add formalization of Arrows and Freyd Categories

[coq-categories.git] / src / FreydCategories.v

diff --git a/src/FreydCategories.v b/src/FreydCategories.v
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+++ b/src/FreydCategories.v
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+(*******************************************************************************)
+(* Freyd 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 InitialTerminal_ch2_2.
+Require Import Subcategories_ch7_1.
+Require Import NaturalTransformations_ch7_4.
+Require Import NaturalIsomorphisms_ch7_5.
+Require Import Coherence_ch7_8.
+Require Import MonoidalCategories_ch7_8.
+
+Class FreydCategory
+{ Ob                          :  Type                                   }
+{ freyd_bobj                  :  Ob -> Ob -> Ob                         }
+{ HomC                        :  Ob -> Ob -> Type                       }
+{ freyd_C                     :  Category Ob HomC                       }
+{ freyd_C_terminal            :  Terminal    freyd_C                    }
+{ freyd_C_binoidal            :  BinoidalCat freyd_C freyd_bobj         }
+( freyd_C_monoidal            :  PreMonoidalCat freyd_C_binoidal 1         ) :=
+{ freyd_K_hom                 :  Ob -> Ob -> Type
+; freyd_K                     :> Category Ob freyd_K_hom
+; freyd_C_cartesian           :  CartesianCat freyd_C_monoidal
+; freyd_K_binoidal            :> BinoidalCat  freyd_K freyd_bobj
+; freyd_K_monoidal            :> PreMonoidalCat  freyd_K_binoidal (one(Terminal:=freyd_C_terminal))
+; freyd_K_braided             :  BraidedCat   freyd_K_monoidal
+; freyd_K_symmetric           :  SymmetricCat freyd_K_braided
+; freyd_F                     :> PreMonoidalFunctor freyd_C_monoidal freyd_K_monoidal (fun x => x)
+; freyd_F_central             :  forall `(f:a~~{freyd_C}~~>b), CentralMorphism(C:=freyd_K) (freyd_F \ f)
+; freyd_F_strict_first        : forall a b,   #(mf_preserves_first(PreMonoidalFunctor:=freyd_F) a b) ~~ id (bin_first a b)
+; freyd_F_strict_second       : forall a b,   #(mf_preserves_second(PreMonoidalFunctor:=freyd_F) a b) ~~ id (bin_second a b)
+; freyd_F_strict_a            : forall a b c, freyd_F \ #(pmon_assoc freyd_C_monoidal a b c) ~~ #(pmon_assoc freyd_K_monoidal _ _ _)
+; freyd_F_strict_cl           :  forall a,     freyd_F \ #(pmon_cancell freyd_C_monoidal a) ~~ #(pmon_cancell freyd_K_monoidal _)
+; freyd_F_strict_cr           :  forall a,     freyd_F \ #(pmon_cancelr freyd_C_monoidal a) ~~ #(pmon_cancelr freyd_K_monoidal _)
+}.
+Coercion freyd_K_monoidal : FreydCategory >-> PreMonoidalCat.
+Coercion freyd_F          : FreydCategory >-> PreMonoidalFunctor.
+Coercion freyd_K          : FreydCategory >-> Category.