Require Import Subcategories_ch7_1.
Require Import NaturalTransformations_ch7_4.
Require Import NaturalIsomorphisms_ch7_5.
+Require Import PreMonoidalCategories.
Require Import MonoidalCategories_ch7_8.
Require Import Coherence_ch7_8.
-Require Import Enrichment_ch2_8.
Require Import RepresentableStructure_ch7_2.
Require Import Reification.
Require Import GeneralizedArrow.
Require Import ReificationCategory.
Require Import ReificationsAndGeneralizedArrows.
Require Import WeakFunctorCategory.
+Require Import BijectionLemma.
+Require Import Enrichments.
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} :
+ Definition M1 {c1 c2 : SMMEs} :
(c1 ~~{ MorphismsOfCategoryOfGeneralizedArrows }~~> c2) ->
(c1 ~~{ MorphismsOfCategoryOfReifications }~~> c2).
intro GA.
destruct GA; [ apply roi_id | idtac ].
apply roi_reif.
- apply reification_from_garrow.
- apply g.
+ apply (reification_from_garrow s1 s2 g).
Defined.
(* I tried really hard to avoid this *)
apply if_id.
destruct g; simpl.
apply if_id.
- unfold mf_f; simpl.
+ simpl.
apply (if_associativity
- ((ga_functor g0 >>>> HomFunctor s0 (mon_i s0))) (ga_functor g) (HomFunctor s2 (me_i s2))).
+ ((ga_functor g0 >>>> HomFunctor s0 (enr_c_i s0))) (ga_functor g) (HomFunctor s2 (enr_c_i s2))).
Defined.
- Definition M2 (c1 c2 : SmallSMMEs.SMMEs) :
+ Definition M2 (c1 c2 : 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).
+ apply (garrow_from_reification s1 (smme_mee s2) s2 r).
Defined.
+ Lemma eqv1 a b (f : a ~~{ MorphismsOfCategoryOfGeneralizedArrows }~~> b)
+ (f' : a ~~{ MorphismsOfCategoryOfGeneralizedArrows }~~> b)
+ (H : generalizedArrowOrIdentityFunc a b f ≃ generalizedArrowOrIdentityFunc a b f') :
+ generalizedArrowOrIdentityFunc a b (M2 a b (M1 f)) ≃ generalizedArrowOrIdentityFunc a b f'.
+ unfold hom in *.
+ set (@roundtrip_garrow_to_garrow a b (smme_mee b) (smme_mon b)) as q.
+ destruct f; simpl in *.
+ apply H.
+ apply if_inv.
+ apply (if_comp (if_inv H)).
+ clear H.
+ apply (if_respects
+ (ga_functor g)
+ (garrow_functor s1 (smme_mee s2) s2 (reification_from_garrow s1 s2 g))
+ (HomFunctor (senr_c s2) (senr_c_i s2))
+ (HomFunctor (senr_c s2) (senr_c_i s2))
+ ).
+ apply q.
+ apply if_id.
+ Qed.
+
+ Lemma eqv2 a b (f : a ~~{ MorphismsOfCategoryOfReifications }~~> b)
+ (f' : a ~~{ MorphismsOfCategoryOfReifications }~~> b)
+ (H : reificationOrIdentityFunc a b f ≃ reificationOrIdentityFunc a b f') :
+ reificationOrIdentityFunc _ _ (M1 (M2 _ _ f)) ≃ reificationOrIdentityFunc _ _ f'.
+ unfold hom in *.
+ set (@roundtrip_reification_to_reification a b (smme_mee b) (smme_mon b)) as q.
+ destruct f; simpl.
+ apply H.
+ apply if_inv.
+ apply (if_comp (if_inv H)).
+ clear H.
+ simpl.
+ simpl in q.
+ simpl in q.
+ apply q.
+ Qed.
+
Lemma M2_respects :
forall a b (f f':a~~{MorphismsOfCategoryOfReifications}~~>b),
f ~~ f' ->
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).
+ 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.
+ eapply if_comp.
+ apply (step1_niso smme (smme_mee smme) (smme_mon smme) x).
+ apply if_inv.
+ apply if_inv.
+ eapply if_comp.
+ apply (if_associativity (RestrictToImage x) (R' smme smme x) (FullImage_InclusionFunctor _)).
+ apply if_inv.
+ eapply if_comp.
+ apply (if_associativity (RestrictToImage x) ((R' smme smme x >>>>
+ ff_functor_section_functor me_homfunctor me_full me_faithful))
+ (HomFunctor (senr_c smme) (senr_c_i smme))).
+ apply (if_respects (RestrictToImage x) (RestrictToImage x)).
+ apply (if_id (RestrictToImage x)).
+ unfold mf_F.
+ eapply if_comp.
+ apply (if_associativity (R' smme smme x) (ff_functor_section_functor me_homfunctor me_full me_faithful)
+ (HomFunctor (senr_c smme) (senr_c_i smme))).
+ set (roundtrip_lemma (me_full(MonicEnrichment:=smme_mee smme))) as q.
+ set (R' smme smme x) as f.
+ set (me_faithful(MonicEnrichment:=smme_mee smme)) as ff.
+ unfold HomFunctor_fullimage in f.
+ unfold mf_F in f.
+ set (q ff _ _ (FullImage x) _ f) as q'.
+ unfold me_homfunctor in q'.
+ exact q'.
+
simpl in *.
destruct f'; simpl in *.
- apply if_inv.
+ simpl in *.
apply if_inv in H.
- apply (if_comp H).
+ eapply if_comp; [ idtac | eapply if_inv; apply 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).
+ eapply if_comp.
+ apply (step1_niso smme (smme_mee smme) (smme_mon smme) r).
+
+ rename r into x.
+ apply if_inv.
+ eapply if_comp.
+ apply (if_associativity (RestrictToImage x) ((R' smme smme x >>>>
+ ff_functor_section_functor me_homfunctor me_full me_faithful))
+ (HomFunctor (senr_c smme) (senr_c_i smme))).
+ apply if_inv.
+ eapply if_comp.
+ apply (if_associativity (RestrictToImage x) (R' smme smme x) (FullImage_InclusionFunctor _)).
+ apply (if_respects (RestrictToImage x) (RestrictToImage x)).
+ apply (if_id (RestrictToImage x)).
+
+ unfold mf_F.
+ apply if_inv.
+
+ eapply if_comp.
+ apply (if_associativity (R' smme smme x) (ff_functor_section_functor me_homfunctor me_full me_faithful)
+ (HomFunctor (senr_c smme) (senr_c_i smme))).
+ set (roundtrip_lemma (me_full(MonicEnrichment:=smme_mee smme))) as q.
+ set (R' smme smme x) as f.
+ set (me_faithful(MonicEnrichment:=smme_mee smme)) as ff.
+ unfold HomFunctor_fullimage in f.
+ unfold mf_F in f.
+ set (q ff _ _ (FullImage x) _ f) as q'.
+ unfold me_homfunctor in q'.
+ exact q'.
+
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)).
+ set (step1_niso s1 (smme_mee s2) s2 r) as q.
+ apply if_inv in q.
+ eapply if_comp.
+ eapply if_comp; [ idtac | apply q ].
+
+ eapply if_comp.
+ apply (if_associativity
+ (RestrictToImage r)
+ (R' s1 s2 r >>>> ff_functor_section_functor me_homfunctor me_full me_faithful)
+ (HomFunctor (senr_c s2) (senr_c_i s2))).
+ apply if_inv.
+ eapply if_comp.
+ apply (if_associativity
+ (RestrictToImage r)
+ (R' s1 s2 r)
+ (FullImage_InclusionFunctor _)).
+ apply (if_respects
+ (RestrictToImage r)
+ (RestrictToImage r)
+ (R' s1 s2 r >>>> FullImage_InclusionFunctor _)
+ (((R' s1 s2 r >>>> ff_functor_section_functor me_homfunctor me_full me_faithful) >>>>
+ HomFunctor (senr_c s2) (senr_c_i s2)))).
+ apply (if_id _).
+ apply if_inv.
+ eapply if_comp.
+ apply (if_associativity
+ (R' s1 s2 r)
+ (ff_functor_section_functor me_homfunctor me_full me_faithful)
+ (HomFunctor (senr_c s2) (senr_c_i s2))).
+ apply roundtrip_lemma.
+
+ apply if_inv.
+ set (step1_niso s1 (smme_mee s2) s2 r0) as q'.
+ apply if_inv in q'.
+ eapply if_comp.
+ eapply if_comp; [ idtac | apply q' ].
+ eapply if_comp.
+ apply (if_associativity
+ (RestrictToImage r0)
+ (R' s1 s2 r0 >>>> ff_functor_section_functor me_homfunctor me_full me_faithful)
+ (HomFunctor (senr_c s2) (senr_c_i s2))).
apply if_inv.
- apply (if_comp (@roundtrip_lemma _ s1 _ _ s2 r0)).
- apply yy'.
+ eapply if_comp.
+ apply (if_associativity
+ (RestrictToImage r0)
+ (R' s1 s2 r0)
+ (FullImage_InclusionFunctor _)).
+ apply (if_respects
+ (RestrictToImage r0)
+ (RestrictToImage r0)
+ (R' s1 s2 r0 >>>> FullImage_InclusionFunctor _)
+ (((R' s1 s2 r0 >>>> ff_functor_section_functor me_homfunctor me_full me_faithful) >>>>
+ HomFunctor (senr_c s2) (senr_c_i s2)))).
+ apply (if_id _).
+ apply if_inv.
+ eapply if_comp.
+ apply (if_associativity
+ (R' s1 s2 r0)
+ (ff_functor_section_functor me_homfunctor me_full me_faithful)
+ (HomFunctor (senr_c s2) (senr_c_i s2))).
+ apply roundtrip_lemma.
+ apply if_inv.
+ apply H.
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.
+ intros; apply (@bijection_lemma _ _ _ _ _ M1_Functor M2); intros.
+ apply M2_respects; auto.
+ unfold fmor; simpl.
+ apply (@eqv1 _ _ f0 f0).
+ apply if_id.
+ unfold fmor; simpl.
+ apply (@eqv2 _ _ f0 f0).
+ apply if_id.
+ 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.
-
+ apply (eqv1 _ _ f f'); auto.
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.
+ apply (eqv2 _ _ f f'); auto.
+ Qed.
End ReificationsIsomorphicToGeneralizedArrows.
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