Library UniMath.CategoryTheory.Monads.RelativeMonads
**********************************************************
Contents:
Reference:
Written by: Benedikt Ahrens (started May 2017)
Extended by: Ralph Matthes (starting August 2018), in particular all on univalence
- Definition of relative monads RelMonad
- Functoriality for relative monads r_lift
- Kleisli category associated to a relative monad Kleisli_precat Kleisli_cat, canonical relative adjunction rKleisli_functors_are_relative_adjoints
- Category of relative monads is univalent if the target category is is_univalent_RelMonad
Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Propositions.
Require Import UniMath.Foundations.Sets.
Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.MoreFoundations.PartA.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.NaturalTransformations.
Require Import UniMath.CategoryTheory.Core.Univalence.
Require Import UniMath.CategoryTheory.FunctorCategory.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.Adjunctions.Core.
Local Open Scope cat.
Section RMonad_def.
Context {C D : precategory_data} {J : functor_data C D}.
Definition RelMonad_data : UU
:= ∑ F : C -> D, (∏ c, D ⟦J c, F c⟧)
× (∏ c d, D ⟦J c, F d⟧ → D ⟦F c, F d⟧).
Definition RelMonad_ob (R : RelMonad_data) (c : C) : D := pr1 R c.
Coercion RelMonad_ob : RelMonad_data >-> Funclass.
Definition r_eta (R : RelMonad_data) c : D ⟦J c, R c⟧ := pr1 (pr2 R) c.
Definition r_bind (R : RelMonad_data) {c d} (f : D⟦J c, R d⟧) : D ⟦R c, R d⟧
:= pr2 (pr2 R) _ _ f.
Definition RelMonad_axioms (R : RelMonad_data) : UU
:= (∏ c, r_bind R (r_eta R c) = identity _ )
× (∏ c d (f : D⟦J c, R d⟧), r_eta R _ · r_bind R f = f)
× (∏ c d e (f : D ⟦J c, R d⟧) (g : D ⟦J d, R e⟧),
r_bind R f · r_bind R g = r_bind R (f · r_bind R g)).
Lemma isaprop_RelMonad_axioms (R : RelMonad_data)(hs : has_homsets D) :
isaprop (RelMonad_axioms R).
Show proof.
Definition r_bind_r_eta {R : RelMonad_data} (X : RelMonad_axioms R)
: ∏ c, r_bind R (r_eta R c) = identity _ := pr1 X.
Definition r_eta_r_bind {R : RelMonad_data} (X : RelMonad_axioms R)
: ∏ c d (f : D⟦J c, R d⟧), r_eta R _ · r_bind R f = f := pr1 (pr2 X).
Definition r_bind_r_bind {R : RelMonad_data} (X : RelMonad_axioms R)
: ∏ c d e (f : D ⟦J c, R d⟧) (g : D ⟦J d, R e⟧),
r_bind R f · r_bind R g = r_bind R (f · r_bind R g)
:= pr2 (pr2 X).
Definition RelMonad : UU := ∑ R : RelMonad_data, RelMonad_axioms R.
Coercion RelMonad_data_from_RelMonad (R : RelMonad) : RelMonad_data := pr1 R.
Coercion RelMonad_axioms_from_RelMonad (R : RelMonad) : RelMonad_axioms R := pr2 R.
Lemma RelMonad_eq (R R' : RelMonad)(hs : has_homsets D) :
pr1 R = pr1 R' -> R = R'.
Show proof.
Context {C D : precategory_data} {J : functor_data C D}.
Definition RelMonad_data : UU
:= ∑ F : C -> D, (∏ c, D ⟦J c, F c⟧)
× (∏ c d, D ⟦J c, F d⟧ → D ⟦F c, F d⟧).
Definition RelMonad_ob (R : RelMonad_data) (c : C) : D := pr1 R c.
Coercion RelMonad_ob : RelMonad_data >-> Funclass.
Definition r_eta (R : RelMonad_data) c : D ⟦J c, R c⟧ := pr1 (pr2 R) c.
Definition r_bind (R : RelMonad_data) {c d} (f : D⟦J c, R d⟧) : D ⟦R c, R d⟧
:= pr2 (pr2 R) _ _ f.
Definition RelMonad_axioms (R : RelMonad_data) : UU
:= (∏ c, r_bind R (r_eta R c) = identity _ )
× (∏ c d (f : D⟦J c, R d⟧), r_eta R _ · r_bind R f = f)
× (∏ c d e (f : D ⟦J c, R d⟧) (g : D ⟦J d, R e⟧),
r_bind R f · r_bind R g = r_bind R (f · r_bind R g)).
Lemma isaprop_RelMonad_axioms (R : RelMonad_data)(hs : has_homsets D) :
isaprop (RelMonad_axioms R).
Show proof.
apply isapropdirprod.
- apply impred_isaprop; intros. apply hs.
- apply isapropdirprod; repeat (apply impred_isaprop; intros); apply hs.
- apply impred_isaprop; intros. apply hs.
- apply isapropdirprod; repeat (apply impred_isaprop; intros); apply hs.
Definition r_bind_r_eta {R : RelMonad_data} (X : RelMonad_axioms R)
: ∏ c, r_bind R (r_eta R c) = identity _ := pr1 X.
Definition r_eta_r_bind {R : RelMonad_data} (X : RelMonad_axioms R)
: ∏ c d (f : D⟦J c, R d⟧), r_eta R _ · r_bind R f = f := pr1 (pr2 X).
Definition r_bind_r_bind {R : RelMonad_data} (X : RelMonad_axioms R)
: ∏ c d e (f : D ⟦J c, R d⟧) (g : D ⟦J d, R e⟧),
r_bind R f · r_bind R g = r_bind R (f · r_bind R g)
:= pr2 (pr2 X).
Definition RelMonad : UU := ∑ R : RelMonad_data, RelMonad_axioms R.
Coercion RelMonad_data_from_RelMonad (R : RelMonad) : RelMonad_data := pr1 R.
Coercion RelMonad_axioms_from_RelMonad (R : RelMonad) : RelMonad_axioms R := pr2 R.
Lemma RelMonad_eq (R R' : RelMonad)(hs : has_homsets D) :
pr1 R = pr1 R' -> R = R'.
Show proof.
intro p.
apply (total2_paths_f p).
apply proofirrelevance.
apply isaprop_RelMonad_axioms.
exact hs.
apply (total2_paths_f p).
apply proofirrelevance.
apply isaprop_RelMonad_axioms.
exact hs.
previous proof
Lemma RelMonad_eq_obsolete (R R' : RelMonad)(hs : has_homsets D) :
pr1 R = pr1 R' -> R = R'.
Show proof.
pr1 R = pr1 R' -> R = R'.
Show proof.
intro p.
apply subtypeInjectivity.
- intro R''.
apply isaprop_RelMonad_axioms.
exact hs.
- assumption.
apply subtypeInjectivity.
- intro R''.
apply isaprop_RelMonad_axioms.
exact hs.
- assumption.
generalize UniMath.CategoryTheory.Monads.KTriples.map
Definition r_lift (R : RelMonad_data) {c d : C} (f : c --> d) : R c --> R d
:= r_bind R (#J f · r_eta R _ ).
End RMonad_def.
:= r_bind R (#J f · r_eta R _ ).
End RMonad_def.
generalize UniMath.CategoryTheory.Monads.KTriples.map_id and
UniMath.CategoryTheory.Monads.KTriples.map_map
Definition is_functor_r_lift {C: precategory_data} {D: precategory} {J : functor C D} (R : RelMonad)
: is_functor (RelMonad_ob R,, @r_lift _ _ J R).
Show proof.
Definition R_functor {C: precategory_data} {D: precategory} {J : functor C D}
(R : RelMonad): functor C D
:= _,, is_functor_r_lift(J:=J) R.
: is_functor (RelMonad_ob R,, @r_lift _ _ J R).
Show proof.
split; [intro c | intros a b c f g]; cbn; unfold r_lift; cbn.
- etrans. apply maponpaths.
etrans. eapply (maponpaths (λ x, x · _ )). apply functor_id.
apply id_left.
apply (r_bind_r_eta R).
- etrans. 2: { eapply pathsinv0; apply (r_bind_r_bind R). }
apply maponpaths.
etrans.
apply map_on_two_paths. apply functor_comp. apply idpath.
etrans.
2: { etrans. 2: apply assoc.
eapply pathsinv0. apply maponpaths. apply (r_eta_r_bind R).
}
apply pathsinv0, assoc.
- etrans. apply maponpaths.
etrans. eapply (maponpaths (λ x, x · _ )). apply functor_id.
apply id_left.
apply (r_bind_r_eta R).
- etrans. 2: { eapply pathsinv0; apply (r_bind_r_bind R). }
apply maponpaths.
etrans.
apply map_on_two_paths. apply functor_comp. apply idpath.
etrans.
2: { etrans. 2: apply assoc.
eapply pathsinv0. apply maponpaths. apply (r_eta_r_bind R).
}
apply pathsinv0, assoc.
Definition R_functor {C: precategory_data} {D: precategory} {J : functor C D}
(R : RelMonad): functor C D
:= _,, is_functor_r_lift(J:=J) R.
generalize UniMath.CategoryTheory.Monads.KTriples.is_nat_trans_η
Definition is_nat_trans_r_eta {C: precategory_data} {D: precategory} {J : functor C D}(R : RelMonad)
: is_nat_trans J (R_functor R) (r_eta R).
Show proof.
Definition r_eta_nat_trans {C: precategory_data} {D: precategory} {J : functor C D}(R : RelMonad)
: nat_trans J (R_functor R) := _ ,, is_nat_trans_r_eta R.
: is_nat_trans J (R_functor R) (r_eta R).
Show proof.
red.
intros c c' f.
unfold R_functor; simpl.
unfold r_lift; simpl.
apply pathsinv0.
apply (r_eta_r_bind R).
intros c c' f.
unfold R_functor; simpl.
unfold r_lift; simpl.
apply pathsinv0.
apply (r_eta_r_bind R).
Definition r_eta_nat_trans {C: precategory_data} {D: precategory} {J : functor C D}(R : RelMonad)
: nat_trans J (R_functor R) := _ ,, is_nat_trans_r_eta R.
generalize UniMath.CategoryTheory.Monads.KTriples.map_bind
Definition r_lift_r_bind {C: precategory_data} {D: precategory} {J : functor C D}(R : RelMonad)(a b c : C) (f : b --> c) (g : J a --> R b)
: r_bind R g · r_lift R f = r_bind R (g · r_lift R f).
Show proof.
: r_bind R g · r_lift R f = r_bind R (g · r_lift R f).
Show proof.
unfold r_lift.
rewrite <- (r_bind_r_bind R).
apply idpath.
rewrite <- (r_bind_r_bind R).
apply idpath.
generalize UniMath.CategoryTheory.Monads.KTriples.bind_map
Definition r_bind_r_lift {C: precategory_data} {D: precategory} {J : functor C D}(R : RelMonad)(a b c : C) (f : J b --> R c) (g : a --> b)
: r_lift R g · r_bind R f = r_bind R (#J g · f).
Show proof.
Arguments RelMonad_data {C} {D} J.
Arguments RelMonad {C} {D} J.
: r_lift R g · r_bind R f = r_bind R (#J g · f).
Show proof.
unfold r_lift.
rewrite (r_bind_r_bind R).
apply maponpaths.
rewrite <- assoc.
apply cancel_precomposition.
apply (r_eta_r_bind R).
rewrite (r_bind_r_bind R).
apply maponpaths.
rewrite <- assoc.
apply cancel_precomposition.
apply (r_eta_r_bind R).
Arguments RelMonad_data {C} {D} J.
Arguments RelMonad {C} {D} J.
analogue of UniMath.CategoryTheory.functor_categories.functor_eq_eq_from_functor_ob_eq
Definition relmonad_eq_eq_from_relmonad_ob_eq {C: precategory_data} {D: precategory} (hs: has_homsets D)
{J : functor C D} (R R' : RelMonad J) (p q : R = R')
(H : base_paths _ _ (base_paths _ _ p) =
base_paths _ _ (base_paths _ _ q)) :
p = q.
Show proof.
{J : functor C D} (R R' : RelMonad J) (p q : R = R')
(H : base_paths _ _ (base_paths _ _ p) =
base_paths _ _ (base_paths _ _ q)) :
p = q.
Show proof.
apply (invmaponpathsweq (total2_paths_equiv _ _ _ )); simpl.
assert (H' : base_paths _ _ p = base_paths _ _ q).
{ apply (invmaponpathsweq (total2_paths_equiv _ _ _ )); simpl.
apply (two_arg_paths_f H), uip.
apply isaset_dirprod.
- apply impred_isaset; intro c. apply hs.
- apply impred_isaset; intro c; apply impred_isaset; intro d; apply impred_isaset; intro f.
apply hs.
}
apply (two_arg_paths_f H'), uip, isasetaprop, isaprop_RelMonad_axioms, hs.
assert (H' : base_paths _ _ p = base_paths _ _ q).
{ apply (invmaponpathsweq (total2_paths_equiv _ _ _ )); simpl.
apply (two_arg_paths_f H), uip.
apply isaset_dirprod.
- apply impred_isaset; intro c. apply hs.
- apply impred_isaset; intro c; apply impred_isaset; intro d; apply impred_isaset; intro f.
apply hs.
}
apply (two_arg_paths_f H'), uip, isasetaprop, isaprop_RelMonad_axioms, hs.
Kleisli precategory associated to a relative monad
Section Kleisli_precat.
Context {C: precategory_data} {D : precategory} {J : functor_data C D}.
Definition Kleisli_precat_ob_mor (R : RelMonad_data J) : precategory_ob_mor :=
precategory_ob_mor_pair (ob C) (λ c d, J c --> R d).
Definition Kleisli_precat_data (R : RelMonad_data J) : precategory_data :=
precategory_data_pair (Kleisli_precat_ob_mor R) (λ c, r_eta R c)
(λ a b c f g, f · r_bind R g).
Lemma Kleisli_precat_is_precat (R : RelMonad J) : is_precategory (Kleisli_precat_data R).
Show proof.
Definition Kleisli_precat (R : RelMonad J) : precategory := (_,, Kleisli_precat_is_precat R).
End Kleisli_precat.
Context {C: precategory_data} {D : precategory} {J : functor_data C D}.
Definition Kleisli_precat_ob_mor (R : RelMonad_data J) : precategory_ob_mor :=
precategory_ob_mor_pair (ob C) (λ c d, J c --> R d).
Definition Kleisli_precat_data (R : RelMonad_data J) : precategory_data :=
precategory_data_pair (Kleisli_precat_ob_mor R) (λ c, r_eta R c)
(λ a b c f g, f · r_bind R g).
Lemma Kleisli_precat_is_precat (R : RelMonad J) : is_precategory (Kleisli_precat_data R).
Show proof.
apply is_precategory_one_assoc_to_two.
do 2 try apply tpair;
try unfold compose; simpl.
- intros a b f.
apply (r_eta_r_bind R).
- intros a b f.
now rewrite (r_bind_r_eta R), id_right.
- intros a b c d f g h.
now rewrite <- assoc, (r_bind_r_bind R).
do 2 try apply tpair;
try unfold compose; simpl.
- intros a b f.
apply (r_eta_r_bind R).
- intros a b f.
now rewrite (r_bind_r_eta R), id_right.
- intros a b c d f g h.
now rewrite <- assoc, (r_bind_r_bind R).
Definition Kleisli_precat (R : RelMonad J) : precategory := (_,, Kleisli_precat_is_precat R).
End Kleisli_precat.
Kleisli category associated to a relative monad
Section Kleisli_cat.
Lemma Kleisli_precat_has_homsets {C : precategory_data} {D : category} {J : functor_data C D} (R : RelMonad J)
(hs : has_homsets D) : has_homsets (Kleisli_precat_data R).
Show proof.
Definition Kleisli_cat {C : precategory_data} {D : category} {J : functor_data C D} (R : RelMonad J)
: category := (Kleisli_precat R,, Kleisli_precat_has_homsets R (homset_property D)).
End Kleisli_cat.
Section MorphismsOfRelativeMonads.
Definition RelMonadMor_data {C D : precategory_data}
{J : functor_data C D} (R R' : RelMonad_data J): UU :=
∏ a : C, R a --> R' a.
Definition RelMonadMor_axioms {C D : precategory_data}
{J : functor_data C D} {R R' : RelMonad_data J} (α : RelMonadMor_data R R') : UU
:= (∏ a : C, r_eta R a · α a = r_eta R' a) ×
(∏ (a b : C) (f : D⟦J a,R b⟧), α a · r_bind R' (f · α b) = (r_bind R f)· α b).
Lemma isaprop_RelMonadMor_axioms {C D : precategory_data}
{J : functor_data C D} {R R' : RelMonad_data J} (α : RelMonadMor_data R R')
(hs : has_homsets D) :
isaprop (RelMonadMor_axioms α).
Show proof.
Lemma Kleisli_precat_has_homsets {C : precategory_data} {D : category} {J : functor_data C D} (R : RelMonad J)
(hs : has_homsets D) : has_homsets (Kleisli_precat_data R).
Show proof.
intros a b.
apply hs.
apply hs.
Definition Kleisli_cat {C : precategory_data} {D : category} {J : functor_data C D} (R : RelMonad J)
: category := (Kleisli_precat R,, Kleisli_precat_has_homsets R (homset_property D)).
End Kleisli_cat.
Section MorphismsOfRelativeMonads.
Definition RelMonadMor_data {C D : precategory_data}
{J : functor_data C D} (R R' : RelMonad_data J): UU :=
∏ a : C, R a --> R' a.
Definition RelMonadMor_axioms {C D : precategory_data}
{J : functor_data C D} {R R' : RelMonad_data J} (α : RelMonadMor_data R R') : UU
:= (∏ a : C, r_eta R a · α a = r_eta R' a) ×
(∏ (a b : C) (f : D⟦J a,R b⟧), α a · r_bind R' (f · α b) = (r_bind R f)· α b).
Lemma isaprop_RelMonadMor_axioms {C D : precategory_data}
{J : functor_data C D} {R R' : RelMonad_data J} (α : RelMonadMor_data R R')
(hs : has_homsets D) :
isaprop (RelMonadMor_axioms α).
Show proof.
apply isapropdirprod; repeat (apply impred_isaprop; intros); apply hs.
generalize UniMath.CategoryTheory.Monads.KTriples.Kleisli_Mor_η
Definition r_eta_α {C D : precategory_data} {J : functor_data C D}
{R R' : RelMonad_data J} {α : RelMonadMor_data R R'}
(X : RelMonadMor_axioms α)
: ∏ a : C, r_eta R a · α a = r_eta R' a := pr1 X.
{R R' : RelMonad_data J} {α : RelMonadMor_data R R'}
(X : RelMonadMor_axioms α)
: ∏ a : C, r_eta R a · α a = r_eta R' a := pr1 X.
generalize UniMath.CategoryTheory.Monads.KTriples.Kleisli_Mor_bind
Definition α_r_bind {C D : precategory_data} {J : functor_data C D}
{R R' : RelMonad_data J} {α : RelMonadMor_data R R'}
(X : RelMonadMor_axioms α)
: ∏ (a b : C) (f : D⟦J a,R b⟧), α a · r_bind R' (f · α b) = (r_bind R f) · α b := pr2 X.
Definition RelMonadMor {C D : precategory_data} {J : functor_data C D}
(R R' : RelMonad_data J) : UU
:= ∑ α : RelMonadMor_data R R', RelMonadMor_axioms α.
Coercion RelMonadMor_data_from_RelMonadMor {C D : precategory_data} {J : functor_data C D}
{R R' : RelMonad_data J} (α : RelMonadMor R R') : RelMonadMor_data R R' := pr1 α.
Coercion RelMonadMor_axioms_from_RelMonadMor {C D : precategory_data} {J : functor_data C D}
{R R' : RelMonad_data J} (α : RelMonadMor R R') :
RelMonadMor_axioms α := pr2 α.
Definition RelMonadMor_map {C D : precategory_data} {J : functor_data C D}
{R R' : RelMonad_data J} (f : RelMonadMor R R') (X : C)
: R X --> R' X
:= (f : RelMonadMor_data _ _ ) X.
Definition RelMonadMor_equiv {C D : precategory_data} (hs : has_homsets D)
{J : functor_data C D} {R R' : RelMonad_data J} (α β : RelMonadMor R R') :
α = β ≃ ((α: RelMonadMor_data R R') = β).
Show proof.
{R R' : RelMonad_data J} {α : RelMonadMor_data R R'}
(X : RelMonadMor_axioms α)
: ∏ (a b : C) (f : D⟦J a,R b⟧), α a · r_bind R' (f · α b) = (r_bind R f) · α b := pr2 X.
Definition RelMonadMor {C D : precategory_data} {J : functor_data C D}
(R R' : RelMonad_data J) : UU
:= ∑ α : RelMonadMor_data R R', RelMonadMor_axioms α.
Coercion RelMonadMor_data_from_RelMonadMor {C D : precategory_data} {J : functor_data C D}
{R R' : RelMonad_data J} (α : RelMonadMor R R') : RelMonadMor_data R R' := pr1 α.
Coercion RelMonadMor_axioms_from_RelMonadMor {C D : precategory_data} {J : functor_data C D}
{R R' : RelMonad_data J} (α : RelMonadMor R R') :
RelMonadMor_axioms α := pr2 α.
Definition RelMonadMor_map {C D : precategory_data} {J : functor_data C D}
{R R' : RelMonad_data J} (f : RelMonadMor R R') (X : C)
: R X --> R' X
:= (f : RelMonadMor_data _ _ ) X.
Definition RelMonadMor_equiv {C D : precategory_data} (hs : has_homsets D)
{J : functor_data C D} {R R' : RelMonad_data J} (α β : RelMonadMor R R') :
α = β ≃ ((α: RelMonadMor_data R R') = β).
Show proof.
apply subtypeInjectivity.
intro a.
apply isaprop_RelMonadMor_axioms.
exact hs.
intro a.
apply isaprop_RelMonadMor_axioms.
exact hs.
generalize UniMath.CategoryTheory.Monads.KTriples.is_nat_trans_kleisli_mor
Definition is_nat_trans_RelMonadMor {C : precategory_data} {D: precategory} {J : functor C D}
{R R' : RelMonad J} (α : RelMonadMor R R'):
is_nat_trans (R_functor R) (R_functor R') (α:RelMonadMor_data R R').
Show proof.
Definition nat_trans_RelMonadMor {C : precategory_data} {D: precategory} {J : functor C D}
{R R' : RelMonad J} (α : RelMonadMor R R'):
nat_trans (R_functor R) (R_functor R') := _ ,, is_nat_trans_RelMonadMor α.
End MorphismsOfRelativeMonads.
Section PrecategoryOfRelativeMonads.
Lemma RelMonad_identity_laws {C : precategory_data} {D : precategory}
{J : functor_data C D} (R : RelMonad_data J):
RelMonadMor_axioms (λ a : C, identity (R a)).
Show proof.
Definition RelMonad_identity {C : precategory_data} {D : precategory}
{J : functor_data C D} (R : RelMonad_data J): RelMonadMor R R
:= _ ,, RelMonad_identity_laws R.
{R R' : RelMonad J} (α : RelMonadMor R R'):
is_nat_trans (R_functor R) (R_functor R') (α:RelMonadMor_data R R').
Show proof.
red.
intros c c' f.
unfold R_functor; simpl. unfold r_lift; simpl.
rewrite <- (α_r_bind α).
rewrite <- assoc.
now rewrite (r_eta_α α).
intros c c' f.
unfold R_functor; simpl. unfold r_lift; simpl.
rewrite <- (α_r_bind α).
rewrite <- assoc.
now rewrite (r_eta_α α).
Definition nat_trans_RelMonadMor {C : precategory_data} {D: precategory} {J : functor C D}
{R R' : RelMonad J} (α : RelMonadMor R R'):
nat_trans (R_functor R) (R_functor R') := _ ,, is_nat_trans_RelMonadMor α.
End MorphismsOfRelativeMonads.
Section PrecategoryOfRelativeMonads.
Lemma RelMonad_identity_laws {C : precategory_data} {D : precategory}
{J : functor_data C D} (R : RelMonad_data J):
RelMonadMor_axioms (λ a : C, identity (R a)).
Show proof.
split; simpl; intros c.
- apply id_right.
- intros. do 2 rewrite id_right; apply id_left.
- apply id_right.
- intros. do 2 rewrite id_right; apply id_left.
Definition RelMonad_identity {C : precategory_data} {D : precategory}
{J : functor_data C D} (R : RelMonad_data J): RelMonadMor R R
:= _ ,, RelMonad_identity_laws R.
generalize UniMath.CategoryTheory.Monads.KTriples.Kleisli_composition_laws
Lemma RelMonad_composition_laws {C : precategory_data} {D : precategory}
{J : functor_data C D} {R R' R'' : RelMonad_data J}
(α : RelMonadMor R R') (α' : RelMonadMor R' R''):
RelMonadMor_axioms (λ a : C, (α : RelMonadMor_data R R') a · (α' : RelMonadMor_data R' R'') a).
Show proof.
Definition RelMonad_composition {C : precategory_data} {D : precategory}
{J : functor_data C D} {R R' R'' : RelMonad_data J}
(α : RelMonadMor R R') (α' : RelMonadMor R' R''):
RelMonadMor R R'' := _ ,, RelMonad_composition_laws α α'.
Definition precategory_RelMonad_ob_mor {C : precategory_data} {D : precategory}
(J : functor_data C D) : precategory_ob_mor :=
precategory_ob_mor_pair (RelMonad J) RelMonadMor.
Definition precategory_RelMonad_data {C : precategory_data} {D : precategory}
(J : functor_data C D) : precategory_data.
Show proof.
Lemma precategory_RelMonad_axioms {C : precategory_data} {D : precategory}
(hs : has_homsets D) (J : functor_data C D) :
is_precategory (precategory_RelMonad_data J).
Show proof.
Definition precategory_RelMonad {C : precategory_data} {D : precategory}
(hs : has_homsets D) (J : functor_data C D): precategory :=
_ ,, precategory_RelMonad_axioms hs J.
Lemma has_homsets_RelMonad {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor_data C D) :
has_homsets (precategory_RelMonad hs J).
Show proof.
Definition category_RelMonad {C : precategory_data} (D : category)
(J : functor_data C D) : category :=
precategory_RelMonad (homset_property D) J,, has_homsets_RelMonad (homset_property D) J.
Definition forgetfunctor_RelMonad {C : precategory} (D : category)
(J : functor C D) :
functor (category_RelMonad D J) (functor_category C D).
Show proof.
Lemma forgetRelMonad_faithful {C : precategory} (D : category)
(J : functor C D) : faithful (forgetfunctor_RelMonad D J).
Show proof.
End PrecategoryOfRelativeMonads.
Section RelativeMonads_saturated.
Definition relmonadmor_weq_nat_trans_fails {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D)(R R': RelMonad J) :
(precategory_RelMonad hs J ⟦R, R'⟧) ≃ [C, D, hs] ⟦R_functor R, R_functor R'⟧.
Show proof.
{J : functor_data C D} {R R' R'' : RelMonad_data J}
(α : RelMonadMor R R') (α' : RelMonadMor R' R''):
RelMonadMor_axioms (λ a : C, (α : RelMonadMor_data R R') a · (α' : RelMonadMor_data R' R'') a).
Show proof.
split; intros; simpl.
- rewrite assoc.
rewrite (r_eta_α α).
apply (r_eta_α α').
- intermediate_path ((α:RelMonadMor_data R R') a ·
((α':RelMonadMor_data R' R'') a ·
r_bind R'' ((f · (α:RelMonadMor_data R R') b) ·
(α':RelMonadMor_data R' R'') b))).
* now repeat rewrite assoc.
* rewrite (α_r_bind α').
rewrite assoc. rewrite (α_r_bind α).
apply pathsinv0. apply assoc.
- rewrite assoc.
rewrite (r_eta_α α).
apply (r_eta_α α').
- intermediate_path ((α:RelMonadMor_data R R') a ·
((α':RelMonadMor_data R' R'') a ·
r_bind R'' ((f · (α:RelMonadMor_data R R') b) ·
(α':RelMonadMor_data R' R'') b))).
* now repeat rewrite assoc.
* rewrite (α_r_bind α').
rewrite assoc. rewrite (α_r_bind α).
apply pathsinv0. apply assoc.
Definition RelMonad_composition {C : precategory_data} {D : precategory}
{J : functor_data C D} {R R' R'' : RelMonad_data J}
(α : RelMonadMor R R') (α' : RelMonadMor R' R''):
RelMonadMor R R'' := _ ,, RelMonad_composition_laws α α'.
Definition precategory_RelMonad_ob_mor {C : precategory_data} {D : precategory}
(J : functor_data C D) : precategory_ob_mor :=
precategory_ob_mor_pair (RelMonad J) RelMonadMor.
Definition precategory_RelMonad_data {C : precategory_data} {D : precategory}
(J : functor_data C D) : precategory_data.
Show proof.
apply (precategory_data_pair (precategory_RelMonad_ob_mor J)).
- intro R.
apply RelMonad_identity.
- intros R R' R'' α α'.
apply (RelMonad_composition α α').
- intro R.
apply RelMonad_identity.
- intros R R' R'' α α'.
apply (RelMonad_composition α α').
Lemma precategory_RelMonad_axioms {C : precategory_data} {D : precategory}
(hs : has_homsets D) (J : functor_data C D) :
is_precategory (precategory_RelMonad_data J).
Show proof.
repeat split; simpl; intros.
- apply (invmap (RelMonadMor_equiv hs _ _ )).
apply funextsec. intros x. apply id_left.
- apply (invmap (RelMonadMor_equiv hs _ _ )).
apply funextsec. intros x. apply id_right.
- apply (invmap (RelMonadMor_equiv hs _ _ )).
apply funextsec. intros x. apply assoc.
- apply (invmap (RelMonadMor_equiv hs _ _ )).
apply funextsec. intros x. apply assoc'.
- apply (invmap (RelMonadMor_equiv hs _ _ )).
apply funextsec. intros x. apply id_left.
- apply (invmap (RelMonadMor_equiv hs _ _ )).
apply funextsec. intros x. apply id_right.
- apply (invmap (RelMonadMor_equiv hs _ _ )).
apply funextsec. intros x. apply assoc.
- apply (invmap (RelMonadMor_equiv hs _ _ )).
apply funextsec. intros x. apply assoc'.
Definition precategory_RelMonad {C : precategory_data} {D : precategory}
(hs : has_homsets D) (J : functor_data C D): precategory :=
_ ,, precategory_RelMonad_axioms hs J.
Lemma has_homsets_RelMonad {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor_data C D) :
has_homsets (precategory_RelMonad hs J).
Show proof.
intros R R'. simpl. unfold RelMonadMor.
apply isaset_total2 .
- apply impred_isaset.
intro. apply hs.
- intro α.
apply isasetaprop.
apply isaprop_RelMonadMor_axioms.
apply hs.
apply isaset_total2 .
- apply impred_isaset.
intro. apply hs.
- intro α.
apply isasetaprop.
apply isaprop_RelMonadMor_axioms.
apply hs.
Definition category_RelMonad {C : precategory_data} (D : category)
(J : functor_data C D) : category :=
precategory_RelMonad (homset_property D) J,, has_homsets_RelMonad (homset_property D) J.
Definition forgetfunctor_RelMonad {C : precategory} (D : category)
(J : functor C D) :
functor (category_RelMonad D J) (functor_category C D).
Show proof.
use mk_functor.
- simpl.
use mk_functor_data.
+ simpl.
exact (λ R : RelMonad J, R_functor R).
+ simpl. intros R R' α.
exact (nat_trans_RelMonadMor α).
- split.
+ red. intros. simpl. apply nat_trans_eq.
* apply D.
* intros; apply idpath.
+ unfold functor_compax. simpl. intros R R' R'' α α'. apply nat_trans_eq.
* apply D.
* intro c. apply idpath.
- simpl.
use mk_functor_data.
+ simpl.
exact (λ R : RelMonad J, R_functor R).
+ simpl. intros R R' α.
exact (nat_trans_RelMonadMor α).
- split.
+ red. intros. simpl. apply nat_trans_eq.
* apply D.
* intros; apply idpath.
+ unfold functor_compax. simpl. intros R R' R'' α α'. apply nat_trans_eq.
* apply D.
* intro c. apply idpath.
Lemma forgetRelMonad_faithful {C : precategory} (D : category)
(J : functor C D) : faithful (forgetfunctor_RelMonad D J).
Show proof.
intros R R'. simpl.
apply isinclbetweensets.
- apply isaset_total2.
+ apply impred_isaset.
intros. apply D.
+ intros. apply isasetaprop. apply isaprop_RelMonadMor_axioms. apply D.
- apply isaset_nat_trans. apply D.
- intros α α' p.
apply RelMonadMor_equiv.
+ apply D.
+ apply funextsec. intro c.
change (nat_trans_RelMonadMor α c = nat_trans_RelMonadMor α' c).
rewrite p.
apply idpath.
apply isinclbetweensets.
- apply isaset_total2.
+ apply impred_isaset.
intros. apply D.
+ intros. apply isasetaprop. apply isaprop_RelMonadMor_axioms. apply D.
- apply isaset_nat_trans. apply D.
- intros α α' p.
apply RelMonadMor_equiv.
+ apply D.
+ apply funextsec. intro c.
change (nat_trans_RelMonadMor α c = nat_trans_RelMonadMor α' c).
rewrite p.
apply idpath.
End PrecategoryOfRelativeMonads.
Section RelativeMonads_saturated.
Definition relmonadmor_weq_nat_trans_fails {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D)(R R': RelMonad J) :
(precategory_RelMonad hs J ⟦R, R'⟧) ≃ [C, D, hs] ⟦R_functor R, R_functor R'⟧.
Show proof.
apply (weqpair nat_trans_RelMonadMor).
use isweq_iso.
- intro α.
exists (nat_trans_data_from_nat_trans α).
split; intros.
Abort.
Definition relmonadmor_eq_type {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D)(R R': RelMonad J) : UU
:= ∑ p : iso (C := [C, D, hs]) (R_functor R) (R_functor R'),
RelMonadMor_axioms (nat_trans_data_from_nat_trans (morphism_from_iso _ _ _ p)).
Definition relmonadmor_ob_eq {C : precategory_data} {D : precategory} (H: is_univalent D)
(J : functor C D)(R R': RelMonad J) :
(R = R') ≃ relmonadmor_eq_type (pr2 H) J R R'.
Proof.
eapply weqcomp.
- apply total2_paths_equiv.
- set (H1 := weqpair _ (pr1 (is_univalent_functor_category C D H) (R_functor R) (R_functor R'))).
Abort.
use isweq_iso.
- intro α.
exists (nat_trans_data_from_nat_trans α).
split; intros.
Abort.
Definition relmonadmor_eq_type {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D)(R R': RelMonad J) : UU
:= ∑ p : iso (C := [C, D, hs]) (R_functor R) (R_functor R'),
RelMonadMor_axioms (nat_trans_data_from_nat_trans (morphism_from_iso _ _ _ p)).
Definition relmonadmor_ob_eq {C : precategory_data} {D : precategory} (H: is_univalent D)
(J : functor C D)(R R': RelMonad J) :
(R = R') ≃ relmonadmor_eq_type (pr2 H) J R R'.
Proof.
eapply weqcomp.
- apply total2_paths_equiv.
- set (H1 := weqpair _ (pr1 (is_univalent_functor_category C D H) (R_functor R) (R_functor R'))).
Abort.
better upstream
Definition functor_iso_pointwise_if_iso' (C : precategory_data) (C' : precategory)
(hs: has_homsets C')
(F G : ob [C, C',hs]) (α: iso F G) :
∏ a : ob C,
iso (pr1 F a) (pr1 G a) :=
λ a, tpair _ _ (is_functor_iso_pointwise_if_iso C C' _ F G (pr1 α) (pr2 α) a).
Lemma idtoiso_functorcat_compute_pointwise' (C : precategory_data) (D : precategory)
(hs: has_homsets D) (F G : ob [C, D, hs])
(p : F = G) (a : ob C) :
functor_iso_pointwise_if_iso' C D _ F G (idtoiso p) a =
idtoiso
(toforallpaths (λ _ : ob C, D) (pr1 (pr1 F)) (pr1 (pr1 G))
(base_paths (pr1 F) (pr1 G) (base_paths F G p)) a).
Proof.
induction p.
apply eq_iso. apply idpath.
(hs: has_homsets C')
(F G : ob [C, C',hs]) (α: iso F G) :
∏ a : ob C,
iso (pr1 F a) (pr1 G a) :=
λ a, tpair _ _ (is_functor_iso_pointwise_if_iso C C' _ F G (pr1 α) (pr2 α) a).
Lemma idtoiso_functorcat_compute_pointwise' (C : precategory_data) (D : precategory)
(hs: has_homsets D) (F G : ob [C, D, hs])
(p : F = G) (a : ob C) :
functor_iso_pointwise_if_iso' C D _ F G (idtoiso p) a =
idtoiso
(toforallpaths (λ _ : ob C, D) (pr1 (pr1 F)) (pr1 (pr1 G))
(base_paths (pr1 F) (pr1 G) (base_paths F G p)) a).
Proof.
induction p.
apply eq_iso. apply idpath.
end of better upstream
a rather trivial observation
Definition is_iso_from_is_relmonadmor_iso {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R')
: is_iso(C := [C, D, hs]) (nat_trans_RelMonadMor (pr1 α)).
Show proof.
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R')
: is_iso(C := [C, D, hs]) (nat_trans_RelMonadMor (pr1 α)).
Show proof.
apply is_iso_from_is_z_iso.
set (H' := iso_inv_after_iso α).
set (H'':= iso_after_iso_inv α).
set (α' := inv_from_iso α).
exists (nat_trans_RelMonadMor α').
split; simpl.
- unfold α'. unfold R_functor.
apply (nat_trans_eq hs).
set (aux := maponpaths pr1 H'). apply toforallpaths in aux.
exact aux.
- unfold α'. unfold R_functor.
apply (nat_trans_eq hs).
set (aux := maponpaths pr1 H''). apply toforallpaths in aux.
exact aux.
set (H' := iso_inv_after_iso α).
set (H'':= iso_after_iso_inv α).
set (α' := inv_from_iso α).
exists (nat_trans_RelMonadMor α').
split; simpl.
- unfold α'. unfold R_functor.
apply (nat_trans_eq hs).
set (aux := maponpaths pr1 H'). apply toforallpaths in aux.
exact aux.
- unfold α'. unfold R_functor.
apply (nat_trans_eq hs).
set (aux := maponpaths pr1 H''). apply toforallpaths in aux.
exact aux.
its immediate consequence
Definition iso_from_is_relmonadmor_iso {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R')
: iso(C := [C, D, hs]) (R_functor R) (R_functor R').
Show proof.
Corollary iso_from_is_relmonadmor_iso_p {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R') (c : C) :
pr1 (pr1 α) c =
functor_iso_pointwise_if_iso' C D
hs (R_functor R) (R_functor R') (iso_from_is_relmonadmor_iso
hs J α) c.
Show proof.
Lemma iso_from_is_relmonadmor_iso_idtoiso {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (p : @paths (precategory_RelMonad hs J) R R'):
iso_from_is_relmonadmor_iso hs J (idtoiso p) =
idtoiso(C := [C, D, hs]) (maponpaths (@R_functor C D J) p).
Show proof.
Definition alternative_inv_to_relmonadmor_iso {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R')
: precategory_RelMonad hs J ⟦R', R⟧.
Show proof.
Lemma alternative_inv_to_relmonadmor_iso_is_inv {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R'):
alternative_inv_to_relmonadmor_iso hs J α = inv_from_iso α.
Show proof.
Corollary iso_from_is_relmonadmor_iso_inv_p {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R') (c : C) :
pr1 (inv_from_iso α) c =
inv_from_iso (functor_iso_pointwise_if_iso' C D
hs (R_functor R) (R_functor R') (iso_from_is_relmonadmor_iso
hs J α) c).
Show proof.
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R')
: iso(C := [C, D, hs]) (R_functor R) (R_functor R').
Show proof.
use (isopair(C := [C, D, hs]) (nat_trans_RelMonadMor (pr1 α))).
exact (is_iso_from_is_relmonadmor_iso hs J α).
exact (is_iso_from_is_relmonadmor_iso hs J α).
Corollary iso_from_is_relmonadmor_iso_p {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R') (c : C) :
pr1 (pr1 α) c =
functor_iso_pointwise_if_iso' C D
hs (R_functor R) (R_functor R') (iso_from_is_relmonadmor_iso
hs J α) c.
Show proof.
apply idpath.
Lemma iso_from_is_relmonadmor_iso_idtoiso {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (p : @paths (precategory_RelMonad hs J) R R'):
iso_from_is_relmonadmor_iso hs J (idtoiso p) =
idtoiso(C := [C, D, hs]) (maponpaths (@R_functor C D J) p).
Show proof.
unfold iso_from_is_relmonadmor_iso.
simpl.
apply eq_iso.
simpl.
apply nat_trans_eq.
- exact hs.
- intro c.
induction p.
apply idpath.
simpl.
apply eq_iso.
simpl.
apply nat_trans_eq.
- exact hs.
- intro c.
induction p.
apply idpath.
Definition alternative_inv_to_relmonadmor_iso {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R')
: precategory_RelMonad hs J ⟦R', R⟧.
Show proof.
use tpair.
- intro c.
exact (inv_from_iso (functor_iso_pointwise_if_iso' C D
hs (R_functor R) (R_functor R') (iso_from_is_relmonadmor_iso
hs J α) c)).
- split.
+ intro c.
apply pathsinv0.
apply iso_inv_on_left.
rewrite <- iso_from_is_relmonadmor_iso_p.
apply pathsinv0.
apply (r_eta_α (RelMonadMor_axioms_from_RelMonadMor (pr1 α))).
+ intros c d f.
apply iso_inv_on_left.
rewrite <- assoc.
apply pathsinv0.
apply iso_inv_on_right.
do 2 rewrite <- iso_from_is_relmonadmor_iso_p.
intermediate_path (pr1 (pr1 α) c · r_bind R' ((f · (inv_from_iso
(functor_iso_pointwise_if_iso' C D hs (R_functor R)
(R_functor R') (iso_from_is_relmonadmor_iso hs J α) d) )) ·
(functor_iso_pointwise_if_iso' C D hs (R_functor R)
(R_functor R') (iso_from_is_relmonadmor_iso hs J α) d ))).
2: { apply cancel_precomposition.
apply maponpaths.
rewrite <- assoc.
rewrite iso_after_iso_inv.
apply id_right.
}
apply pathsinv0.
rewrite <- iso_from_is_relmonadmor_iso_p.
apply (α_r_bind (RelMonadMor_axioms_from_RelMonadMor (pr1 α))).
- intro c.
exact (inv_from_iso (functor_iso_pointwise_if_iso' C D
hs (R_functor R) (R_functor R') (iso_from_is_relmonadmor_iso
hs J α) c)).
- split.
+ intro c.
apply pathsinv0.
apply iso_inv_on_left.
rewrite <- iso_from_is_relmonadmor_iso_p.
apply pathsinv0.
apply (r_eta_α (RelMonadMor_axioms_from_RelMonadMor (pr1 α))).
+ intros c d f.
apply iso_inv_on_left.
rewrite <- assoc.
apply pathsinv0.
apply iso_inv_on_right.
do 2 rewrite <- iso_from_is_relmonadmor_iso_p.
intermediate_path (pr1 (pr1 α) c · r_bind R' ((f · (inv_from_iso
(functor_iso_pointwise_if_iso' C D hs (R_functor R)
(R_functor R') (iso_from_is_relmonadmor_iso hs J α) d) )) ·
(functor_iso_pointwise_if_iso' C D hs (R_functor R)
(R_functor R') (iso_from_is_relmonadmor_iso hs J α) d ))).
2: { apply cancel_precomposition.
apply maponpaths.
rewrite <- assoc.
rewrite iso_after_iso_inv.
apply id_right.
}
apply pathsinv0.
rewrite <- iso_from_is_relmonadmor_iso_p.
apply (α_r_bind (RelMonadMor_axioms_from_RelMonadMor (pr1 α))).
Lemma alternative_inv_to_relmonadmor_iso_is_inv {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R'):
alternative_inv_to_relmonadmor_iso hs J α = inv_from_iso α.
Show proof.
apply inv_iso_unique'.
unfold precomp_with.
apply RelMonadMor_equiv.
- exact hs.
- apply funextsec.
intro c.
unfold alternative_inv_to_relmonadmor_iso.
simpl.
apply pathsinv0.
apply iso_inv_on_left.
rewrite id_left.
intermediate_path (pr1 (pr1 α) c).
{ apply idpath. }
rewrite iso_from_is_relmonadmor_iso_p.
apply idpath.
unfold precomp_with.
apply RelMonadMor_equiv.
- exact hs.
- apply funextsec.
intro c.
unfold alternative_inv_to_relmonadmor_iso.
simpl.
apply pathsinv0.
apply iso_inv_on_left.
rewrite id_left.
intermediate_path (pr1 (pr1 α) c).
{ apply idpath. }
rewrite iso_from_is_relmonadmor_iso_p.
apply idpath.
Corollary iso_from_is_relmonadmor_iso_inv_p {C : precategory_data} {D : precategory} (hs: has_homsets D)
(J : functor C D) {R R': RelMonad J} (α : iso(C := precategory_RelMonad hs J) R R') (c : C) :
pr1 (inv_from_iso α) c =
inv_from_iso (functor_iso_pointwise_if_iso' C D
hs (R_functor R) (R_functor R') (iso_from_is_relmonadmor_iso
hs J α) c).
Show proof.
rewrite <- alternative_inv_to_relmonadmor_iso_is_inv.
apply idpath.
apply idpath.
the other direction, first the inverse monad morphism
Definition inv_relmonadmor_from_is_iso {C : precategory_data} {D : precategory}
(hs: has_homsets D) (J : functor C D){R R': RelMonad J}
(α : precategory_RelMonad hs J ⟦R, R'⟧)
: is_iso(C := [C, D, hs]) (nat_trans_RelMonadMor α) → precategory_RelMonad hs J ⟦R', R⟧.
Show proof.
(hs: has_homsets D) (J : functor C D){R R': RelMonad J}
(α : precategory_RelMonad hs J ⟦R, R'⟧)
: is_iso(C := [C, D, hs]) (nat_trans_RelMonadMor α) → precategory_RelMonad hs J ⟦R', R⟧.
Show proof.
intro T.
set (fiso := isopair(C := [C, D, hs]) (nat_trans_RelMonadMor α) T).
set (finv := inv_from_iso fiso).
exists (pr1 finv).
unfold finv.
split.
- intros c.
unfold fiso.
rewrite <- nat_trans_inv_pointwise_inv_after_p.
apply pathsinv0.
apply iso_inv_on_left.
simpl.
apply pathsinv0.
apply (r_eta_α (RelMonadMor_axioms_from_RelMonadMor α)).
- intros a b f.
unfold fiso.
do 2 rewrite <- nat_trans_inv_pointwise_inv_after_p.
apply iso_inv_on_left.
rewrite <- assoc.
apply pathsinv0.
apply iso_inv_on_right.
simpl.
etrans.
{ apply pathsinv0.
apply (α_r_bind (RelMonadMor_axioms_from_RelMonadMor α)). }
apply cancel_precomposition.
apply maponpaths.
rewrite <- assoc.
etrans.
2: { apply id_right. }
apply cancel_precomposition.
apply iso_inv_on_right.
apply pathsinv0.
apply id_right.
set (fiso := isopair(C := [C, D, hs]) (nat_trans_RelMonadMor α) T).
set (finv := inv_from_iso fiso).
exists (pr1 finv).
unfold finv.
split.
- intros c.
unfold fiso.
rewrite <- nat_trans_inv_pointwise_inv_after_p.
apply pathsinv0.
apply iso_inv_on_left.
simpl.
apply pathsinv0.
apply (r_eta_α (RelMonadMor_axioms_from_RelMonadMor α)).
- intros a b f.
unfold fiso.
do 2 rewrite <- nat_trans_inv_pointwise_inv_after_p.
apply iso_inv_on_left.
rewrite <- assoc.
apply pathsinv0.
apply iso_inv_on_right.
simpl.
etrans.
{ apply pathsinv0.
apply (α_r_bind (RelMonadMor_axioms_from_RelMonadMor α)). }
apply cancel_precomposition.
apply maponpaths.
rewrite <- assoc.
etrans.
2: { apply id_right. }
apply cancel_precomposition.
apply iso_inv_on_right.
apply pathsinv0.
apply id_right.
verification that the proposed inverse monad morphism is suitable
Definition is_relmonadmor_iso_from_is_iso {C : precategory_data} {D : precategory}
(hs: has_homsets D) (J : functor C D) {R R': RelMonad J}
(α : precategory_RelMonad hs J ⟦R, R'⟧)
: is_iso(C := [C, D, hs]) (nat_trans_RelMonadMor α) → is_iso α.
Show proof.
(hs: has_homsets D) (J : functor C D) {R R': RelMonad J}
(α : precategory_RelMonad hs J ⟦R, R'⟧)
: is_iso(C := [C, D, hs]) (nat_trans_RelMonadMor α) → is_iso α.
Show proof.
intro T.
apply is_iso_from_is_z_iso.
exists (inv_relmonadmor_from_is_iso hs J α T).
split; simpl.
-
apply RelMonadMor_equiv.
+ exact hs.
+ simpl.
set (aux := iso_inv_after_iso (isopair(C := [C, D, hs]) (nat_trans_RelMonadMor α) T)).
apply (maponpaths pr1) in aux.
exact aux.
- apply RelMonadMor_equiv.
+ exact hs.
+ simpl.
set (aux := iso_after_iso_inv (isopair(C := [C, D, hs]) (nat_trans_RelMonadMor α) T)).
apply (maponpaths pr1) in aux.
exact aux.
apply is_iso_from_is_z_iso.
exists (inv_relmonadmor_from_is_iso hs J α T).
split; simpl.
-
apply RelMonadMor_equiv.
+ exact hs.
+ simpl.
set (aux := iso_inv_after_iso (isopair(C := [C, D, hs]) (nat_trans_RelMonadMor α) T)).
apply (maponpaths pr1) in aux.
exact aux.
- apply RelMonadMor_equiv.
+ exact hs.
+ simpl.
set (aux := iso_after_iso_inv (isopair(C := [C, D, hs]) (nat_trans_RelMonadMor α) T)).
apply (maponpaths pr1) in aux.
exact aux.
its immediate consequence
Definition relmonadmor_iso_from_is_iso {C : precategory_data} {D : precategory}
(hs: has_homsets D) (J : functor C D)(R R': RelMonad J)
(α : precategory_RelMonad hs J ⟦R, R'⟧)
: is_iso(C := [C, D, hs]) (nat_trans_RelMonadMor α) → iso(C := precategory_RelMonad hs J) R R'.
Show proof.
Definition relmonadmor_iso_first_iso {C : precategory_data} {D : precategory}
(hs: has_homsets D) (J : functor C D)(R R': RelMonad J)
: iso(C := precategory_RelMonad hs J) R R' ≃ ∑ α : R_functor R ⟹ R_functor R', is_iso(C := [C, D, hs]) α.
Show proof.
Lemma pr1_pr1_relmonadmor_eq_from_relmonadmor_iso_idtoiso_aux {C : precategory_data} {D : precategory}
(hs: has_homsets D) (J : functor C D) {R R': RelMonad J}
(p: @paths (precategory_RelMonad hs J) R R'):
base_paths (pr1 (R_functor R)) (pr1 (R_functor R'))
(base_paths (R_functor R) (R_functor R') (maponpaths R_functor p)) =
base_paths (pr1 R) (pr1 R') (base_paths R R' p).
Show proof.
Lemma pr1_pr1_relmonadmor_eq_from_relmonadmor_iso_idtoiso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(p: @paths (precategory_RelMonad (pr2 H) J) R R'):
pr1_pr1_relmonadmor_eq_from_relmonadmor_iso H J (idtoiso p) =
base_paths (pr1 R) (pr1 R') (base_paths R R' p).
Show proof.
Lemma transport_of_relmonad_η_is_pointwise {C : precategory_data} {D : precategory}
(J : functor C D) (F0 G0 : ob C -> ob D)
(F1 : ∏ a : ob C, J a --> F0 a)
(gamma : F0 = G0 )
(c: ob C) :
transportf (fun x : ob C -> ob D => ∏ a, D ⟦J a, x a⟧) gamma F1 c =
transportf (fun d: ob D => D ⟦J c, d⟧) (toforallpaths (λ _ : ob C, D) F0 G0 gamma c) (F1 c).
Show proof.
Lemma transport_of_relmonad_bind_is_pointwise {C : precategory_data} {D : precategory}
(J : functor C D) (F0 G0 : ob C -> ob D)
(F1 : ∏ a b : ob C , D ⟦J a, F0 b⟧ → D ⟦F0 a, F0 b⟧)
(gamma : F0 = G0 )
(c d : ob C) (f : J c --> G0 d) :
transportf (fun x : ob C -> ob D => ∏ a b : ob C , D ⟦J a, x b⟧ → D ⟦x a, x b⟧) gamma F1 c d f =
double_transport (toforallpaths (λ _ : ob C, D) F0 G0 gamma c)
(toforallpaths (λ _ : ob C, D) F0 G0 gamma d)
(F1 c d (transportb (fun x : ob D => D ⟦J c, x⟧)
(toforallpaths (λ _ : ob C, D) F0 G0 gamma d) f)).
Show proof.
(hs: has_homsets D) (J : functor C D)(R R': RelMonad J)
(α : precategory_RelMonad hs J ⟦R, R'⟧)
: is_iso(C := [C, D, hs]) (nat_trans_RelMonadMor α) → iso(C := precategory_RelMonad hs J) R R'.
Show proof.
intro T.
use (isopair α).
exact (is_relmonadmor_iso_from_is_iso hs J α T).
use (isopair α).
exact (is_relmonadmor_iso_from_is_iso hs J α T).
Definition relmonadmor_iso_first_iso {C : precategory_data} {D : precategory}
(hs: has_homsets D) (J : functor C D)(R R': RelMonad J)
: iso(C := precategory_RelMonad hs J) R R' ≃ ∑ α : R_functor R ⟹ R_functor R', is_iso(C := [C, D, hs]) α.
Show proof.
unfold iso.
Abort.
Definition pr1_pr1_relmonadmor_eq_from_relmonadmor_iso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J} :
iso(C := precategory_RelMonad (pr2 H) J) R R' -> pr1 (pr1 R) = pr1 (pr1 R').
Proof.
intro α.
change (pr1 (pr1 (R_functor R)) = pr1 (pr1 (R_functor R'))).
do 2 apply maponpaths.
set (H1 := weqpair _ (pr1 (is_univalent_functor_category C D H) (R_functor R) (R_functor R'))).
apply H1.
apply (iso_from_is_relmonadmor_iso (pr2 H) J α).
Abort.
Definition pr1_pr1_relmonadmor_eq_from_relmonadmor_iso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J} :
iso(C := precategory_RelMonad (pr2 H) J) R R' -> pr1 (pr1 R) = pr1 (pr1 R').
Proof.
intro α.
change (pr1 (pr1 (R_functor R)) = pr1 (pr1 (R_functor R'))).
do 2 apply maponpaths.
set (H1 := weqpair _ (pr1 (is_univalent_functor_category C D H) (R_functor R) (R_functor R'))).
apply H1.
apply (iso_from_is_relmonadmor_iso (pr2 H) J α).
Lemma pr1_pr1_relmonadmor_eq_from_relmonadmor_iso_idtoiso_aux {C : precategory_data} {D : precategory}
(hs: has_homsets D) (J : functor C D) {R R': RelMonad J}
(p: @paths (precategory_RelMonad hs J) R R'):
base_paths (pr1 (R_functor R)) (pr1 (R_functor R'))
(base_paths (R_functor R) (R_functor R') (maponpaths R_functor p)) =
base_paths (pr1 R) (pr1 R') (base_paths R R' p).
Show proof.
unfold base_paths.
etrans.
{ apply maponpaths.
apply (maponpathscomp (@R_functor C D J) (@functor_data_from_functor C D)).
}
etrans.
{ apply (maponpathscomp ((functor_data_from_functor C D ∘ R_functor)%functions) (@functor_on_objects C D)). }
etrans.
2: { apply pathsinv0.
apply (maponpathscomp (@RelMonad_data_from_RelMonad C D J) (@RelMonad_ob C D J)).
}
apply idpath.
etrans.
{ apply maponpaths.
apply (maponpathscomp (@R_functor C D J) (@functor_data_from_functor C D)).
}
etrans.
{ apply (maponpathscomp ((functor_data_from_functor C D ∘ R_functor)%functions) (@functor_on_objects C D)). }
etrans.
2: { apply pathsinv0.
apply (maponpathscomp (@RelMonad_data_from_RelMonad C D J) (@RelMonad_ob C D J)).
}
apply idpath.
Lemma pr1_pr1_relmonadmor_eq_from_relmonadmor_iso_idtoiso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(p: @paths (precategory_RelMonad (pr2 H) J) R R'):
pr1_pr1_relmonadmor_eq_from_relmonadmor_iso H J (idtoiso p) =
base_paths (pr1 R) (pr1 R') (base_paths R R' p).
Show proof.
unfold pr1_pr1_relmonadmor_eq_from_relmonadmor_iso.
simpl.
rewrite (iso_from_is_relmonadmor_iso_idtoiso (pr2 H) J).
rewrite functor_eq_from_functor_iso_idtoiso.
apply pr1_pr1_relmonadmor_eq_from_relmonadmor_iso_idtoiso_aux.
simpl.
rewrite (iso_from_is_relmonadmor_iso_idtoiso (pr2 H) J).
rewrite functor_eq_from_functor_iso_idtoiso.
apply pr1_pr1_relmonadmor_eq_from_relmonadmor_iso_idtoiso_aux.
Lemma transport_of_relmonad_η_is_pointwise {C : precategory_data} {D : precategory}
(J : functor C D) (F0 G0 : ob C -> ob D)
(F1 : ∏ a : ob C, J a --> F0 a)
(gamma : F0 = G0 )
(c: ob C) :
transportf (fun x : ob C -> ob D => ∏ a, D ⟦J a, x a⟧) gamma F1 c =
transportf (fun d: ob D => D ⟦J c, d⟧) (toforallpaths (λ _ : ob C, D) F0 G0 gamma c) (F1 c).
Show proof.
induction gamma.
apply idpath.
apply idpath.
Lemma transport_of_relmonad_bind_is_pointwise {C : precategory_data} {D : precategory}
(J : functor C D) (F0 G0 : ob C -> ob D)
(F1 : ∏ a b : ob C , D ⟦J a, F0 b⟧ → D ⟦F0 a, F0 b⟧)
(gamma : F0 = G0 )
(c d : ob C) (f : J c --> G0 d) :
transportf (fun x : ob C -> ob D => ∏ a b : ob C , D ⟦J a, x b⟧ → D ⟦x a, x b⟧) gamma F1 c d f =
double_transport (toforallpaths (λ _ : ob C, D) F0 G0 gamma c)
(toforallpaths (λ _ : ob C, D) F0 G0 gamma d)
(F1 c d (transportb (fun x : ob D => D ⟦J c, x⟧)
(toforallpaths (λ _ : ob C, D) F0 G0 gamma d) f)).
Show proof.
induction gamma.
apply idpath.
apply idpath.
a preparation for the following lemma
Lemma isotoid_functorcat_pointwise_aux (C : precategory_data) (D : precategory)
(H : is_univalent D) (F G : ob [C, D, (pr2 H)]) (p: F = G) (c: C) :
let α := idtoiso p in
toforallpaths (fun _ : ob C => ob D) (pr1 (pr1 F)) (pr1 (pr1 G))
(maponpaths pr1
(maponpaths pr1
(isotoid [C, D, pr2 H]
(is_univalent_functor_category C D H) α))) c
= isotoid D H
(functor_iso_pointwise_if_iso C D (pr2 H) F G α (pr2 α) c).
Show proof.
(H : is_univalent D) (F G : ob [C, D, (pr2 H)]) (p: F = G) (c: C) :
let α := idtoiso p in
toforallpaths (fun _ : ob C => ob D) (pr1 (pr1 F)) (pr1 (pr1 G))
(maponpaths pr1
(maponpaths pr1
(isotoid [C, D, pr2 H]
(is_univalent_functor_category C D H) α))) c
= isotoid D H
(functor_iso_pointwise_if_iso C D (pr2 H) F G α (pr2 α) c).
Show proof.
induction p.
cbn delta in *.
unfold functor_iso_pointwise_if_iso.
rewrite isotoid_idtoiso.
unfold idtoiso.
simpl.
apply idtoiso_inj; try assumption.
rewrite idtoiso_isotoid.
simpl.
apply eq_iso.
apply idpath.
cbn delta in *.
unfold functor_iso_pointwise_if_iso.
rewrite isotoid_idtoiso.
unfold idtoiso.
simpl.
apply idtoiso_inj; try assumption.
rewrite idtoiso_isotoid.
simpl.
apply eq_iso.
apply idpath.
general lemma on univalence of functor category
Lemma isotoid_functorcat_pointwise (C : precategory_data) (D : precategory) (H : is_univalent D)
(F G : ob [C, D, (pr2 H)]) (α: iso F G) (c: C) :
toforallpaths (fun _ : ob C => ob D) (pr1 (pr1 F)) (pr1 (pr1 G))
(maponpaths pr1
(maponpaths pr1
(isotoid [C, D, pr2 H]
(is_univalent_functor_category C D H) α))) c
= isotoid D H
(functor_iso_pointwise_if_iso' C D (pr2 H) F G α c).
Show proof.
Definition η_relmonadmor_eq_from_relmonadmor_iso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(α: iso(C := precategory_RelMonad (pr2 H) J) R R')
: transportf (fun x : ob C -> ob D => ∏ c, D ⟦J c, x c⟧)
(pr1_pr1_relmonadmor_eq_from_relmonadmor_iso H J α) (pr1 (pr2 (pr1 R))) =
pr1 (pr2 (pr1 R')).
Show proof.
(F G : ob [C, D, (pr2 H)]) (α: iso F G) (c: C) :
toforallpaths (fun _ : ob C => ob D) (pr1 (pr1 F)) (pr1 (pr1 G))
(maponpaths pr1
(maponpaths pr1
(isotoid [C, D, pr2 H]
(is_univalent_functor_category C D H) α))) c
= isotoid D H
(functor_iso_pointwise_if_iso' C D (pr2 H) F G α c).
Show proof.
assert (aux := isotoid_functorcat_pointwise_aux C D H F G
(isotoid [C, D, pr2 H] (is_univalent_functor_category C D H) α)).
rewrite idtoiso_isotoid in aux.
apply aux.
(isotoid [C, D, pr2 H] (is_univalent_functor_category C D H) α)).
rewrite idtoiso_isotoid in aux.
apply aux.
Definition η_relmonadmor_eq_from_relmonadmor_iso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(α: iso(C := precategory_RelMonad (pr2 H) J) R R')
: transportf (fun x : ob C -> ob D => ∏ c, D ⟦J c, x c⟧)
(pr1_pr1_relmonadmor_eq_from_relmonadmor_iso H J α) (pr1 (pr2 (pr1 R))) =
pr1 (pr2 (pr1 R')).
Show proof.
apply funextsec; intro c.
rewrite transport_of_relmonad_η_is_pointwise.
unfold pr1_pr1_relmonadmor_eq_from_relmonadmor_iso.
simpl.
rewrite <- idtoiso_postcompose.
simpl.
change (pr1 (pr2 (pr1 R')) c) with (r_eta R' c).
change (pr1 (pr2 (pr1 R)) c) with (r_eta R c).
intermediate_path (r_eta R c · (pr1 (pr1 α): RelMonadMor_data _ _) c).
2: { set (X := RelMonadMor_axioms_from_RelMonadMor (pr1 α)).
exact (pr1 X c).
}
apply cancel_precomposition.
set (isor := iso_from_is_relmonadmor_iso (pr2 H) J α).
set (isor_p := functor_iso_pointwise_if_iso' C D (pr2 H) _ _ isor c).
change (pr1 (pr1 α) c) with (pr1 isor_p).
apply maponpaths.
unfold precategory_data_from_precategory in isor.
simpl in isor.
unfold precategory_data_from_precategory; simpl.
fold isor.
intermediate_path (idtoiso (isotoid _ H isor_p)).
2: { apply idtoiso_isotoid. }
apply maponpaths.
change (functor_eq_from_functor_iso
(pr2 H) H (R_functor R)
(R_functor R') isor) with (isotoid _ (is_univalent_functor_category C D H) isor).
apply (isotoid_functorcat_pointwise C D H (R_functor R) (R_functor R')).
rewrite transport_of_relmonad_η_is_pointwise.
unfold pr1_pr1_relmonadmor_eq_from_relmonadmor_iso.
simpl.
rewrite <- idtoiso_postcompose.
simpl.
change (pr1 (pr2 (pr1 R')) c) with (r_eta R' c).
change (pr1 (pr2 (pr1 R)) c) with (r_eta R c).
intermediate_path (r_eta R c · (pr1 (pr1 α): RelMonadMor_data _ _) c).
2: { set (X := RelMonadMor_axioms_from_RelMonadMor (pr1 α)).
exact (pr1 X c).
}
apply cancel_precomposition.
set (isor := iso_from_is_relmonadmor_iso (pr2 H) J α).
set (isor_p := functor_iso_pointwise_if_iso' C D (pr2 H) _ _ isor c).
change (pr1 (pr1 α) c) with (pr1 isor_p).
apply maponpaths.
unfold precategory_data_from_precategory in isor.
simpl in isor.
unfold precategory_data_from_precategory; simpl.
fold isor.
intermediate_path (idtoiso (isotoid _ H isor_p)).
2: { apply idtoiso_isotoid. }
apply maponpaths.
change (functor_eq_from_functor_iso
(pr2 H) H (R_functor R)
(R_functor R') isor) with (isotoid _ (is_univalent_functor_category C D H) isor).
apply (isotoid_functorcat_pointwise C D H (R_functor R) (R_functor R')).
the dual of UniMath.MoreFoundations.PartA.transportf_transpose
Lemma transportb_transpose {X : UU} {P : X → UU} {x x' : X}
(e : x' = x) (y : P x) (y' : P x') :
transportf P e y' = y -> y' = transportb P e y.
Show proof.
(e : x' = x) (y : P x) (y' : P x') :
transportf P e y' = y -> y' = transportb P e y.
Show proof.
intro H; induction e; exact H.
the following lemma should also be put upstream
Lemma transportb_isotoid (C : precategory) (H : is_univalent C)
(a b b' : ob C) (p : iso b b') (f : a --> b') :
transportb (λ b0 : C, a --> b0) (isotoid C H p) f = f · inv_from_iso p.
Show proof.
Definition bind_relmonadmor_eq_from_relmonadmor_iso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(α: iso(C := precategory_RelMonad (pr2 H) J) R R')
: transportf (fun x : ob C -> ob D => ∏ c d, D ⟦J c, x d⟧ → D ⟦x c, x d⟧)
(pr1_pr1_relmonadmor_eq_from_relmonadmor_iso H J α) (pr2 (pr2 (pr1 R))) =
pr2 (pr2 (pr1 R')).
Show proof.
Definition relmonad_eq_from_relmonad_iso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(α : iso(C := precategory_RelMonad (pr2 H) J) R R')
: R = R'.
Show proof.
Definition relmonad_eq_from_relmonad_iso_obsolete {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(α : iso(C := precategory_RelMonad (pr2 H) J) R R')
: R = R'.
Show proof.
Lemma relmonad_eq_from_relmonad_iso_idtoiso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J} (p: R = R') :
relmonad_eq_from_relmonad_iso H J (idtoiso(C := precategory_RelMonad (pr2 H) J) p) = p.
Show proof.
Lemma idtoiso_relmonad_eq_from_relmonad_iso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(α : iso(C := precategory_RelMonad (pr2 H) J) R R') :
idtoiso(C := precategory_RelMonad (pr2 H) J) (relmonad_eq_from_relmonad_iso H J α) = α.
Show proof.
Definition relmonadmor_idtoiso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D)(R R': RelMonad J) :
(R = R') ≃ iso(C := precategory_RelMonad (pr2 H) J) R R'.
Show proof.
Lemma isweq_idtoiso_RelMonad {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D)(R R': RelMonad J)
: isweq (@idtoiso (precategory_RelMonad (pr2 H) J) R R').
Show proof.
Lemma is_univalent_RelMonad {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D)(R R': RelMonad J)
: is_univalent (precategory_RelMonad (pr2 H) J).
Show proof.
End RelativeMonads_saturated.
Section RelAdjunctionWithKleisliCategory.
(a b b' : ob C) (p : iso b b') (f : a --> b') :
transportb (λ b0 : C, a --> b0) (isotoid C H p) f = f · inv_from_iso p.
Show proof.
apply pathsinv0.
apply transportb_transpose.
change (precategory_morphisms a) with (λ b0 : C, a --> b0).
rewrite transportf_isotoid'.
rewrite <- assoc.
rewrite iso_after_iso_inv.
apply id_right.
apply transportb_transpose.
change (precategory_morphisms a) with (λ b0 : C, a --> b0).
rewrite transportf_isotoid'.
rewrite <- assoc.
rewrite iso_after_iso_inv.
apply id_right.
Definition bind_relmonadmor_eq_from_relmonadmor_iso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(α: iso(C := precategory_RelMonad (pr2 H) J) R R')
: transportf (fun x : ob C -> ob D => ∏ c d, D ⟦J c, x d⟧ → D ⟦x c, x d⟧)
(pr1_pr1_relmonadmor_eq_from_relmonadmor_iso H J α) (pr2 (pr2 (pr1 R))) =
pr2 (pr2 (pr1 R')).
Show proof.
apply funextsec; intro c. apply funextsec; intro d. apply funextsec; intro f.
rewrite transport_of_relmonad_bind_is_pointwise.
unfold pr1_pr1_relmonadmor_eq_from_relmonadmor_iso.
simpl.
rewrite double_transport_idtoiso.
rewrite <- assoc.
set (isor := iso_from_is_relmonadmor_iso (pr2 H) J α).
unfold precategory_data_from_precategory in isor.
simpl in isor.
unfold precategory_data_from_precategory; simpl.
fold isor.
change (functor_eq_from_functor_iso
(pr2 H) H (R_functor R)
(R_functor R') isor) with (isotoid _ (is_univalent_functor_category C D H) isor).
do 2 rewrite (isotoid_functorcat_pointwise C D H (R_functor R) (R_functor R')).
do 2 rewrite idtoiso_isotoid.
change (pr2 (pr2 (pr1 R')) c d f) with (r_bind R' f).
change (pr2 (pr2 (pr1 R)) c d) with (r_bind(c:=c)(d:=d) R).
rewrite (transportb_isotoid D H).
do 2 rewrite <- (iso_from_is_relmonadmor_iso_inv_p (pr2 H) J α).
rewrite <- (iso_from_is_relmonadmor_iso_p (pr2 H) J α).
assert (aux := α_r_bind (RelMonadMor_axioms_from_RelMonadMor (inv_from_iso α)) c d f).
etrans.
{ apply assoc. }
etrans.
{ apply cancel_postcomposition.
apply aux.
}
intermediate_path (r_bind R' f · identity _).
2: { apply id_right. }
etrans.
{ apply pathsinv0. apply assoc. }
apply cancel_precomposition.
assert (aux2 := iso_after_iso_inv α).
apply (maponpaths pr1) in aux2.
apply toforallpaths in aux2.
apply aux2.
rewrite transport_of_relmonad_bind_is_pointwise.
unfold pr1_pr1_relmonadmor_eq_from_relmonadmor_iso.
simpl.
rewrite double_transport_idtoiso.
rewrite <- assoc.
set (isor := iso_from_is_relmonadmor_iso (pr2 H) J α).
unfold precategory_data_from_precategory in isor.
simpl in isor.
unfold precategory_data_from_precategory; simpl.
fold isor.
change (functor_eq_from_functor_iso
(pr2 H) H (R_functor R)
(R_functor R') isor) with (isotoid _ (is_univalent_functor_category C D H) isor).
do 2 rewrite (isotoid_functorcat_pointwise C D H (R_functor R) (R_functor R')).
do 2 rewrite idtoiso_isotoid.
change (pr2 (pr2 (pr1 R')) c d f) with (r_bind R' f).
change (pr2 (pr2 (pr1 R)) c d) with (r_bind(c:=c)(d:=d) R).
rewrite (transportb_isotoid D H).
do 2 rewrite <- (iso_from_is_relmonadmor_iso_inv_p (pr2 H) J α).
rewrite <- (iso_from_is_relmonadmor_iso_p (pr2 H) J α).
assert (aux := α_r_bind (RelMonadMor_axioms_from_RelMonadMor (inv_from_iso α)) c d f).
etrans.
{ apply assoc. }
etrans.
{ apply cancel_postcomposition.
apply aux.
}
intermediate_path (r_bind R' f · identity _).
2: { apply id_right. }
etrans.
{ apply pathsinv0. apply assoc. }
apply cancel_precomposition.
assert (aux2 := iso_after_iso_inv α).
apply (maponpaths pr1) in aux2.
apply toforallpaths in aux2.
apply aux2.
Definition relmonad_eq_from_relmonad_iso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(α : iso(C := precategory_RelMonad (pr2 H) J) R R')
: R = R'.
Show proof.
apply RelMonad_eq.
- exact (pr2 H).
- set (Hob := pr1_pr1_relmonadmor_eq_from_relmonadmor_iso H J α).
apply (total2_paths_f Hob).
apply dirprodeq.
+ intermediate_path (transportf (λ F : C → D, ∏ c : C, D ⟦ J c, F c ⟧) Hob
(pr1 (pr2 (pr1 R)))).
{ apply pathsinv0. apply (transport_map (fun F => dirprod_pr1(X := ∏ c : C, D ⟦ J c, F c ⟧)(Y := ∏ c d : C, D ⟦ J c, F d ⟧ → D ⟦ F c, F d ⟧))). }
apply (η_relmonadmor_eq_from_relmonadmor_iso H J α).
+ intermediate_path (transportf (λ F : C → D, ∏ c d : C, D ⟦ J c, F d ⟧ → D ⟦ F c, F d ⟧) Hob
(pr2 (pr2 (pr1 R)))).
{ apply pathsinv0. apply (transport_map (fun F => dirprod_pr2(X := ∏ c : C, D ⟦ J c, F c ⟧)(Y := ∏ c d : C, D ⟦ J c, F d ⟧ → D ⟦ F c, F d ⟧))). }
apply (bind_relmonadmor_eq_from_relmonadmor_iso H J α).
- exact (pr2 H).
- set (Hob := pr1_pr1_relmonadmor_eq_from_relmonadmor_iso H J α).
apply (total2_paths_f Hob).
apply dirprodeq.
+ intermediate_path (transportf (λ F : C → D, ∏ c : C, D ⟦ J c, F c ⟧) Hob
(pr1 (pr2 (pr1 R)))).
{ apply pathsinv0. apply (transport_map (fun F => dirprod_pr1(X := ∏ c : C, D ⟦ J c, F c ⟧)(Y := ∏ c d : C, D ⟦ J c, F d ⟧ → D ⟦ F c, F d ⟧))). }
apply (η_relmonadmor_eq_from_relmonadmor_iso H J α).
+ intermediate_path (transportf (λ F : C → D, ∏ c d : C, D ⟦ J c, F d ⟧ → D ⟦ F c, F d ⟧) Hob
(pr2 (pr2 (pr1 R)))).
{ apply pathsinv0. apply (transport_map (fun F => dirprod_pr2(X := ∏ c : C, D ⟦ J c, F c ⟧)(Y := ∏ c d : C, D ⟦ J c, F d ⟧ → D ⟦ F c, F d ⟧))). }
apply (bind_relmonadmor_eq_from_relmonadmor_iso H J α).
Definition relmonad_eq_from_relmonad_iso_obsolete {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(α : iso(C := precategory_RelMonad (pr2 H) J) R R')
: R = R'.
Show proof.
set (Hob := pr1_pr1_relmonadmor_eq_from_relmonadmor_iso H J α).
assert (η_eq := η_relmonadmor_eq_from_relmonadmor_iso H J α).
assert (bind_eq := bind_relmonadmor_eq_from_relmonadmor_iso H J α).
fold Hob in η_eq, bind_eq.
induction R as [[F [e b]] a].
induction R' as [[F' [e' b']] a'].
simpl in Hob.
induction Hob.
cbn in η_eq, bind_eq.
apply RelMonad_eq.
+ exact (pr2 H).
+ simpl.
apply maponpaths.
apply pathsdirprod; assumption.
assert (η_eq := η_relmonadmor_eq_from_relmonadmor_iso H J α).
assert (bind_eq := bind_relmonadmor_eq_from_relmonadmor_iso H J α).
fold Hob in η_eq, bind_eq.
induction R as [[F [e b]] a].
induction R' as [[F' [e' b']] a'].
simpl in Hob.
induction Hob.
cbn in η_eq, bind_eq.
apply RelMonad_eq.
+ exact (pr2 H).
+ simpl.
apply maponpaths.
apply pathsdirprod; assumption.
Lemma relmonad_eq_from_relmonad_iso_idtoiso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J} (p: R = R') :
relmonad_eq_from_relmonad_iso H J (idtoiso(C := precategory_RelMonad (pr2 H) J) p) = p.
Show proof.
apply relmonad_eq_eq_from_relmonad_ob_eq.
- apply (pr2 H).
- unfold relmonad_eq_from_relmonad_iso.
unfold RelMonad_eq.
rewrite base_total2_paths.
rewrite base_total2_paths.
apply pr1_pr1_relmonadmor_eq_from_relmonadmor_iso_idtoiso.
- apply (pr2 H).
- unfold relmonad_eq_from_relmonad_iso.
unfold RelMonad_eq.
rewrite base_total2_paths.
rewrite base_total2_paths.
apply pr1_pr1_relmonadmor_eq_from_relmonadmor_iso_idtoiso.
Lemma idtoiso_relmonad_eq_from_relmonad_iso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D) {R R': RelMonad J}
(α : iso(C := precategory_RelMonad (pr2 H) J) R R') :
idtoiso(C := precategory_RelMonad (pr2 H) J) (relmonad_eq_from_relmonad_iso H J α) = α.
Show proof.
apply eq_iso.
apply RelMonadMor_equiv.
- exact (pr2 H).
-
apply funextsec; intro c.
etrans.
{ apply iso_from_is_relmonadmor_iso_p. }
rewrite (iso_from_is_relmonadmor_iso_idtoiso (pr2 H) J (relmonad_eq_from_relmonad_iso H J α)).
rewrite idtoiso_functorcat_compute_pointwise'.
unfold relmonad_eq_from_relmonad_iso.
unfold RelMonad_eq.
rewrite pr1_pr1_relmonadmor_eq_from_relmonadmor_iso_idtoiso_aux; try apply (pr2 H).
rewrite base_total2_paths.
rewrite base_total2_paths.
intermediate_path (pr1 (idtoiso
(isotoid D H (functor_iso_pointwise_if_iso' C D (pr2 H) _ _ (iso_from_is_relmonadmor_iso (pr2 H) J α) c)))).
2: { rewrite idtoiso_isotoid.
apply idpath.
}
apply maponpaths.
apply maponpaths.
apply isotoid_functorcat_pointwise.
apply RelMonadMor_equiv.
- exact (pr2 H).
-
apply funextsec; intro c.
etrans.
{ apply iso_from_is_relmonadmor_iso_p. }
rewrite (iso_from_is_relmonadmor_iso_idtoiso (pr2 H) J (relmonad_eq_from_relmonad_iso H J α)).
rewrite idtoiso_functorcat_compute_pointwise'.
unfold relmonad_eq_from_relmonad_iso.
unfold RelMonad_eq.
rewrite pr1_pr1_relmonadmor_eq_from_relmonadmor_iso_idtoiso_aux; try apply (pr2 H).
rewrite base_total2_paths.
rewrite base_total2_paths.
intermediate_path (pr1 (idtoiso
(isotoid D H (functor_iso_pointwise_if_iso' C D (pr2 H) _ _ (iso_from_is_relmonadmor_iso (pr2 H) J α) c)))).
2: { rewrite idtoiso_isotoid.
apply idpath.
}
apply maponpaths.
apply maponpaths.
apply isotoid_functorcat_pointwise.
Definition relmonadmor_idtoiso {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D)(R R': RelMonad J) :
(R = R') ≃ iso(C := precategory_RelMonad (pr2 H) J) R R'.
Show proof.
apply (weqpair (@idtoiso (precategory_RelMonad (pr2 H) J) R R')).
use isweq_iso.
- exact (relmonad_eq_from_relmonad_iso H J).
- intro p. exact (relmonad_eq_from_relmonad_iso_idtoiso H J p).
- intro α. exact (idtoiso_relmonad_eq_from_relmonad_iso H J α).
use isweq_iso.
- exact (relmonad_eq_from_relmonad_iso H J).
- intro p. exact (relmonad_eq_from_relmonad_iso_idtoiso H J p).
- intro α. exact (idtoiso_relmonad_eq_from_relmonad_iso H J α).
Lemma isweq_idtoiso_RelMonad {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D)(R R': RelMonad J)
: isweq (@idtoiso (precategory_RelMonad (pr2 H) J) R R').
Show proof.
apply (isweqhomot (relmonadmor_idtoiso H J R R')).
- intro p. induction p.
apply idpath.
- apply (pr2 _ ).
- intro p. induction p.
apply idpath.
- apply (pr2 _ ).
Lemma is_univalent_RelMonad {C : precategory_data} {D : precategory}
(H: is_univalent D) (J : functor C D)(R R': RelMonad J)
: is_univalent (precategory_RelMonad (pr2 H) J).
Show proof.
split.
- apply isweq_idtoiso_RelMonad.
- clear R R'. intros R R'.
apply (has_homsets_RelMonad (pr2 H) J).
- apply isweq_idtoiso_RelMonad.
- clear R R'. intros R R'.
apply (has_homsets_RelMonad (pr2 H) J).
End RelativeMonads_saturated.
Section RelAdjunctionWithKleisliCategory.
The canonical relative adjunction between J and the Kleisli category of a J-relative monad
This is the obvious generalization of the material in UniMath.CategoryTheory.Monads.KleisliCategory.
Definition Left_rKleisli_functor_data {C: precategory_data} {D : precategory}
{J : functor_data C D} (R: RelMonad J) :
functor_data C (Kleisli_precat R).
Show proof.
use mk_functor_data.
- apply idfun.
- intros a b f; unfold idfun.
exact (#J f · (r_eta R) b).
- apply idfun.
- intros a b f; unfold idfun.
exact (#J f · (r_eta R) b).
Lemma Left_rKleisli_is_functor {C: precategory_data} {D : precategory}
{J : functor C D} (R: RelMonad J) :
is_functor (Left_rKleisli_functor_data R).
Show proof.
split.
- intro a.
unfold Left_rKleisli_functor_data; simpl.
rewrite functor_id.
apply id_left.
- intros a b c f g.
unfold Left_rKleisli_functor_data; simpl.
unfold compose at 3; simpl.
rewrite functor_comp.
do 2 (rewrite <- assoc).
apply cancel_precomposition.
apply pathsinv0.
apply (r_eta_r_bind R).
- intro a.
unfold Left_rKleisli_functor_data; simpl.
rewrite functor_id.
apply id_left.
- intros a b c f g.
unfold Left_rKleisli_functor_data; simpl.
unfold compose at 3; simpl.
rewrite functor_comp.
do 2 (rewrite <- assoc).
apply cancel_precomposition.
apply pathsinv0.
apply (r_eta_r_bind R).
Definition Left_rKleisli_functor {C: precategory_data} {D : precategory}
{J : functor C D} (R: RelMonad J) :
functor C (Kleisli_precat R)
:= (Left_rKleisli_functor_data R,,Left_rKleisli_is_functor R).
Definition Right_rKleisli_functor_data {C: precategory_data} {D : precategory}
{J : functor_data C D} (R: RelMonad J):
functor_data (Kleisli_precat R) D.
Show proof.
use mk_functor_data.
- exact R.
- intros a b.
apply r_bind.
- exact R.
- intros a b.
apply r_bind.
Lemma Right_rKleisli_is_functor {C: precategory_data} {D : precategory}
{J : functor C D} (R: RelMonad J) :
is_functor (Right_rKleisli_functor_data R).
Show proof.
use tpair.
- intro a.
unfold Right_rKleisli_functor_data; unfold identity;
unfold functor_on_morphisms; simpl.
apply (r_bind_r_eta R).
- intros a b c f g; simpl.
apply pathsinv0.
apply (r_bind_r_bind R).
- intro a.
unfold Right_rKleisli_functor_data; unfold identity;
unfold functor_on_morphisms; simpl.
apply (r_bind_r_eta R).
- intros a b c f g; simpl.
apply pathsinv0.
apply (r_bind_r_bind R).
Definition Right_rKleisli_functor {C: precategory_data} {D : precategory}
{J : functor C D} (R: RelMonad J) :
functor (Kleisli_precat R) D
:= (Right_rKleisli_functor_data R,,Right_rKleisli_is_functor R).
Composition of the left and right Kleisli functors is equal to R as a functor
Definition rKleisli_functor_left_right_compose {C: precategory} {D : precategory}
(hs : has_homsets D) {J : functor C D} (R: RelMonad J) :
(Left_rKleisli_functor R) ∙ (Right_rKleisli_functor R) = R_functor R.
Show proof.
use functor_eq.
- exact hs.
- use functor_data_eq_from_nat_trans.
+ intro a; apply idpath.
+ intros a b f; simpl.
rewrite id_right.
rewrite id_left.
apply idpath.
- exact hs.
- use functor_data_eq_from_nat_trans.
+ intro a; apply idpath.
+ intros a b f; simpl.
rewrite id_right.
rewrite id_left.
apply idpath.
Showing that these functors are relative adjoints
Definition rKleisli_functors_are_relative_adjoints {C: precategory_data} {D : precategory}
(hs : has_homsets D) {J : functor C D} (R: RelMonad J) :
are_relative_adjoints J (Left_rKleisli_functor R) (Right_rKleisli_functor R).
Show proof.
use tpair.
- intros a b.
use tpair.
+ simpl.
apply idfun.
+ simpl.
apply idisweq.
- unfold idfun; split.
+ intros a b f c h.
simpl.
unfold compose at 1; simpl.
rewrite <- assoc.
apply cancel_precomposition.
apply (r_eta_r_bind R).
+ intros a b f c k; simpl.
reflexivity.
- intros a b.
use tpair.
+ simpl.
apply idfun.
+ simpl.
apply idisweq.
- unfold idfun; split.
+ intros a b f c h.
simpl.
unfold compose at 1; simpl.
rewrite <- assoc.
apply cancel_precomposition.
apply (r_eta_r_bind R).
+ intros a b f c k; simpl.
reflexivity.
End RelAdjunctionWithKleisliCategory.