pr2 (pr2 T))).
Definition Monad'_data (C : category) := ∑ (T : raw_Monad_data C), Monad'_data_laws T.
Definition Monad'_data_to_Monad_data {C : category} (T : Monad'_data C) : disp_Monad_data (_,, pr1 (pr1 (pr2 T))) :=
((pr2 (pr1 (pr2 (pr1 T))),, (pr2 (pr1 (pr2 T))))),,
(pr2 (pr2 (pr1 T)),, (pr2 (pr2 T))).
Definition Monad' (C : category) := ∑ (T : Monad'_data C),
(disp_Monad_laws (Monad'_data_to_Monad_data T)).
End Monad'_def.
Section Monad_Monad'_equiv.
Definition Monad'_to_Monad {C : category} (T : Monad' C) : Monad C :=
(_,,(Monad'_data_to_Monad_data (pr1 T),, pr2 T)).
Definition Monad_to_raw_data {C : category} (T : Monad C) : raw_Monad_data C.
Proof.
use tpair.
- exact (functor_on_objects T).
- use tpair.
+ use tpair.
* exact (@functor_on_morphisms C C T).
* exact (μ T).
+ exact (η T).
Defined.
Definition Monad_to_Monad'_data {C : category} (T : Monad C) : Monad'_data C :=
(Monad_to_raw_data T,, ((pr2 (T : functor C C),, (pr2 (μ T))),, pr2 (η T))).
Definition Monad_to_Monad' {C : category} (T : Monad C) : Monad' C :=
(Monad_to_Monad'_data T,, pr22 T).
Definition Monad'_to_Monad_to_Monad' {C : category} (T : Monad' C) :
Monad_to_Monad' (Monad'_to_Monad T) = T := (idpath T).
Definition Monad_to_Monad'_to_Monad {C : category} (T : Monad C) :
Monad'_to_Monad (Monad_to_Monad' T) = T := (idpath T).
End Monad_Monad'_equiv.
Lemma Monad'_eq_raw_data (C : category) (T T' : Monad' C) :
pr1 (pr1 T) = pr1 (pr1 T') -> T = T'.
Proof.
intro e.
apply subtypePath.
- intro. now apply isaprop_disp_Monad_laws.
- apply subtypePath.
+ intro. apply isapropdirprod.
* apply isapropdirprod.
-- apply (isaprop_is_functor C C), homset_property.
-- apply (isaprop_is_nat_trans C C), homset_property.
* apply (isaprop_is_nat_trans C C), homset_property.
+ apply e.
Qed.
Lemma Monad_eq_raw_data (C : category) (T T' : Monad C) :
Monad_to_raw_data T = Monad_to_raw_data T' -> T = T'.
Proof.
intro e.
apply (invmaponpathsweq (_,, (isweq_iso _ _ (@Monad_to_Monad'_to_Monad C)
(@Monad'_to_Monad_to_Monad' C)))).
now apply (Monad'_eq_raw_data C).
Qed.
End Monad_eq_helper.
Section Monads_from_adjunctions.
Definition functor_from_adjunction {C D : category}
{L : functor C D} {R : functor D C} (H : are_adjoints L R) : functor C C := L∙R.
Definition Monad_data_from_adjunction {C D : category} {L : functor C D}
{R : functor D C} (H : are_adjoints L R) : disp_Monad_data (functor_from_adjunction H).
Proof.
use tpair.
- exact (pre_whisker L (post_whisker (adjcounit H) R)).
- exact (adjunit H).
Defined.
Lemma Monad_laws_from_adjunction {C D : category} {L : functor C D}
{R : functor D C} (H : are_adjoints L R) : disp_Monad_laws (Monad_data_from_adjunction H).
Proof.
cbn.
use make_dirprod.
+ use make_dirprod.
* intro c; cbn.
apply triangle_id_right_ad.
* intro c; cbn.
rewrite <- functor_id.
rewrite <- functor_comp.
apply maponpaths.
apply triangle_id_left_ad.
+ intro c; cbn.
do 2 (rewrite <- functor_comp).
apply maponpaths.
apply (nat_trans_ax ((counit_from_are_adjoints H))).
Qed.
Definition Monad_from_adjunction {C D : category} {L : functor C D}
{R : functor D C} (H : are_adjoints L R) : Monad C.
Proof.
exists (functor_from_adjunction H).
exact (Monad_data_from_adjunction H,, Monad_laws_from_adjunction H).
Defined.
End Monads_from_adjunctions.
Definition Monad'_to_Monad {C : category} (T : Monad' C) : Monad C :=
(_,,(Monad'_data_to_Monad_data (pr1 T),, pr2 T)).
Definition Monad_to_raw_data {C : category} (T : Monad C) : raw_Monad_data C.
Proof.
use tpair.
- exact (functor_on_objects T).
- use tpair.
+ use tpair.
* exact (@functor_on_morphisms C C T).
* exact (μ T).
+ exact (η T).
Defined.
Definition Monad_to_Monad'_data {C : category} (T : Monad C) : Monad'_data C :=
(Monad_to_raw_data T,, ((pr2 (T : functor C C),, (pr2 (μ T))),, pr2 (η T))).
Definition Monad_to_Monad' {C : category} (T : Monad C) : Monad' C :=
(Monad_to_Monad'_data T,, pr22 T).
Definition Monad'_to_Monad_to_Monad' {C : category} (T : Monad' C) :
Monad_to_Monad' (Monad'_to_Monad T) = T := (idpath T).
Definition Monad_to_Monad'_to_Monad {C : category} (T : Monad C) :
Monad'_to_Monad (Monad_to_Monad' T) = T := (idpath T).
End Monad_Monad'_equiv.
Lemma Monad'_eq_raw_data (C : category) (T T' : Monad' C) :
pr1 (pr1 T) = pr1 (pr1 T') -> T = T'.
Proof.
intro e.
apply subtypePath.
- intro. now apply isaprop_disp_Monad_laws.
- apply subtypePath.
+ intro. apply isapropdirprod.
* apply isapropdirprod.
-- apply (isaprop_is_functor C C), homset_property.
-- apply (isaprop_is_nat_trans C C), homset_property.
* apply (isaprop_is_nat_trans C C), homset_property.
+ apply e.
Qed.
Lemma Monad_eq_raw_data (C : category) (T T' : Monad C) :
Monad_to_raw_data T = Monad_to_raw_data T' -> T = T'.
Proof.
intro e.
apply (invmaponpathsweq (_,, (isweq_iso _ _ (@Monad_to_Monad'_to_Monad C)
(@Monad'_to_Monad_to_Monad' C)))).
now apply (Monad'_eq_raw_data C).
Qed.
End Monad_eq_helper.
Section Monads_from_adjunctions.
Definition functor_from_adjunction {C D : category}
{L : functor C D} {R : functor D C} (H : are_adjoints L R) : functor C C := L∙R.
Definition Monad_data_from_adjunction {C D : category} {L : functor C D}
{R : functor D C} (H : are_adjoints L R) : disp_Monad_data (functor_from_adjunction H).
Proof.
use tpair.
- exact (pre_whisker L (post_whisker (adjcounit H) R)).
- exact (adjunit H).
Defined.
Lemma Monad_laws_from_adjunction {C D : category} {L : functor C D}
{R : functor D C} (H : are_adjoints L R) : disp_Monad_laws (Monad_data_from_adjunction H).
Proof.
cbn.
use make_dirprod.
+ use make_dirprod.
* intro c; cbn.
apply triangle_id_right_ad.
* intro c; cbn.
rewrite <- functor_id.
rewrite <- functor_comp.
apply maponpaths.
apply triangle_id_left_ad.
+ intro c; cbn.
do 2 (rewrite <- functor_comp).
apply maponpaths.
apply (nat_trans_ax ((counit_from_are_adjoints H))).
Qed.
Definition Monad_from_adjunction {C D : category} {L : functor C D}
{R : functor D C} (H : are_adjoints L R) : Monad C.
Proof.
exists (functor_from_adjunction H).
exact (Monad_data_from_adjunction H,, Monad_laws_from_adjunction H).
Defined.
End Monads_from_adjunctions.