src/Reification.v

   1 (*********************************************************************************************************************************)
   2 (* Reification:                                                                                                                  *)
   3 (*                                                                                                                               *)
   4 (*   A reification is a functor R from one enrichING category A to another enrichING category B which forms a commuting square   *)
   5 (*   with every pair of hom-functors Hom(X,-):a->A and Hom(Y,-):b->B up to natural isomorphism.                                  *)
   6 (*                                                                                                                               *)
   7 (*********************************************************************************************************************************)
   8
   9 Generalizable All Variables.
  10 Require Import Preamble.
  11 Require Import General.
  12 Require Import Categories_ch1_3.
  13 Require Import Functors_ch1_4.
  14 Require Import Isomorphisms_ch1_5.
  15 Require Import ProductCategories_ch1_6_1.
  16 Require Import OppositeCategories_ch1_6_2.
  17 Require Import Enrichment_ch2_8.
  18 Require Import Subcategories_ch7_1.
  19 Require Import NaturalTransformations_ch7_4.
  20 Require Import NaturalIsomorphisms_ch7_5.
  21 Require Import MonoidalCategories_ch7_8.
  22 Require Import Coherence_ch7_8.
  23 Require Import Enrichment_ch2_8.
  24 Require Import RepresentableStructure_ch7_2.
  25
  26 Opaque RepresentableFunctor.
  27 Opaque functor_comp.
  28 Structure Reification (K:Enrichment) (C:Enrichment) (CI:C) :=
  29 { reification_r_obj     : K -> K -> C
  30 ; reification_rstar_obj : enr_v K -> enr_v C
  31 ; reification_r         : forall k:K, Functor K C (reification_r_obj k)
  32 ; reification_rstar_f   :             Functor (enr_v K) (enr_v C) reification_rstar_obj
  33 ; reification_rstar     :             MonoidalFunctor (enr_v_mon K) (enr_v_mon C) reification_rstar_f
  34 ; reification_commutes  : forall k, reification_r k >>>> RepresentableFunctor C CI <~~~> RepresentableFunctor K k >>>> reification_rstar_f
  35 }.
  36 Transparent RepresentableFunctor.
  37 Transparent functor_comp.
  38
  39 Coercion reification_rstar : Reification >-> MonoidalFunctor.
  40 Implicit Arguments Reification                [ ].
  41 Implicit Arguments reification_r_obj          [ K C CI ].
  42 Implicit Arguments reification_r              [ K C CI ].
  43 Implicit Arguments reification_rstar          [ K C CI ].
  44 Implicit Arguments reification_rstar_f        [ K C CI ].
  45 Implicit Arguments reification_commutes       [ K C CI ].
  46 Implicit Arguments reification_rstar_obj      [ K C CI ].