FreydCategories: add strictness requirement for unit object

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

diff --git a/src/FreydCategories.v b/src/FreydCategories.v
index c50e732..8b921df 100644 (file)
--- a/src/FreydCategories.v
+++ b/src/FreydCategories.v
@@ -3,8 +3,7 @@
 (*******************************************************************************)
 
 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.
@@ -14,25 +13,29 @@ Require Import Subcategories_ch7_1.
 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