Require Import ReificationsEquivalentToGeneralizedArrows.
Require Import WeakFunctorCategory.
-
Section ReificationsIsomorphicToGeneralizedArrows.
- Definition M1 (c1 c2 : SmallSMMEs.SMMEs) :
+ Definition M1 {c1 c2 : SmallSMMEs.SMMEs} :
(c1 ~~{ MorphismsOfCategoryOfGeneralizedArrows }~~> c2) ->
(c1 ~~{ MorphismsOfCategoryOfReifications }~~> c2).
intro GA.
apply g.
Defined.
+ Lemma invert_ga : forall (a: SMME)
+ (f:a~~{MorphismsOfCategoryOfGeneralizedArrows}~~>a),
+ (f = gaoi_id _) \/ (exists f', f = gaoi_ga _ _ f').
+ admit.
+ Qed.
+
+ Lemma invert_reif : forall (a: SMME)
+ (f:a~~{MorphismsOfCategoryOfReifications}~~>a),
+ (f = roi_id _) \/ (exists f', f = roi_reif _ _ f').
+ admit.
+ 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 >>>> RepresentableFunctor s0 (mon_i s0))) (ga_functor g) (RepresentableFunctor s2 (me_i s2))).
+ Defined.
+
Definition M2 (c1 c2 : SmallSMMEs.SMMEs) :
(c1 ~~{ MorphismsOfCategoryOfReifications }~~> c2) ->
(c1 ~~{ MorphismsOfCategoryOfGeneralizedArrows }~~> c2).
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.
- * Oh my, this is massively embarrassing. In the paper I claim
- * that Generalized Arrows form a category and Reifications form a
- * category, when in fact they form merely a SEMIcategory (see
- * http://ncatlab.org/nlab/show/semicategory) since we cannot be sure that the identity functor on the
+ 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 >>>> RepresentableFunctor s0 (mon_i s0)) >>>> garrow_functor s0 s2 r0))
+ (F1:=(garrow_functor s1 s2 (compose_reifications s1 s0 s2 r r0)))
+ (G0:=(RepresentableFunctor s2 (mon_i s2)))
+ (G1:=(RepresentableFunctor s2 (mon_i s2))));
+ [ idtac | apply if_id ].
+ admit.
+ Defined.
Theorem ReificationsAreGArrows : IsomorphicCategories CategoryOfGeneralizedArrows CategoryOfReifications.
- apply WeakFunctorCategoriesIsomorphic with (M1:=M1) (M2:=M2).
- destruct g.
- intros.
- simpl.
- simpl in H.
- split f.
- destruct f.
- dependent destruction f.
- intros until g.
- destruct f.
- simpl.
- inversion g.
- destruct f as [ ] _eqn.
- destruct g as [ ] _eqn.
- destruct g.
- subst.
- simpl.
- case g.
- simpl.
- inversion g; subst; intros.
- destruct g.
- Qed.
+ 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:=RepresentableFunctor s2 (mon_i s2))(G1:=RepresentableFunctor 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.