(*********************************************************************************************************************************) (* ReificationsEquivalentToGeneralizedArrows: *) (* *) (* The category of generalized arrows and the category of reifications are equivalent categories. *) (* *) (*********************************************************************************************************************************) Generalizable All Variables. Require Import Preamble. Require Import General. Require Import Categories_ch1_3. Require Import Functors_ch1_4. Require Import Isomorphisms_ch1_5. Require Import ProductCategories_ch1_6_1. Require Import OppositeCategories_ch1_6_2. Require Import Enrichment_ch2_8. Require Import Subcategories_ch7_1. Require Import NaturalTransformations_ch7_4. Require Import NaturalIsomorphisms_ch7_5. 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 GeneralizedArrowFromReification. Require Import ReificationFromGeneralizedArrow. Require Import ReificationCategory. Require Import GeneralizedArrowCategory. Section ReificationsEquivalentToGeneralizedArrows. Ltac if_transitive := match goal with [ |- ?A ≃ ?B ] => refine (@if_comp _ _ _ _ _ _ _ A _ _ _ B _ _) end. Lemma roundtrip_lemma' `{C:Category}`{D:Category}`{E:Category} {Gobj}(G:Functor E D Gobj) G_full G_faithful {Fobj}(F:Functor C (FullImage G) Fobj) : ((F >>>> ff_functor_section_functor G G_full G_faithful) >>>> G) ≃ (F >>>> InclusionFunctor _ _). if_transitive. apply (if_associativity F (ff_functor_section_functor G _ _) G). apply if_respects. apply if_id. if_transitive; [ idtac | apply if_left_identity ]. apply (if_comp(F2:=(ff_functor_section_functor G G_full G_faithful) >>>> RestrictToImage G >>>> InclusionFunctor _ _)). apply if_inv. apply (if_associativity (ff_functor_section_functor G G_full G_faithful) (RestrictToImage G) (InclusionFunctor D (FullImage G))). apply if_respects. apply ff_functor_section_splits_niso. apply if_id. Qed. Definition roundtrip_lemma (K:SurjectiveEnrichment) (C:MonicMonoidalEnrichment) (reification : Reification K C (me_i C)) := roundtrip_lemma' (RepresentableFunctor C (me_i C)) (ffme_mf_full C) (ffme_mf_faithful C) (step1_functor K C reification). Lemma roundtrip_reification_to_reification (K:SurjectiveEnrichment) (C:MonicMonoidalEnrichment) (reification : Reification K C (me_i C)) : reification ≃ reification_from_garrow K C (garrow_from_reification K C reification). simpl. unfold mon_f. unfold garrow_functor. apply (if_comp(F2:=(step1_functor K C reification >>>> InclusionFunctor _ (FullImage (RepresentableFunctor C (me_i C)))))). apply step1_niso. apply (if_inv (roundtrip_lemma K C reification)). Qed. (* FIXME: show that the R-functors are naturally isomorphic as well; should follow pretty easily from the proof for Rstar *) Lemma roundtrip_garrow_to_garrow (K:SurjectiveEnrichment) (C:MonicMonoidalEnrichment) (garrow : GeneralizedArrow K C) : garrow ≃ garrow_from_reification K C (reification_from_garrow K C garrow). apply (ffc_functor_weakly_monic _ (ffme_mf_conservative C)). apply if_inv. apply (if_comp(F2:=(step1_functor K C (reification_from_garrow K C garrow) >>>> InclusionFunctor _ (FullImage (RepresentableFunctor C (me_i C)))))). unfold mf_f. unfold garrow_from_reification. unfold garrow_functor. unfold step2_functor. apply roundtrip_lemma. apply if_inv. apply (step1_niso K C (reification_from_garrow K C garrow)). Qed. Theorem ReificationsAreGArrows : IsomorphicCategories CategoryOfReifications CategoryOfGeneralizedArrows. admit. Qed. End ReificationsEquivalentToGeneralizedArrows.