add MonoidalFullSubcategory_Monoidal

[coq-categories.git] / src / MonoidalCategories_ch7_8.v

diff --git a/src/MonoidalCategories_ch7_8.v b/src/MonoidalCategories_ch7_8.v
index 8355319..733dc5f 100644 (file)
--- a/src/MonoidalCategories_ch7_8.v
+++ b/src/MonoidalCategories_ch7_8.v
@@ -1,5 +1,5 @@
 Generalizable All Variables.
-Require Import Preamble.
+Require Import Notations.
 Require Import Categories_ch1_3.
 Require Import Functors_ch1_4.
 Require Import Isomorphisms_ch1_5.
@@ -46,6 +46,46 @@ Definition MonoidalFunctorsCompose
    (PMF23 :MonoidalFunctor PM2 PM3 PMFF23)
    := PreMonoidalFunctorsCompose PMF12 PMF23.
 
+Class MonoidalNaturalIsomorphism
+ `{C1:MonoidalCat}`{C2:MonoidalCat}
+ `(F1:!MonoidalFunctor(fobj:=fobj1) C1 C2 Func1)
+ `(F2:!MonoidalFunctor(fobj:=fobj2) C1 C2 Func2) :=
+{ mni_ni        : NaturalIsomorphism F1 F2
+; mni_commutes1 : forall A B,
+                   #(ni_iso (mf_first(PreMonoidalFunctor:=F1) B) A)  >>> #(ni_iso mni_ni (A⊗B)) ~~
+                   #(ni_iso mni_ni _) ⋉ _ >>> _ ⋊ #(ni_iso mni_ni _) >>> #(ni_iso (mf_first(PreMonoidalFunctor:=F2) B) A)
+; mni_commutes2 : forall A B,
+                   #(ni_iso (mf_second(PreMonoidalFunctor:=F1) A) B)  >>> #(ni_iso mni_ni (A⊗B)) ~~
+                   #(ni_iso mni_ni _) ⋉ _ >>> _ ⋊ #(ni_iso mni_ni _) >>> #(ni_iso (mf_second(PreMonoidalFunctor:=F2) A) B)
+}.
+Notation "F <~~⊗~~> G" := (@MonoidalNaturalIsomorphism _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ F _ _ G) : category_scope.
+
+(* an equivalence of categories via monoidal functors, but the natural iso isn't necessarily a monoidal natural iso *)
+Structure MonoidalEquivalence `{C1:MonoidalCat} `{C2:MonoidalCat} :=
+{ meqv_forward_fobj  : C1 -> C2
+; meqv_forward       : Functor C1 C2 meqv_forward_fobj
+; meqv_forward_mon   : PreMonoidalFunctor _ _ meqv_forward
+; meqv_backward_fobj : C2 -> C1
+; meqv_backward      : Functor C2 C1 meqv_backward_fobj
+; meqv_backward_mon  : PreMonoidalFunctor _ _ meqv_backward
+; meqv_comp1         : meqv_forward  >>>> meqv_backward ≃ functor_id _
+; meqv_comp2         : meqv_backward >>>> meqv_forward  ≃ functor_id _
+}.
+
+(* a monoidally-natural equivalence of categories *)
+(*
+Structure MonoidalNaturalEquivalence `{C1:MonoidalCat} `{C2:MonoidalCat} :=
+{ mneqv_forward_fobj  : C1 -> C2
+; mneqv_forward       : Functor C1 C2 mneqv_forward_fobj
+; mneqv_forward_mon   : PreMonoidalFunctor _ _ mneqv_forward
+; mneqv_backward_fobj : C2 -> C1
+; mneqv_backward      : Functor C2 C1 mneqv_backward_fobj
+; mneqv_backward_mon  : PreMonoidalFunctor _ _ mneqv_backward
+; mneqv_comp1         : mneqv_forward_mon  >>⊗>> mneqv_backward_mon <~~⊗~~> premonoidal_id _
+; mneqv_comp2         : mneqv_backward_mon >>⊗>> mneqv_forward_mon  <~~⊗~~> premonoidal_id _
+}.
+*)
+
 Section BinoidalCat_from_Bifunctor.
   Context `{C:Category}{Fobj}(F:Functor (C ×× C) C Fobj).
   Definition BinoidalCat_from_Bifunctor_first (a:C) : Functor C C (fun b => Fobj (pair_obj b a)).
@@ -138,6 +178,28 @@ Section Bifunctor_from_MonoidalCat.
 End Bifunctor_from_MonoidalCat.
 
 
+Instance MonoidalFullSubcategory_Monoidal `(mc:MonoidalCat) {Pobj} Pobj_unit Pobj_closed V
+  : MonoidalCat (PreMonoidalFullSubcategory_PreMonoidal(Pobj:=Pobj) mc V Pobj_unit Pobj_closed).
+  apply Build_MonoidalCat.
+  apply Build_CommutativeCat.
+  intros.
+  idtac.
+  apply Build_CentralMorphism; intros.
+    destruct a.
+    destruct b.
+    destruct c.
+    destruct d.
+    simpl.
+    apply (centralmor_first(CentralMorphism:=commutative_central(CommutativeCat:=mon_commutative(MonoidalCat:=mc)) f)).
+
+    destruct a.
+    destruct b.
+    destruct c.
+    destruct d.
+    simpl.
+    apply (centralmor_first(CentralMorphism:=commutative_central(CommutativeCat:=mon_commutative(MonoidalCat:=mc)) g)).
+    Defined.
+
 Class DiagonalCat `(mc:MonoidalCat) :=
 { copy          :  forall (a:mc),   a~~{mc}~~>(bin_obj(BinoidalCat:=mc) a a)
 ; copy_natural1 :  forall {a}{b}(f:a~~{mc}~~>b)(c:mc), copy _ >>> f ⋉ a >>> b ⋊ f ~~ f >>> copy _
@@ -146,7 +208,7 @@ Class DiagonalCat `(mc:MonoidalCat) :=
 }.
 
 Class CartesianCat `(mc:MonoidalCat) :=
-{ car_terminal  :> TerminalObject mc pmon_I
+{ car_terminal  :> TerminalObject mc (pmon_I mc)
 ; car_diagonal  :  DiagonalCat mc
 ; car_law1      :  forall {a}, id a ~~ (copy(DiagonalCat:=car_diagonal) a) >>> (drop a  ⋉ a) >>> (#(pmon_cancell _))
 ; car_law2      :  forall {a}, id a ~~ (copy(DiagonalCat:=car_diagonal) a) >>> (a ⋊ drop  a) >>> (#(pmon_cancelr _))