(*******************************************************************************)
Generalizable All Variables.
-Require Import Preamble.
-Require Import General.
+Require Import Notations.
Require Import Categories_ch1_3.
Require Import Functors_ch1_4.
Require Import Isomorphisms_ch1_5.
Require Import NaturalTransformations_ch7_4.
Require Import NaturalIsomorphisms_ch7_5.
Require Import Coherence_ch7_8.
+Require Import BinoidalCategories.
+Require Import PreMonoidalCategories.
+Require Import PreMonoidalCenter.
Require Import MonoidalCategories_ch7_8.
(* A Freyd Category is... *)
Class FreydCategory
- (* a cartesian category C *)
- `(C:CartesianCat(Ob:=Ob)(C:=C1)(Fobj:=fun x => bobj (fst_obj _ _ x) (snd_obj _ _ x)))
+ (* a cartesian category C *)
+ `(C:CartesianCat(Ob:=Ob)(C:=C1)(bin_obj':=bobj))
- (* a premonoidal category K with the same objects (its unit object is 1 because the functor is both IIO and strict) *)
- `(K:PreMonoidalCat(Ob:=Ob)(bin_obj':=bobj)(I:=@one _ _ _ C))
+ (* a premonoidal category K with the same objects (its unit object is 1 because the functor is both IIO and strict) *)
+ `(K:PreMonoidalCat(Ob:=Ob)(bin_obj':=bobj)(I:=pmon_I C))
- (* an identity-on-objects functor J:C->Z(K) *)
-:= { fc_J : Functor C (CenterMonoidal K) (fun x => exist _ x I)
+ (* an identity-on-objects functor J:C->Z(K) *)
+:= { fc_J : Functor C (Center_is_Monoidal K) (fun x => x)
- (* which is not only monoidal *)
- ; fc_mf_J : MonoidalFunctor C (CenterMonoidal K) fc_J
+ (* which is a monoidal functor *)
+ ; fc_mf_J : MonoidalFunctor C (Center_is_Monoidal K) fc_J
- (* but in fact strict (meaning the functor's coherence maps are identities) *)
- ; fc_strict : forall a, iso_forward (mf_coherence fc_mf_J a) ~~ id _
+ (* and strict (meaning the functor's coherence maps are identities) *)
+ ; fc_strict_first : forall a b, #(mf_first(PreMonoidalFunctor:=fc_mf_J) a b) ~~ id _ (* mf_consistent gives us mf_second *)
+ ; fc_strict_id : #(mf_i(PreMonoidalFunctor:=fc_mf_J)) ~~ id _
; fc_C := C
; fc_K := K