Require Import NaturalTransformations_ch7_4.
Require Import NaturalIsomorphisms_ch7_5.
Require Import Coherence_ch7_8.
+Require Import BinoidalCategories.
+Require Import PreMonoidalCategories.
(******************************************************************************)
-(* Chapter 7.8: (Pre)Monoidal Categories *)
+(* Chapter 7.8: Monoidal Categories *)
(******************************************************************************)
-Class BinoidalCat
-`( C : Category )
-( bin_obj' : C -> C -> C ) :=
-{ bin_obj := bin_obj' where "a ⊗ b" := (bin_obj a b)
-; bin_first : forall a:C, Functor C C (fun x => x⊗a)
-; bin_second : forall a:C, Functor C C (fun x => a⊗x)
-; bin_c := C
-}.
-Coercion bin_c : BinoidalCat >-> Category.
-Notation "a ⊗ b" := (@bin_obj _ _ _ _ _ a b) : category_scope.
-Notation "C ⋊ f" := (@fmor _ _ _ _ _ _ _ (@bin_second _ _ _ _ _ C) _ _ f) : category_scope.
-Notation "g ⋉ C" := (@fmor _ _ _ _ _ _ _ (@bin_first _ _ _ _ _ C) _ _ g) : category_scope.
-Notation "C ⋊ -" := (@bin_second _ _ _ _ _ C) : category_scope.
-Notation "- ⋉ C" := (@bin_first _ _ _ _ _ C) : category_scope.
-
-Class CentralMorphism `{BinoidalCat}`(f:a~>b) : Prop :=
-{ centralmor_first : forall `(g:c~>d), (f ⋉ _ >>> _ ⋊ g) ~~ (_ ⋊ g >>> f ⋉ _)
-; centralmor_second : forall `(g:c~>d), (g ⋉ _ >>> _ ⋊ f) ~~ (_ ⋊ f >>> g ⋉ _)
-}.
-
-(* the central morphisms of a category constitute a subcategory *)
-Definition Center `(bc:BinoidalCat) : SubCategory bc (fun _ => True) (fun _ _ _ _ f => CentralMorphism f).
- apply Build_SubCategory; intros; apply Build_CentralMorphism; intros.
- abstract (setoid_rewrite (fmor_preserves_id(bin_first c));
- setoid_rewrite (fmor_preserves_id(bin_first d));
- setoid_rewrite left_identity; setoid_rewrite right_identity; reflexivity).
- abstract (setoid_rewrite (fmor_preserves_id(bin_second c));
- setoid_rewrite (fmor_preserves_id(bin_second d));
- setoid_rewrite left_identity; setoid_rewrite right_identity; reflexivity).
- abstract (setoid_rewrite <- (fmor_preserves_comp(bin_first c0));
- setoid_rewrite associativity;
- setoid_rewrite centralmor_first;
- setoid_rewrite <- associativity;
- setoid_rewrite centralmor_first;
- setoid_rewrite associativity;
- setoid_rewrite <- (fmor_preserves_comp(bin_first d));
- reflexivity).
- abstract (setoid_rewrite <- (fmor_preserves_comp(bin_second d));
- setoid_rewrite <- associativity;
- setoid_rewrite centralmor_second;
- setoid_rewrite associativity;
- setoid_rewrite centralmor_second;
- setoid_rewrite <- associativity;
- setoid_rewrite <- (fmor_preserves_comp(bin_second c0));
- reflexivity).
- Qed.
-
-Class CommutativeCat `(BinoidalCat) :=
-{ commutative_central : forall `(f:a~>b), CentralMorphism f
-; commutative_morprod := fun `(f:a~>b)`(g:a~>b) => f ⋉ _ >>> _ ⋊ g
-}.
-Notation "f × g" := (commutative_morprod f g).
-
-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)).
- apply Build_Functor with (fmor:=(fun a0 b (f:a0~~{C}~~>b) =>
- @fmor _ _ _ _ _ _ _ F (pair_obj a0 a) (pair_obj b a) (pair_mor (pair_obj a0 a) (pair_obj b a) f (id a)))); intros; simpl;
- [ abstract (set (fmor_respects(F)) as q; apply q; split; simpl; auto)
- | abstract (set (fmor_preserves_id(F)) as q; apply q)
- | abstract (etransitivity;
- [ set (@fmor_preserves_comp _ _ _ _ _ _ _ F) as q; apply q
- | set (fmor_respects(F)) as q; apply q ];
- split; simpl; auto) ].
- Defined.
- Definition BinoidalCat_from_Bifunctor_second (a:C) : Functor C C (fun b => Fobj (pair_obj a b)).
- apply Build_Functor with (fmor:=(fun a0 b (f:a0~~{C}~~>b) =>
- @fmor _ _ _ _ _ _ _ F (pair_obj a a0) (pair_obj a b) (pair_mor (pair_obj a a0) (pair_obj a b) (id a) f))); intros;
- [ abstract (set (fmor_respects(F)) as q; apply q; split; simpl; auto)
- | abstract (set (fmor_preserves_id(F)) as q; apply q)
- | abstract (etransitivity;
- [ set (@fmor_preserves_comp _ _ _ _ _ _ _ F) as q; apply q
- | set (fmor_respects(F)) as q; apply q ];
- split; simpl; auto) ].
- Defined.
-
- Definition BinoidalCat_from_Bifunctor : BinoidalCat C (fun a b => Fobj (pair_obj a b)).
- refine {| bin_first := BinoidalCat_from_Bifunctor_first
- ; bin_second := BinoidalCat_from_Bifunctor_second
- |}.
- Defined.
-
- Lemma Bifunctors_Create_Commutative_Binoidal_Categories : CommutativeCat BinoidalCat_from_Bifunctor.
- abstract (intros; apply Build_CommutativeCat; intros; apply Build_CentralMorphism; intros; simpl; (
- etransitivity; [ apply (fmor_preserves_comp(F)) | idtac ]; symmetry;
- etransitivity; [ apply (fmor_preserves_comp(F)) | idtac ];
- apply (fmor_respects(F));
- split;
- [ etransitivity; [ apply left_identity | symmetry; apply right_identity ]
- | etransitivity; [ apply right_identity | symmetry; apply left_identity ] ])).
- Defined.
-
-End BinoidalCat_from_Bifunctor.
-
-(* not in Awodey *)
-Class PreMonoidalCat `(bc:BinoidalCat(C:=C))(I:C) :=
-{ pmon_I := I
-; pmon_bin := bc
-; pmon_cat := C
-; pmon_assoc : forall a b, (bin_second a >>>> bin_first b) <~~~> (bin_first b >>>> bin_second a)
-; pmon_cancelr : (bin_first I) <~~~> functor_id C
-; pmon_cancell : (bin_second I) <~~~> functor_id C
-; pmon_pentagon : Pentagon (fun a b c f => f ⋉ c) (fun a b c f => c ⋊ f) (fun a b c => #((pmon_assoc a c) b))
-; pmon_triangle : Triangle (fun a b c f => f ⋉ c) (fun a b c f => c ⋊ f) (fun a b c => #((pmon_assoc a c) b))
- (fun a => #(pmon_cancell a)) (fun a => #(pmon_cancelr a))
-; pmon_assoc_rr : forall a b, (bin_first (a⊗b)) <~~~> (bin_first a >>>> bin_first b)
-; pmon_assoc_ll : forall a b, (bin_second (a⊗b)) <~~~> (bin_second b >>>> bin_second a)
-; pmon_coherent_r : forall a c d:C, #(pmon_assoc_rr c d a) ~~ #(pmon_assoc a d c)⁻¹
-; pmon_coherent_l : forall a c d:C, #(pmon_assoc_ll c a d) ~~ #(pmon_assoc c d a)
-}.
-(*
- * Premonoidal categories actually have three associators (the "f"
- * indicates the position in which the operation is natural:
- *
- * assoc : (A ⋊ f) ⋉ C <-> A ⋊ (f ⋉ C)
- * assoc_rr : (f ⋉ B) ⋉ C <-> f ⋉ (B ⊗ C)
- * assoc_ll : (A ⋊ B) ⋊ f <-> (A ⊗ B) ⋊ f
- *
- * Fortunately, in a monoidal category these are all the same natural
- * isomorphism (and in any case -- monoidal or not -- the objects in
- * the left column are all the same and the objects in the right
- * column are all the same). This formalization assumes that is the
- * case even for premonoidal categories with non-central maps, in
- * order to keep the complexity manageable. I don't know much about
- * the consequences of having them and letting them be different; you
- * might need extra versions of the triangle/pentagon diagrams.
- *)
-
-Implicit Arguments pmon_cancell [ Ob Hom C bin_obj' bc I ].
-Implicit Arguments pmon_cancelr [ Ob Hom C bin_obj' bc I ].
-Implicit Arguments pmon_assoc [ Ob Hom C bin_obj' bc I ].
-Coercion pmon_bin : PreMonoidalCat >-> BinoidalCat.
-
-(* this turns out to be Exercise VII.1.1 from Mac Lane's CWM *)
-Lemma MacLane_ex_VII_1_1 `{mn:PreMonoidalCat(I:=EI)} a b
- : #((pmon_cancelr mn) (a ⊗ b)) ~~ #((pmon_assoc mn a EI) b) >>> (a ⋊-) \ #((pmon_cancelr mn) b).
- set (pmon_pentagon EI EI a b) as penta. unfold pmon_pentagon in penta.
- set (pmon_triangle a b) as tria. unfold pmon_triangle in tria.
- apply (fmor_respects(bin_second EI)) in tria.
- set (@fmor_preserves_comp) as fpc.
- setoid_rewrite <- fpc in tria.
- set (ni_commutes (pmon_assoc mn a b)) as xx.
- (* FIXME *)
- Admitted.
-
-(* Formalized Definition 3.10 *)
-Class PreMonoidalFunctor
-`(PM1:PreMonoidalCat(C:=C1)(I:=I1))
-`(PM2:PreMonoidalCat(C:=C2)(I:=I2))
- (fobj : C1 -> C2 ) :=
-{ mf_F :> Functor C1 C2 fobj
-; mf_preserves_i : mf_F I1 ≅ I2
-; mf_preserves_first : forall a, bin_first a >>>> mf_F <~~~> mf_F >>>> bin_first (mf_F a)
-; mf_preserves_second : forall a, bin_second a >>>> mf_F <~~~> mf_F >>>> bin_second (mf_F a)
-; mf_preserves_center : forall `(f:a~>b), CentralMorphism f -> CentralMorphism (mf_F \ f)
-}.
-Coercion mf_F : PreMonoidalFunctor >-> Functor.
-
-(*******************************************************************************)
-(* Braided and Symmetric Categories *)
-
-Class BraidedCat `(mc:PreMonoidalCat) :=
-{ br_niso : forall a, bin_first a <~~~> bin_second a
-; br_swap := fun a b => ni_iso (br_niso b) a
-; triangleb : forall a:C, #(pmon_cancelr mc a) ~~ #(br_swap a (pmon_I(PreMonoidalCat:=mc))) >>> #(pmon_cancell mc a)
-; hexagon1 : forall {a b c}, #(pmon_assoc mc _ _ _) >>> #(br_swap a _) >>> #(pmon_assoc mc _ _ _)
- ~~ #(br_swap _ _) ⋉ c >>> #(pmon_assoc mc _ _ _) >>> b ⋊ #(br_swap _ _)
-; hexagon2 : forall {a b c}, #(pmon_assoc mc _ _ _)⁻¹ >>> #(br_swap _ c) >>> #(pmon_assoc mc _ _ _)⁻¹
- ~~ a ⋊ #(br_swap _ _) >>> #(pmon_assoc mc _ _ _)⁻¹ >>> #(br_swap _ _) ⋉ b
-}.
-
-Class SymmetricCat `(bc:BraidedCat) :=
-{ symcat_swap : forall a b:C, #((br_swap(BraidedCat:=bc)) a b) ~~ #(br_swap _ _)⁻¹
-}.
-
-(* Definition 7.23 *)
Class MonoidalCat `{C:Category}{Fobj:prod_obj C C -> C}{F:Functor (C ×× C) C Fobj}(I:C) :=
{ mon_f := F
; mon_i := I
Implicit Arguments mon_c [Ob Hom C Fobj F I].
Implicit Arguments MonoidalCat [Ob Hom ].
+(* TO DO: show that the endofunctors on any given category form a monoidal category *)
+Section MonoidalFunctor.
+ Context `(m1:MonoidalCat(C:=C1)) `(m2:MonoidalCat(C:=C2)).
+ Class MonoidalFunctor {Mobj:C1->C2} (mf_F:Functor C1 C2 Mobj) :=
+ { mf_f := mf_F where "f ⊕⊕ g" := (@fmor _ _ _ _ _ _ _ m2 _ _ (pair_mor (pair_obj _ _) (pair_obj _ _) f g))
+ ; mf_coherence : (mf_F **** mf_F) >>>> (mon_f m2) <~~~> (mon_f m1) >>>> mf_F
+ ; mf_phi := fun a b => #(mf_coherence (pair_obj a b))
+ ; mf_id : (mon_i m2) ≅ (mf_F (mon_i m1))
+ ; mf_cancelr : forall a, #(mon_cancelr(MonoidalCat:=m2) (mf_F a)) ~~
+ (id (mf_F a)) ⊕⊕ #mf_id >>> mf_phi a (mon_i _) >>> mf_F \ #(mon_cancelr a)
+ ; mf_cancell : forall b, #(mon_cancell (mf_F b)) ~~
+ #mf_id ⊕⊕ (id (mf_F b)) >>> mf_phi (mon_i _) b >>> mf_F \ #(mon_cancell b)
+ ; mf_assoc : forall a b c, (mf_phi a b) ⊕⊕ (id (mf_F c)) >>> (mf_phi _ c) >>>
+ (mf_F \ #(mon_assoc (pair_obj (pair_obj a b) c) )) ~~
+ #(mon_assoc (pair_obj (pair_obj _ _) _) ) >>>
+ (id (mf_F a)) ⊕⊕ (mf_phi b c) >>> (mf_phi a _)
+ }.
+End MonoidalFunctor.
+Coercion mf_f : MonoidalFunctor >-> Functor.
+Implicit Arguments mf_coherence [ Ob Hom C1 Fobj F I0 m1 Ob0 Hom0 C2 Fobj0 F0 I1 m2 Mobj mf_F ].
+Implicit Arguments mf_id [ Ob Hom C1 Fobj F I0 m1 Ob0 Hom0 C2 Fobj0 F0 I1 m2 Mobj mf_F ].
+
+Section MonoidalFunctorsCompose.
+ Context `(m1:MonoidalCat).
+ Context `(m2:MonoidalCat).
+ Context `(m3:MonoidalCat).
+ Context {f1obj}(f1:@Functor _ _ m1 _ _ m2 f1obj).
+ Context {f2obj}(f2:@Functor _ _ m2 _ _ m3 f2obj).
+ Context (mf1:MonoidalFunctor m1 m2 f1).
+ Context (mf2:MonoidalFunctor m2 m3 f2).
+
+ Lemma mf_compose_coherence : (f1 >>>> f2) **** (f1 >>>> f2) >>>> m3 <~~~> m1 >>>> (f1 >>>> f2).
+ set (mf_coherence mf1) as mc1.
+ set (mf_coherence mf2) as mc2.
+ set (@ni_comp) as q.
+ set (q _ _ _ _ _ _ _ ((f1 >>>> f2) **** (f1 >>>> f2) >>>> m3) _ ((f1 **** f1 >>>> m2) >>>> f2) _ (m1 >>>> (f1 >>>> f2))) as qq.
+ apply qq; clear qq; clear q.
+ apply (@ni_comp _ _ _ _ _ _ _ _ _ (f1 **** f1 >>>> (f2 **** f2 >>>> m3)) _ _).
+ apply (@ni_comp _ _ _ _ _ _ _ _ _ ((f1 **** f1 >>>> f2 **** f2) >>>> m3) _ _).
+ eapply ni_respects.
+ apply ni_prod_comp.
+ apply ni_id.
+ apply ni_associativity.
+ apply ni_inv.
+ eapply ni_comp.
+ apply (ni_associativity (f1 **** f1) m2 f2).
+ apply (ni_respects (F0:=f1 **** f1)(F1:=f1 **** f1)(G0:=(m2 >>>> f2))(G1:=(f2 **** f2 >>>> m3))).
+ apply ni_id.
+ apply ni_inv.
+ apply mc2.
+ apply ni_inv.
+ eapply ni_comp.
+ eapply ni_inv.
+ apply (ni_associativity m1 f1 f2).
+ apply ni_respects.
+ apply ni_inv.
+ apply mc1.
+ apply ni_id.
+ Defined.
+
+ Instance MonoidalFunctorsCompose : MonoidalFunctor m1 m3 (f1 >>>> f2) :=
+ { mf_id := id_comp (mf_id mf2) (functors_preserve_isos f2 (mf_id mf1))
+ ; mf_coherence := mf_compose_coherence
+ }.
+ admit.
+ admit.
+ admit.
+ Defined.
+
+End MonoidalFunctorsCompose.
+
Section MonoidalCat_is_PreMonoidal.
Context `(M:MonoidalCat).
Definition mon_bin_M := BinoidalCat_from_Bifunctor (mon_f M).
Existing Instance mon_bin_M.
+
Lemma mon_pmon_assoc : forall a b, (bin_second a >>>> bin_first b) <~~~> (bin_first b >>>> bin_second a).
intros.
set (fun c => mon_assoc (pair_obj (pair_obj a c) b)) as qq.
Hint Extern 1 => apply MonoidalCat_all_central.
Coercion MonoidalCat_is_PreMonoidal : MonoidalCat >-> PreMonoidalCat.
-(* Later: generalized to premonoidal categories *)
+(* Later: generalize to premonoidal categories *)
Class DiagonalCat `(mc:MonoidalCat(F:=F)) :=
{ copy_nt : forall a, functor_id _ ~~~> func_diagonal >>>> F
; copy : forall (a:mc), a~~{mc}~~>(bin_obj(BinoidalCat:=mc) a a)
(* for non-cartesian braided diagonal categories we also need: copy >> swap == copy *)
}.
-(* TO DO: show that the endofunctors on any given category form a monoidal category *)
-Section MonoidalFunctor.
- Context `(m1:MonoidalCat(C:=C1)) `(m2:MonoidalCat(C:=C2)).
- Class MonoidalFunctor {Mobj:C1->C2} (mf_F:Functor C1 C2 Mobj) :=
- { mf_f := mf_F where "f ⊕⊕ g" := (@fmor _ _ _ _ _ _ _ m2 _ _ (pair_mor (pair_obj _ _) (pair_obj _ _) f g))
- ; mf_coherence : (mf_F **** mf_F) >>>> (mon_f m2) <~~~> (mon_f m1) >>>> mf_F
- ; mf_phi := fun a b => #(mf_coherence (pair_obj a b))
- ; mf_id : (mon_i m2) ≅ (mf_F (mon_i m1))
- ; mf_cancelr : forall a, #(mon_cancelr(MonoidalCat:=m2) (mf_F a)) ~~
- (id (mf_F a)) ⊕⊕ #mf_id >>> mf_phi a (mon_i _) >>> mf_F \ #(mon_cancelr a)
- ; mf_cancell : forall b, #(mon_cancell (mf_F b)) ~~
- #mf_id ⊕⊕ (id (mf_F b)) >>> mf_phi (mon_i _) b >>> mf_F \ #(mon_cancell b)
- ; mf_assoc : forall a b c, (mf_phi a b) ⊕⊕ (id (mf_F c)) >>> (mf_phi _ c) >>>
- (mf_F \ #(mon_assoc (pair_obj (pair_obj a b) c) )) ~~
- #(mon_assoc (pair_obj (pair_obj _ _) _) ) >>>
- (id (mf_F a)) ⊕⊕ (mf_phi b c) >>> (mf_phi a _)
- }.
-End MonoidalFunctor.
-Coercion mf_f : MonoidalFunctor >-> Functor.
-Implicit Arguments mf_coherence [ Ob Hom C1 Fobj F I0 m1 Ob0 Hom0 C2 Fobj0 F0 I1 m2 Mobj mf_F ].
-Implicit Arguments mf_id [ Ob Hom C1 Fobj F I0 m1 Ob0 Hom0 C2 Fobj0 F0 I1 m2 Mobj mf_F ].
-
-Section MonoidalFunctorsCompose.
- Context `(m1:MonoidalCat).
- Context `(m2:MonoidalCat).
- Context `(m3:MonoidalCat).
- Context {f1obj}(f1:@Functor _ _ m1 _ _ m2 f1obj).
- Context {f2obj}(f2:@Functor _ _ m2 _ _ m3 f2obj).
- Context (mf1:MonoidalFunctor m1 m2 f1).
- Context (mf2:MonoidalFunctor m2 m3 f2).
-
- Lemma mf_compose_coherence : (f1 >>>> f2) **** (f1 >>>> f2) >>>> m3 <~~~> m1 >>>> (f1 >>>> f2).
- set (mf_coherence mf1) as mc1.
- set (mf_coherence mf2) as mc2.
- set (@ni_comp) as q.
- set (q _ _ _ _ _ _ _ ((f1 >>>> f2) **** (f1 >>>> f2) >>>> m3) _ ((f1 **** f1 >>>> m2) >>>> f2) _ (m1 >>>> (f1 >>>> f2))) as qq.
- apply qq; clear qq; clear q.
- apply (@ni_comp _ _ _ _ _ _ _ _ _ (f1 **** f1 >>>> (f2 **** f2 >>>> m3)) _ _).
- apply (@ni_comp _ _ _ _ _ _ _ _ _ ((f1 **** f1 >>>> f2 **** f2) >>>> m3) _ _).
- eapply ni_respects.
- apply ni_prod_comp.
- apply ni_id.
- apply ni_associativity.
- apply ni_inv.
- eapply ni_comp.
- apply (ni_associativity (f1 **** f1) m2 f2).
- apply (ni_respects (F0:=f1 **** f1)(F1:=f1 **** f1)(G0:=(m2 >>>> f2))(G1:=(f2 **** f2 >>>> m3))).
- apply ni_id.
- apply ni_inv.
- apply mc2.
- apply ni_inv.
- eapply ni_comp.
- eapply ni_inv.
- apply (ni_associativity m1 f1 f2).
- apply ni_respects.
- apply ni_inv.
- apply mc1.
- apply ni_id.
- Qed.
-
- Instance MonoidalFunctorsCompose : MonoidalFunctor m1 m3 (f1 >>>> f2) :=
- { mf_id := id_comp (mf_id mf2) (functors_preserve_isos f2 (mf_id mf1))
- ; mf_coherence := mf_compose_coherence
- }.
- admit.
- admit.
- admit.
- Defined.
-
-End MonoidalFunctorsCompose.
-
Class CartesianCat `(mc:MonoidalCat) :=
{ car_terminal : Terminal mc
; car_one : (@one _ _ _ car_terminal) ≅ (mon_i mc)
Coercion car_diagonal : CartesianCat >-> DiagonalCat.
Coercion car_terminal : CartesianCat >-> Terminal.
Coercion car_mn : CartesianCat >-> MonoidalCat.
-
-Section CenterMonoidal.
-
- Context `(mc:PreMonoidalCat(I:=pI)).
-
- Definition CenterMonoidal_Fobj : (Center mc) ×× (Center mc) -> Center mc.
- intro.
- destruct X as [a b].
- destruct a as [a apf].
- destruct b as [b bpf].
- exists (a ⊗ b); auto.
- Defined.
-
- Definition CenterMonoidal_F : Functor _ _ CenterMonoidal_Fobj.
- admit.
- Defined.
-
- Definition CenterMonoidal : MonoidalCat _ _ CenterMonoidal_F (exist _ pI I).
- admit.
- Defined.
-
-End CenterMonoidal.
-
projects / coq-categories.git / blobdiff