(*********************************************************************************************************************************)
-(* ReificationsEquivalentToGeneralizedArrows: *)
+(* ReificationsIsomorphicToGeneralizedArrows: *)
(* *)
-(* The category of generalized arrows and the category of reifications are equivalent categories. *)
+(* The category of generalized arrows and the category of reifications are isomorphic categories. *)
(* *)
(*********************************************************************************************************************************)
Require Import ReificationFromGeneralizedArrow.
Require Import ReificationCategory.
Require Import GeneralizedArrowCategory.
-Require Import ReificationsEquivalentToGeneralizedArrows.
+Require Import ReificationCategory.
+Require Import ReificationsAndGeneralizedArrows.
Require Import WeakFunctorCategory.
Section ReificationsIsomorphicToGeneralizedArrows.
+ Lemma step1_lemma (s0 s1 s2:SmallSMMEs.SMMEs)(r01:Reification s0 s1 (me_i s1))(r12:Reification s1 s2 (me_i s2)) :
+ (step1_functor s0 s1 r01 >>>>
+ InclusionFunctor (enr_v s1) (FullImage (HomFunctor s1 (me_i s1)))) >>>> step1_functor s1 s2 r12
+ ≃ step1_functor s0 s2 (compose_reifications s0 s1 s2 r01 r12).
+ admit.
+ Defined.
+
+ Definition M1 {c1 c2 : SmallSMMEs.SMMEs} :
+ (c1 ~~{ MorphismsOfCategoryOfGeneralizedArrows }~~> c2) ->
+ (c1 ~~{ MorphismsOfCategoryOfReifications }~~> c2).
+ intro GA.
+ destruct GA; [ apply roi_id | idtac ].
+ apply roi_reif.
+ apply reification_from_garrow.
+ apply g.
+ Defined.
+
+ (* I tried really hard to avoid this *)
+ Require Import Coq.Logic.Eqdep.
+
+ Inductive Heq : forall {A}{B}, A -> B -> Prop :=
+ heq : forall {A} (a:A), Heq a a.
+
+ Lemma invert_ga' : forall (a b: SMME)
+ (f:a~~{MorphismsOfCategoryOfGeneralizedArrows}~~>b), a=b ->
+ (Heq f (gaoi_id a)) \/ (exists f', Heq f (gaoi_ga a b f')).
+ intros.
+ destruct f.
+ left; apply heq.
+ subst; right.
+ exists g.
+ apply heq.
+ Defined.
+
+ Lemma invert_ga : forall (a: SMME)
+ (f:a~~{MorphismsOfCategoryOfGeneralizedArrows}~~>a),
+ (f = gaoi_id _) \/ (exists f', f = gaoi_ga _ _ f').
+ intros.
+ set (invert_ga' a a f (refl_equal a)) as q.
+ destruct q.
+ left.
+ inversion H.
+ apply inj_pairT2 in H2.
+ apply inj_pairT2 in H1.
+ subst; auto.
+ right.
+ destruct H.
+ exists x.
+ inversion H.
+ apply inj_pairT2 in H2.
+ apply inj_pairT2 in H1.
+ subst; auto.
+ Qed.
+
+ Lemma invert_reif' : forall (a b: SMME)
+ (f:a~~{MorphismsOfCategoryOfReifications}~~>b), a=b ->
+ (Heq f (roi_id a)) \/ (exists f', Heq f (roi_reif a b f')).
+ intros.
+ destruct f.
+ left; apply heq.
+ subst; right.
+ exists r.
+ apply heq.
+ Defined.
+
+ Lemma invert_reif : forall (a: SMME)
+ (f:a~~{MorphismsOfCategoryOfReifications}~~>a),
+ (f = roi_id _) \/ (exists f', f = roi_reif _ _ f').
+ intros.
+ set (invert_reif' a a f (refl_equal a)) as q.
+ destruct q.
+ left.
+ inversion H.
+ apply inj_pairT2 in H2.
+ apply inj_pairT2 in H1.
+ subst; auto.
+ right.
+ destruct H.
+ exists x.
+ inversion H.
+ apply inj_pairT2 in H2.
+ apply inj_pairT2 in H1.
+ subst; auto.
+ Qed.
+
+ Definition M1_Functor : Functor MorphismsOfCategoryOfGeneralizedArrows MorphismsOfCategoryOfReifications (fun x => x).
+ refine {| fmor := fun a b f => M1 f |}.
+ intros.
+ unfold hom in *.
+ unfold eqv in *.
+ simpl in *.
+ destruct f.
+ set (invert_ga _ f') as q.
+ destruct q; subst.
+ apply if_id.
+ simpl in *.
+ destruct H0; subst.
+ apply H.
+ simpl in *.
+ destruct f'; simpl in *.
+ apply H.
+ apply H.
+ intros; simpl.
+ apply if_id.
+ intros.
+ simpl.
+ destruct f; simpl.
+ apply if_id.
+ destruct g; simpl.
+ apply if_id.
+ unfold mf_f; simpl.
+ apply (if_associativity
+ ((ga_functor g0 >>>> HomFunctor s0 (mon_i s0))) (ga_functor g) (HomFunctor s2 (me_i s2))).
+ Defined.
+
+ Definition M2 (c1 c2 : SmallSMMEs.SMMEs) :
+ (c1 ~~{ MorphismsOfCategoryOfReifications }~~> c2) ->
+ (c1 ~~{ MorphismsOfCategoryOfGeneralizedArrows }~~> c2).
+ intro RE.
+ destruct RE; [ apply gaoi_id | idtac ].
+ apply gaoi_ga.
+ apply (garrow_from_reification s1 s2 r).
+ Defined.
+
+ Lemma M2_respects :
+ forall a b (f f':a~~{MorphismsOfCategoryOfReifications}~~>b),
+ f ~~ f' ->
+ M2 a b f ~~ M2 a b f'.
+ intros.
+ unfold hom in *.
+ unfold eqv in *.
+ simpl in *.
+ destruct f.
+ set (invert_reif _ f') as q.
+ destruct q; subst.
+ apply if_id.
+ simpl in *.
+ destruct H0; subst.
+ simpl in *.
+ unfold garrow_functor.
+ unfold step2_functor.
+ apply (if_comp H).
+ clear H.
+ apply (if_comp (@step1_niso _ smme _ _ smme x)).
+ apply if_inv.
+ apply (@roundtrip_lemma _ smme _ _ smme x).
+ simpl in *.
+ destruct f'; simpl in *.
+ apply if_inv.
+ apply if_inv in H.
+ apply (if_comp H).
+ clear H.
+ unfold garrow_functor.
+ unfold step2_functor.
+ apply (if_comp (@step1_niso _ smme _ _ smme r)).
+ apply if_inv.
+ apply (@roundtrip_lemma _ smme _ _ smme r).
+ simpl in *.
+ unfold garrow_functor.
+ unfold step2_functor.
+ apply if_inv in H.
+ set (if_comp H (@step1_niso _ s1 _ _ s2 r)) as yy.
+ set (if_comp (if_inv (@step1_niso _ s1 _ _ s2 r0)) yy) as yy'.
+ apply (if_comp (@roundtrip_lemma _ s1 _ _ s2 r)).
+ apply if_inv.
+ apply (if_comp (@roundtrip_lemma _ s1 _ _ s2 r0)).
+ apply yy'.
+ Qed.
+
+ Definition M2_Functor : Functor MorphismsOfCategoryOfReifications MorphismsOfCategoryOfGeneralizedArrows (fun x => x).
+ refine {| fmor := fun a b f => M2 _ _ f |}.
+ apply M2_respects.
+ intros; simpl; apply if_id.
+ intros.
+ simpl.
+ destruct f; simpl.
+ apply if_id.
+ destruct g; simpl.
+ apply if_id.
+ unfold mf_f; simpl.
+ apply (if_respects
+ (F0:=((garrow_functor s1 s0 r >>>> HomFunctor s0 (mon_i s0)) >>>> garrow_functor s0 s2 r0))
+ (F1:=(garrow_functor s1 s2 (compose_reifications s1 s0 s2 r r0)))
+ (G0:=(HomFunctor s2 (mon_i s2)))
+ (G1:=(HomFunctor s2 (mon_i s2))));
+ [ idtac | apply if_id ].
+ eapply if_comp.
+ idtac.
+ apply (if_associativity (garrow_functor s1 s0 r) (HomFunctor s0 (mon_i s0)) (garrow_functor s0 s2 r0)).
+ idtac.
+ unfold garrow_functor at 1.
+ unfold step2_functor at 1.
+ set (roundtrip_lemma'
+ (HomFunctor (enr_c (smme_e s0)) (me_i (smme_mon s0)))
+ (ffme_mf_full (smme_mee s0)) (ffme_mf_faithful (smme_mee s0))
+ (step1_functor (smme_see s1) (smme_mee s0) r)
+ ) as q.
+ set (if_respects
+ (G0:=garrow_functor (smme_see s0) (smme_mee s2) r0)
+ (G1:=garrow_functor (smme_see s0) (smme_mee s2) r0)
+ q (if_id _)) as q'.
+ apply (if_comp q').
+ clear q' q.
+ unfold garrow_functor at 2.
+ unfold garrow_functor at 1.
+ eapply if_comp.
+ eapply if_inv.
+ apply (if_associativity _ (step1_functor s0 s2 r0) (step2_functor s2)).
+ apply (if_respects
+ (G0:=step2_functor s2)
+ (G1:=step2_functor s2)); [ idtac | apply if_id ].
+ apply step1_lemma.
+ Defined.
+
+ Theorem ReificationsAreGArrows : IsomorphicCategories CategoryOfGeneralizedArrows CategoryOfReifications.
+ refine {| ic_f := M1_Functor ; ic_g := M2_Functor |}.
+ unfold EqualFunctors; intros; apply heq_morphisms_intro; unfold eqv in *; simpl in *.
+ unfold hom in *.
+ set (@roundtrip_garrow_to_garrow _ a _ _ b) as q.
+ destruct f; simpl in *.
+ apply H.
+ apply if_inv.
+ apply (if_comp (if_inv H)).
+ clear H.
+ unfold mf_f in q.
+ apply (if_respects(F0:=ga_functor g)(F1:=garrow_functor s1 s2 (reification_from_garrow s1 s2 g))
+ (G0:=HomFunctor s2 (mon_i s2))(G1:=HomFunctor s2 (mon_i s2))).
+ apply q.
+ apply if_id.
+
+ unfold EqualFunctors; intros; apply heq_morphisms_intro; unfold eqv in *; simpl in *.
+ unfold hom in *.
+ set (@roundtrip_reification_to_reification _ a _ _ b) as q.
+ destruct f; simpl.
+ apply H.
+ apply if_inv.
+ apply (if_comp (if_inv H)).
+ clear H.
+ simpl.
+ unfold mf_f; simpl.
+ simpl in q.
+ unfold mf_f in q.
+ simpl in q.
+ apply q.
+ Qed.
+
End ReificationsIsomorphicToGeneralizedArrows.
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