Require Import Categories_ch1_3.
Require Import Functors_ch1_4.
Require Import Isomorphisms_ch1_5.
-Require Import ProductCategories_ch1_6_1.
+Require Import EpicMonic_ch2_1.
Require Import InitialTerminal_ch2_2.
Require Import Subcategories_ch7_1.
Require Import NaturalTransformations_ch7_4.
(* 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)
+{ 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)
+; pmon_assoc_central : forall a b c, CentralMorphism #(pmon_assoc a b c)
+; pmon_cancelr_central : forall a , CentralMorphism #(pmon_cancelr a)
+; pmon_cancell_central : forall a , CentralMorphism #(pmon_cancell a)
}.
(*
* Premonoidal categories actually have three associators (the "f"
* 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 ].
+Implicit Arguments pmon_cancell [ Ob Hom C bin_obj' bc I PreMonoidalCat ].
+Implicit Arguments pmon_cancelr [ Ob Hom C bin_obj' bc I PreMonoidalCat ].
+Implicit Arguments pmon_assoc [ Ob Hom C bin_obj' bc I PreMonoidalCat ].
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.
+Lemma MacLane_ex_VII_1_1 `{mn:PreMonoidalCat(I:=EI)} b a
+ :
+ let α := fun a b c => #((pmon_assoc a c) b)
+ in α a b EI >>> _ ⋊ #(pmon_cancelr _) ~~ #(pmon_cancelr _).
+
+ intros. simpl in α.
+
+ (* following Mac Lane's hint, we aim for (λ >>> α >>> λ×1)~~(λ >>> λ) *)
+ set (epic _ (iso_epic (pmon_cancelr ((a⊗b)⊗EI)))) as q.
+ apply q.
+ clear q.
+
+ (* next, we show that the hint goal above is implied by the bottom-left 1/5th of the big whiteboard diagram *)
+ set (ni_commutes pmon_cancelr (α a b EI)) as q.
+ setoid_rewrite <- associativity.
+ setoid_rewrite q.
+ clear q.
+ setoid_rewrite associativity.
+
+ set (ni_commutes pmon_cancelr (a ⋊ #(pmon_cancelr b))) as q.
+ simpl in q.
+ setoid_rewrite q.
+ clear q.
+
+ set (ni_commutes pmon_cancelr (#(pmon_cancelr (a⊗b)))) as q.
+ simpl in q.
+ setoid_rewrite q.
+ clear q.
+
+ setoid_rewrite <- associativity.
+ apply comp_respects; try reflexivity.
+
+ (* now we carry out the proof in the whiteboard diagram, starting from the pentagon diagram *)
+
+ (* top 2/5ths *)
+ assert (α (a⊗b) EI EI >>> α _ _ _ >>> (_ ⋊ (_ ⋊ #(pmon_cancell _))) ~~ #(pmon_cancelr _) ⋉ _ >>> α _ _ _).
+ set (pmon_triangle (a⊗b) EI) as tria.
+ simpl in tria.
+ unfold α; simpl.
+ setoid_rewrite tria.
+ clear tria.
+ setoid_rewrite associativity.
+ apply comp_respects; try reflexivity.
+ set (ni_commutes (pmon_assoc_ll a b) #(pmon_cancell EI)) as x.
+ simpl in x.
+ setoid_rewrite pmon_coherent_l in x.
+ apply x.
+
+ (* bottom 3/5ths *)
+ assert (((#((pmon_assoc a EI) b) ⋉ EI >>> #((pmon_assoc a EI) (b ⊗ EI))) >>>
+ a ⋊ #((pmon_assoc b EI) EI)) >>> a ⋊ (b ⋊ #(pmon_cancell EI))
+ ~~ α _ _ _ ⋉ _ >>> (_ ⋊ #(pmon_cancelr _)) ⋉ _ >>> α _ _ _).
+
+ unfold α; simpl.
+ repeat setoid_rewrite associativity.
+ apply comp_respects; try reflexivity.
+
+ set (ni_commutes (pmon_assoc a EI) (#(pmon_cancelr b) )) as x.
+ simpl in x.
+ setoid_rewrite <- associativity.
+ simpl in x.
+ setoid_rewrite <- x.
+ clear x.
+
+ setoid_rewrite associativity.
+ apply comp_respects; try reflexivity.
+ setoid_rewrite (fmor_preserves_comp (a⋊-)).
+ apply (fmor_respects (a⋊-)).
+
+ set (pmon_triangle b EI) as tria.
+ simpl in tria.
+ symmetry.
+ apply tria.
+
+ set (pmon_pentagon a b EI EI) as penta. unfold pmon_pentagon in penta. simpl in penta.
+
+ set (@comp_respects _ _ _ _ _ _ _ _ penta (a ⋊ (b ⋊ #(pmon_cancell EI))) (a ⋊ (b ⋊ #(pmon_cancell EI)))) as qq.
+ unfold α in H.
+ setoid_rewrite H in qq.
+ unfold α in H0.
+ setoid_rewrite H0 in qq.
+ clear H0 H.
+
+ unfold α.
+ apply (monic _ (iso_monic ((pmon_assoc a EI) b))).
+ apply qq.
+ clear qq penta.
+ reflexivity.
+ Qed.
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)
+`(PM1 : PreMonoidalCat(C:=C1)(I:=I1))
+`(PM2 : PreMonoidalCat(C:=C2)(I:=I2))
+ {fobj : C1 -> C2 }
+ (F : Functor C1 C2 fobj ) :=
+{ mf_F := F
+; mf_i : I2 ≅ mf_F I1
+; mf_first : ∀ a, mf_F >>>> bin_first (mf_F a) <~~~> bin_first a >>>> mf_F
+; mf_second : ∀ a, mf_F >>>> bin_second (mf_F a) <~~~> bin_second a >>>> mf_F
+; mf_consistent : ∀ a b, #(mf_first a b) ~~ #(mf_second b a)
+; mf_center : forall `(f:a~>b), CentralMorphism f -> CentralMorphism (mf_F \ f)
+; mf_cancell : ∀ b, #(pmon_cancell _) ~~ #mf_i ⋉ _ >>> #(mf_first b I1) >>> mf_F \ #(pmon_cancell b)
+; mf_cancelr : ∀ a, #(pmon_cancelr _) ~~ _ ⋊ #mf_i >>> #(mf_second a I1) >>> mf_F \ #(pmon_cancelr a)
+; mf_assoc : ∀ a b c, #(pmon_assoc _ _ _) >>> _ ⋊ #(mf_first _ _) >>> #(mf_second _ _) ~~
+ #(mf_second _ _) ⋉ _ >>> #(mf_first _ _) >>> mf_F \ #(pmon_assoc a c b)
}.
Coercion mf_F : PreMonoidalFunctor >-> Functor.
+Section PreMonoidalFunctorsCompose.
+ Context
+ `{PM1 :PreMonoidalCat(C:=C1)(I:=I1)}
+ `{PM2 :PreMonoidalCat(C:=C2)(I:=I2)}
+ {fobj12:C1 -> C2 }
+ {PMFF12:Functor C1 C2 fobj12 }
+ (PMF12 :PreMonoidalFunctor PM1 PM2 PMFF12)
+ `{PM3 :PreMonoidalCat(C:=C3)(I:=I3)}
+ {fobj23:C2 -> C3 }
+ {PMFF23:Functor C2 C3 fobj23 }
+ (PMF23 :PreMonoidalFunctor PM2 PM3 PMFF23).
+
+ Definition compose_mf := PMF12 >>>> PMF23.
+
+ Definition compose_mf_i : I3 ≅ PMF23 (PMF12 I1).
+ eapply iso_comp.
+ apply (mf_i(PreMonoidalFunctor:=PMF23)).
+ apply functors_preserve_isos.
+ apply (mf_i(PreMonoidalFunctor:=PMF12)).
+ Defined.
+
+ Definition compose_mf_first a : compose_mf >>>> bin_first (compose_mf a) <~~~> bin_first a >>>> compose_mf.
+ set (mf_first(PreMonoidalFunctor:=PMF12) a) as mf_first12.
+ set (mf_first(PreMonoidalFunctor:=PMF23) (PMF12 a)) as mf_first23.
+ unfold functor_fobj in *; simpl in *.
+ unfold compose_mf.
+ eapply ni_comp.
+ apply (ni_associativity PMF12 PMF23 (- ⋉fobj23 (fobj12 a))).
+ eapply ni_comp.
+ apply (ni_respects1 PMF12 (PMF23 >>>> - ⋉fobj23 (fobj12 a)) (- ⋉fobj12 a >>>> PMF23)).
+ apply mf_first23.
+ clear mf_first23.
+
+ eapply ni_comp.
+ eapply ni_inv.
+ apply (ni_associativity PMF12 (- ⋉fobj12 a) PMF23).
+
+ apply ni_inv.
+ eapply ni_comp.
+ eapply ni_inv.
+ eapply (ni_associativity _ PMF12 PMF23).
+
+ apply ni_respects2.
+ apply ni_inv.
+ apply mf_first12.
+ Defined.
+
+ Definition compose_mf_second a : compose_mf >>>> bin_second (compose_mf a) <~~~> bin_second a >>>> compose_mf.
+ set (mf_second(PreMonoidalFunctor:=PMF12) a) as mf_second12.
+ set (mf_second(PreMonoidalFunctor:=PMF23) (PMF12 a)) as mf_second23.
+ unfold functor_fobj in *; simpl in *.
+ unfold compose_mf.
+ eapply ni_comp.
+ apply (ni_associativity PMF12 PMF23 (fobj23 (fobj12 a) ⋊-)).
+ eapply ni_comp.
+ apply (ni_respects1 PMF12 (PMF23 >>>> fobj23 (fobj12 a) ⋊-) (fobj12 a ⋊- >>>> PMF23)).
+ apply mf_second23.
+ clear mf_second23.
+
+ eapply ni_comp.
+ eapply ni_inv.
+ apply (ni_associativity PMF12 (fobj12 a ⋊ -) PMF23).
+
+ apply ni_inv.
+ eapply ni_comp.
+ eapply ni_inv.
+ eapply (ni_associativity (a ⋊-) PMF12 PMF23).
+
+ apply ni_respects2.
+ apply ni_inv.
+ apply mf_second12.
+ Defined.
+
+ (* this proof is really gross; I will write a better one some other day *)
+ Lemma mf_associativity_comp :
+ ∀a b c : C1,
+ (#((pmon_assoc (compose_mf a) (compose_mf c)) (fobj23 (fobj12 b))) >>>
+ compose_mf a ⋊ #((compose_mf_first c) b)) >>>
+ #((compose_mf_second a) (b ⊗ c)) ~~
+ (#((compose_mf_second a) b) ⋉ compose_mf c >>>
+ #((compose_mf_first c) (a ⊗ b))) >>> compose_mf \ #((pmon_assoc a c) b).
+ intros; intros.
+ unfold compose_mf_second; simpl.
+ unfold compose_mf_first; simpl.
+ unfold functor_comp; simpl.
+ unfold ni_respects1.
+ unfold functor_fobj; simpl.
+
+ set (mf_first (fobj12 c)) as m'.
+ assert (mf_first (fobj12 c)=m'). reflexivity.
+ destruct m'; simpl.
+
+ set (mf_second (fobj12 a)) as m.
+ assert (mf_second (fobj12 a)=m). reflexivity.
+ destruct m; simpl.
+
+ Implicit Arguments id [[Ob][Hom][Category][a]].
+ idtac.
+
+ symmetry.
+ etransitivity.
+ repeat setoid_rewrite <- fmor_preserves_comp.
+ repeat setoid_rewrite fmor_preserves_id.
+ repeat setoid_rewrite left_identity.
+ repeat setoid_rewrite right_identity.
+ reflexivity.
+ symmetry.
+ etransitivity.
+ repeat setoid_rewrite <- fmor_preserves_comp.
+ repeat setoid_rewrite fmor_preserves_id.
+ repeat setoid_rewrite left_identity.
+ repeat setoid_rewrite right_identity.
+ reflexivity.
+
+ assert ( (#((pmon_assoc (fobj23 (fobj12 a)) (fobj23 (fobj12 c)))
+ (fobj23 (fobj12 b))) >>>
+ fobj23 (fobj12 a)
+ ⋊ (
+ (#(ni_iso (fobj12 b)) >>> ( (PMF23 \ #((mf_first c) b) ))))) >>>
+ (
+ (#(ni_iso0 (fobj12 (b ⊗ c))) >>>
+ ((PMF23 \ #((mf_second a) (b ⊗ c)))))) ~~
+ ((
+ (#(ni_iso0 (fobj12 b)) >>> ( (PMF23 \ #((mf_second a) b) ))))
+ ⋉ fobj23 (fobj12 c) >>>
+ (
+ (#(ni_iso (fobj12 (a ⊗ b))) >>>
+ ( (PMF23 \ #((mf_first c) (a ⊗ b))))))) >>>
+ PMF23 \ (PMF12 \ #((pmon_assoc a c) b))
+ ).
+
+ repeat setoid_rewrite associativity.
+ setoid_rewrite (fmor_preserves_comp PMF23).
+ unfold functor_comp in *.
+ unfold functor_fobj in *.
+ simpl in *.
+ rename ni_commutes into ni_commutes7.
+ set (mf_assoc(PreMonoidalFunctor:=PMF12)) as q.
+ set (ni_commutes7 _ _ (#((mf_second a) b))) as q'.
+ simpl in q'.
+ repeat setoid_rewrite associativity.
+ symmetry.
+ setoid_rewrite <- (fmor_preserves_comp (-⋉ fobj23 (fobj12 c))).
+ repeat setoid_rewrite <- associativity.
+ setoid_rewrite juggle1.
+ setoid_rewrite <- q'.
+ repeat setoid_rewrite associativity.
+ setoid_rewrite fmor_preserves_comp.
+ idtac.
+ unfold functor_fobj in *.
+ simpl in *.
+ repeat setoid_rewrite <- associativity.
+ setoid_rewrite <- q.
+ clear q.
+ repeat setoid_rewrite <- fmor_preserves_comp.
+ repeat setoid_rewrite <- associativity.
+ apply comp_respects; try reflexivity.
+
+ set (mf_assoc(PreMonoidalFunctor:=PMF23) (fobj12 a) (fobj12 b) (fobj12 c)) as q.
+ unfold functor_fobj in *.
+ simpl in *.
+
+ rewrite H in q.
+ rewrite H0 in q.
+ simpl in q.
+ repeat setoid_rewrite <- associativity.
+ repeat setoid_rewrite <- associativity in q.
+ setoid_rewrite <- q.
+ clear q.
+ unfold functor_fobj; simpl.
+
+ repeat setoid_rewrite associativity.
+ apply comp_respects; try reflexivity.
+ apply comp_respects; try reflexivity.
+ auto.
+
+ repeat setoid_rewrite associativity.
+ repeat setoid_rewrite associativity in H1.
+ repeat setoid_rewrite <- fmor_preserves_comp in H1.
+ repeat setoid_rewrite associativity in H1.
+ apply H1.
+ Qed.
+ Implicit Arguments id [[Ob][Hom][Category]].
+
+ (* this proof is really gross; I will write a better one some other day *)
+ Instance PreMonoidalFunctorsCompose : PreMonoidalFunctor PM1 PM3 compose_mf :=
+ { mf_i := compose_mf_i
+ ; mf_first := compose_mf_first
+ ; mf_second := compose_mf_second }.
+
+ intros; unfold compose_mf_first; unfold compose_mf_second.
+ set (mf_first (PMF12 a)) as x in *.
+ set (mf_second (PMF12 b)) as y in *.
+ assert (x=mf_first (PMF12 a)). reflexivity.
+ assert (y=mf_second (PMF12 b)). reflexivity.
+ destruct x.
+ destruct y.
+ simpl.
+ repeat setoid_rewrite left_identity.
+ repeat setoid_rewrite right_identity.
+ set (mf_consistent (PMF12 a) (PMF12 b)) as later.
+ apply comp_respects; try reflexivity.
+ rewrite <- H in later.
+ rewrite <- H0 in later.
+ simpl in later.
+ apply later.
+ apply fmor_respects.
+ apply mf_consistent.
+
+ intros.
+ simpl.
+ apply mf_center.
+ apply mf_center.
+ auto.
+
+ intros.
+ unfold compose_mf_first; simpl.
+ set (mf_first (PMF12 b)) as m.
+ assert (mf_first (PMF12 b)=m). reflexivity.
+ destruct m.
+ simpl.
+ unfold functor_fobj; simpl.
+ repeat setoid_rewrite <- fmor_preserves_comp.
+ repeat setoid_rewrite left_identity.
+ repeat setoid_rewrite right_identity.
+
+ set (mf_cancell b) as y.
+ set (mf_cancell (fobj12 b)) as y'.
+ unfold functor_fobj in *.
+ setoid_rewrite y in y'.
+ clear y.
+ setoid_rewrite <- fmor_preserves_comp in y'.
+ setoid_rewrite <- fmor_preserves_comp in y'.
+ etransitivity.
+ apply y'.
+ clear y'.
+
+ repeat setoid_rewrite <- associativity.
+ apply comp_respects; try reflexivity.
+ apply comp_respects; try reflexivity.
+ repeat setoid_rewrite associativity.
+ apply comp_respects; try reflexivity.
+
+ set (ni_commutes _ _ #mf_i) as x.
+ unfold functor_comp in x.
+ unfold functor_fobj in x.
+ simpl in x.
+ rewrite H.
+ simpl.
+ apply x.
+
+ intros.
+ unfold compose_mf_second; simpl.
+ set (mf_second (PMF12 a)) as m.
+ assert (mf_second (PMF12 a)=m). reflexivity.
+ destruct m.
+ simpl.
+ unfold functor_fobj; simpl.
+ repeat setoid_rewrite <- fmor_preserves_comp.
+ repeat setoid_rewrite left_identity.
+ repeat setoid_rewrite right_identity.
+
+ set (mf_cancelr a) as y.
+ set (mf_cancelr (fobj12 a)) as y'.
+ unfold functor_fobj in *.
+ setoid_rewrite y in y'.
+ clear y.
+ setoid_rewrite <- fmor_preserves_comp in y'.
+ setoid_rewrite <- fmor_preserves_comp in y'.
+ etransitivity.
+ apply y'.
+ clear y'.
+
+ repeat setoid_rewrite <- associativity.
+ apply comp_respects; try reflexivity.
+ apply comp_respects; try reflexivity.
+ repeat setoid_rewrite associativity.
+ apply comp_respects; try reflexivity.
+
+ set (ni_commutes _ _ #mf_i) as x.
+ unfold functor_comp in x.
+ unfold functor_fobj in x.
+ simpl in x.
+ rewrite H.
+ simpl.
+ apply x.
+
+ apply mf_associativity_comp.
+
+ Defined.
+
+End PreMonoidalFunctorsCompose.
+
+
(*******************************************************************************)
(* 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
+; triangleb : forall a:C, #(pmon_cancelr a) ~~ #(br_swap a (pmon_I(PreMonoidalCat:=mc))) >>> #(pmon_cancell a)
+; hexagon1 : forall {a b c}, #(pmon_assoc _ _ _) >>> #(br_swap a _) >>> #(pmon_assoc _ _ _)
+ ~~ #(br_swap _ _) ⋉ c >>> #(pmon_assoc _ _ _) >>> b ⋊ #(br_swap _ _)
+; hexagon2 : forall {a b c}, #(pmon_assoc _ _ _)⁻¹ >>> #(br_swap _ c) >>> #(pmon_assoc _ _ _)⁻¹
+ ~~ a ⋊ #(br_swap _ _) >>> #(pmon_assoc _ _ _)⁻¹ >>> #(br_swap _ _) ⋉ b
}.
Class SymmetricCat `(bc:BraidedCat) :=
{ symcat_swap : forall a b:C, #((br_swap(BraidedCat:=bc)) a b) ~~ #(br_swap _ _)⁻¹
}.
+
+
+(* a wide subcategory inherits the premonoidal structure if it includes all of the coherence maps *)
+Section PreMonoidalWideSubcategory.
+
+ Context `(pm:PreMonoidalCat(I:=pmI)).
+ Context {Pmor}(S:WideSubcategory pm Pmor).
+ Context (Pmor_first : forall {a}{b}{c}{f}(pf:Pmor a b f), Pmor _ _ (f ⋉ c)).
+ Context (Pmor_second : forall {a}{b}{c}{f}(pf:Pmor a b f), Pmor _ _ (c ⋊ f)).
+ Context (Pmor_assoc : forall {a}{b}{c}, Pmor _ _ #(pmon_assoc a c b)).
+ Context (Pmor_unassoc: forall {a}{b}{c}, Pmor _ _ #(pmon_assoc a c b)⁻¹).
+ Context (Pmor_cancell: forall {a}, Pmor _ _ #(pmon_cancell a)).
+ Context (Pmor_uncancell: forall {a}, Pmor _ _ #(pmon_cancell a)⁻¹).
+ Context (Pmor_cancelr: forall {a}, Pmor _ _ #(pmon_cancelr a)).
+ Context (Pmor_uncancelr: forall {a}, Pmor _ _ #(pmon_cancelr a)⁻¹).
+ Implicit Arguments Pmor_first [[a][b][c][f]].
+ Implicit Arguments Pmor_second [[a][b][c][f]].
+
+ Definition PreMonoidalWideSubcategory_first_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y), (bin_obj' x a)~~{S}~~>(bin_obj' y a).
+ unfold hom; simpl; intros.
+ destruct f.
+ simpl in *.
+ exists (bin_first(BinoidalCat:=pm) a \ x0).
+ apply Pmor_first; auto.
+ Defined.
+
+ Definition PreMonoidalWideSubcategory_second_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y), (bin_obj' a x)~~{S}~~>(bin_obj' a y).
+ unfold hom; simpl; intros.
+ destruct f.
+ simpl in *.
+ exists (bin_second(BinoidalCat:=pm) a \ x0).
+ apply Pmor_second; auto.
+ Defined.
+
+ Instance PreMonoidalWideSubcategory_first (a:S) : Functor S S (fun x => bin_obj' x a) :=
+ { fmor := fun x y f => PreMonoidalWideSubcategory_first_fmor a f }.
+ unfold PreMonoidalWideSubcategory_first_fmor; intros; destruct f; destruct f'; simpl in *.
+ apply (fmor_respects (bin_first(BinoidalCat:=pm) a)); auto.
+ unfold PreMonoidalWideSubcategory_first_fmor; intros; simpl in *.
+ apply (fmor_preserves_id (bin_first(BinoidalCat:=pm) a)); auto.
+ unfold PreMonoidalWideSubcategory_first_fmor; intros; destruct f; destruct g; simpl in *.
+ apply (fmor_preserves_comp (bin_first(BinoidalCat:=pm) a)); auto.
+ Defined.
+
+ Instance PreMonoidalWideSubcategory_second (a:S) : Functor S S (fun x => bin_obj' a x) :=
+ { fmor := fun x y f => PreMonoidalWideSubcategory_second_fmor a f }.
+ unfold PreMonoidalWideSubcategory_second_fmor; intros; destruct f; destruct f'; simpl in *.
+ apply (fmor_respects (bin_second(BinoidalCat:=pm) a)); auto.
+ unfold PreMonoidalWideSubcategory_second_fmor; intros; simpl in *.
+ apply (fmor_preserves_id (bin_second(BinoidalCat:=pm) a)); auto.
+ unfold PreMonoidalWideSubcategory_second_fmor; intros; destruct f; destruct g; simpl in *.
+ apply (fmor_preserves_comp (bin_second(BinoidalCat:=pm) a)); auto.
+ Defined.
+
+ Instance PreMonoidalWideSubcategory_is_Binoidal : BinoidalCat S bin_obj' :=
+ { bin_first := PreMonoidalWideSubcategory_first
+ ; bin_second := PreMonoidalWideSubcategory_second }.
+
+ Definition PreMonoidalWideSubcategory_assoc_iso
+ : forall a b c, Isomorphic(C:=S) (bin_obj' (bin_obj' a b) c) (bin_obj' a (bin_obj' b c)).
+ intros.
+ refine {| iso_forward := existT _ _ (Pmor_assoc a b c) ; iso_backward := existT _ _ (Pmor_unassoc a b c) |}.
+ simpl; apply iso_comp1.
+ simpl; apply iso_comp2.
+ Defined.
+
+ Definition PreMonoidalWideSubcategory_assoc
+ : forall a b,
+ (PreMonoidalWideSubcategory_second a >>>> PreMonoidalWideSubcategory_first b) <~~~>
+ (PreMonoidalWideSubcategory_first b >>>> PreMonoidalWideSubcategory_second a).
+ intros.
+ apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _ (PreMonoidalWideSubcategory_second a >>>>
+ PreMonoidalWideSubcategory_first b) (PreMonoidalWideSubcategory_first b >>>>
+ PreMonoidalWideSubcategory_second a) (fun c => PreMonoidalWideSubcategory_assoc_iso a c b)).
+ intros; simpl.
+ unfold PreMonoidalWideSubcategory_second_fmor; simpl.
+ destruct f; simpl.
+ set (ni_commutes (pmon_assoc(PreMonoidalCat:=pm) a b) x) as q.
+ apply q.
+ Defined.
+
+ Definition PreMonoidalWideSubcategory_assoc_ll
+ : forall a b,
+ PreMonoidalWideSubcategory_second (a⊗b) <~~~>
+ PreMonoidalWideSubcategory_second b >>>> PreMonoidalWideSubcategory_second a.
+ intros.
+ apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
+ (PreMonoidalWideSubcategory_second (a⊗b))
+ (PreMonoidalWideSubcategory_second b >>>> PreMonoidalWideSubcategory_second a)
+ (fun c => PreMonoidalWideSubcategory_assoc_iso a b c)).
+ intros; simpl.
+ unfold PreMonoidalWideSubcategory_second_fmor; simpl.
+ destruct f; simpl.
+ set (ni_commutes (pmon_assoc_ll(PreMonoidalCat:=pm) a b) x) as q.
+ unfold functor_comp in q; simpl in q.
+ set (pmon_coherent_l(PreMonoidalCat:=pm)) as q'.
+ setoid_rewrite q' in q.
+ apply q.
+ Defined.
+
+ Definition PreMonoidalWideSubcategory_assoc_rr
+ : forall a b,
+ PreMonoidalWideSubcategory_first (a⊗b) <~~~>
+ PreMonoidalWideSubcategory_first a >>>> PreMonoidalWideSubcategory_first b.
+ intros.
+ apply ni_inv.
+ apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
+ (PreMonoidalWideSubcategory_first a >>>> PreMonoidalWideSubcategory_first b)
+ (PreMonoidalWideSubcategory_first (a⊗b))
+ (fun c => PreMonoidalWideSubcategory_assoc_iso c a b)).
+ intros; simpl.
+ unfold PreMonoidalWideSubcategory_second_fmor; simpl.
+ destruct f; simpl.
+ set (ni_commutes (pmon_assoc_rr(PreMonoidalCat:=pm) a b) x) as q.
+ unfold functor_comp in q; simpl in q.
+ set (pmon_coherent_r(PreMonoidalCat:=pm)) as q'.
+ setoid_rewrite q' in q.
+ apply iso_shift_right' in q.
+ apply iso_shift_left.
+ symmetry.
+ setoid_rewrite iso_inv_inv in q.
+ setoid_rewrite associativity.
+ apply q.
+ Defined.
+
+ Definition PreMonoidalWideSubcategory_cancelr_iso : forall a, Isomorphic(C:=S) (bin_obj' a pmI) a.
+ intros.
+ refine {| iso_forward := existT _ _ (Pmor_cancelr a) ; iso_backward := existT _ _ (Pmor_uncancelr a) |}.
+ simpl; apply iso_comp1.
+ simpl; apply iso_comp2.
+ Defined.
+
+ Definition PreMonoidalWideSubcategory_cancell_iso : forall a, Isomorphic(C:=S) (bin_obj' pmI a) a.
+ intros.
+ refine {| iso_forward := existT _ _ (Pmor_cancell a) ; iso_backward := existT _ _ (Pmor_uncancell a) |}.
+ simpl; apply iso_comp1.
+ simpl; apply iso_comp2.
+ Defined.
+
+ Definition PreMonoidalWideSubcategory_cancelr : PreMonoidalWideSubcategory_first pmI <~~~> functor_id _.
+ apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
+ (PreMonoidalWideSubcategory_first pmI) (functor_id _) PreMonoidalWideSubcategory_cancelr_iso).
+ intros; simpl.
+ unfold PreMonoidalWideSubcategory_first_fmor; simpl.
+ destruct f; simpl.
+ apply (ni_commutes (pmon_cancelr(PreMonoidalCat:=pm)) x).
+ Defined.
+
+ Definition PreMonoidalWideSubcategory_cancell : PreMonoidalWideSubcategory_second pmI <~~~> functor_id _.
+ apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
+ (PreMonoidalWideSubcategory_second pmI) (functor_id _) PreMonoidalWideSubcategory_cancell_iso).
+ intros; simpl.
+ unfold PreMonoidalWideSubcategory_second_fmor; simpl.
+ destruct f; simpl.
+ apply (ni_commutes (pmon_cancell(PreMonoidalCat:=pm)) x).
+ Defined.
+
+ Instance PreMonoidalWideSubcategory_PreMonoidal : PreMonoidalCat PreMonoidalWideSubcategory_is_Binoidal pmI :=
+ { pmon_assoc := PreMonoidalWideSubcategory_assoc
+ ; pmon_assoc_rr := PreMonoidalWideSubcategory_assoc_rr
+ ; pmon_assoc_ll := PreMonoidalWideSubcategory_assoc_ll
+ ; pmon_cancelr := PreMonoidalWideSubcategory_cancelr
+ ; pmon_cancell := PreMonoidalWideSubcategory_cancell
+ }.
+ apply Build_Pentagon.
+ intros; unfold PreMonoidalWideSubcategory_assoc; simpl.
+ set (pmon_pentagon(PreMonoidalCat:=pm) a b c) as q.
+ simpl in q.
+ apply q.
+ apply Build_Triangle.
+ intros; unfold PreMonoidalWideSubcategory_assoc;
+ unfold PreMonoidalWideSubcategory_cancelr; unfold PreMonoidalWideSubcategory_cancell; simpl.
+ set (pmon_triangle(PreMonoidalCat:=pm) a b) as q.
+ simpl in q.
+ apply q.
+ intros.
+
+ set (pmon_triangle(PreMonoidalCat:=pm)) as q.
+ apply q.
+
+ intros; simpl; reflexivity.
+ intros; simpl; reflexivity.
+
+ intros; simpl.
+ apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
+ apply (pmon_assoc_central(PreMonoidalCat:=pm) a b c).
+ apply (pmon_assoc_central(PreMonoidalCat:=pm) a b c).
+
+ intros; simpl.
+ apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
+ apply (pmon_cancelr_central(PreMonoidalCat:=pm) a).
+ apply (pmon_cancelr_central(PreMonoidalCat:=pm) a).
+
+ intros; simpl.
+ apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
+ apply (pmon_cancell_central(PreMonoidalCat:=pm) a).
+ apply (pmon_cancell_central(PreMonoidalCat:=pm) a).
+ Defined.
+
+End PreMonoidalWideSubcategory.
+
+
+(* a full subcategory inherits the premonoidal structure if it includes the unit object and is closed under object-pairing *)
+(*
+Section PreMonoidalFullSubcategory.
+
+ Context `(pm:PreMonoidalCat(I:=pmI)).
+ Context {Pobj}(S:FullSubcategory pm Pobj).
+ Context (Pobj_unit:Pobj pmI).
+ Context (Pobj_closed:forall {a}{b}, Pobj a -> Pobj b -> Pobj (a⊗b)).
+ Implicit Arguments Pobj_closed [[a][b]].
+
+ Definition PreMonoidalFullSubcategory_bobj (x y:S) :=
+ existT Pobj _ (Pobj_closed (projT2 x) (projT2 y)).
+
+ Definition PreMonoidalFullSubcategory_first_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y),
+ (PreMonoidalFullSubcategory_bobj x a)~~{S}~~>(PreMonoidalFullSubcategory_bobj y a).
+ unfold hom; simpl; intros.
+ destruct a as [a apf].
+ destruct x as [x xpf].
+ destruct y as [y ypf].
+ simpl in *.
+ apply (f ⋉ a).
+ Defined.
+
+ Definition PreMonoidalFullSubcategory_second_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y),
+ (PreMonoidalFullSubcategory_bobj a x)~~{S}~~>(PreMonoidalFullSubcategory_bobj a y).
+ unfold hom; simpl; intros.
+ destruct a as [a apf].
+ destruct x as [x xpf].
+ destruct y as [y ypf].
+ simpl in *.
+ apply (a ⋊ f).
+ Defined.
+
+ Instance PreMonoidalFullSubcategory_first (a:S)
+ : Functor S S (fun x => PreMonoidalFullSubcategory_bobj x a) :=
+ { fmor := fun x y f => PreMonoidalFullSubcategory_first_fmor a f }.
+ unfold PreMonoidalFullSubcategory_first_fmor; intros; destruct a; destruct a0; destruct b; simpl in *.
+ apply (fmor_respects (-⋉x)); auto.
+ unfold PreMonoidalFullSubcategory_first_fmor; intros; destruct a; destruct a0; simpl in *.
+ apply (fmor_preserves_id (-⋉x)); auto.
+ unfold PreMonoidalFullSubcategory_first_fmor; intros;
+ destruct a; destruct a0; destruct b; destruct c; simpl in *.
+ apply (fmor_preserves_comp (-⋉x)); auto.
+ Defined.
+
+ Instance PreMonoidalFullSubcategory_second (a:S)
+ : Functor S S (fun x => PreMonoidalFullSubcategory_bobj a x) :=
+ { fmor := fun x y f => PreMonoidalFullSubcategory_second_fmor a f }.
+ unfold PreMonoidalFullSubcategory_second_fmor; intros; destruct a; destruct a0; destruct b; simpl in *.
+ apply (fmor_respects (x⋊-)); auto.
+ unfold PreMonoidalFullSubcategory_second_fmor; intros; destruct a; destruct a0; simpl in *.
+ apply (fmor_preserves_id (x⋊-)); auto.
+ unfold PreMonoidalFullSubcategory_second_fmor; intros;
+ destruct a; destruct a0; destruct b; destruct c; simpl in *.
+ apply (fmor_preserves_comp (x⋊-)); auto.
+ Defined.
+
+ Instance PreMonoidalFullSubcategory_is_Binoidal : BinoidalCat S PreMonoidalFullSubcategory_bobj :=
+ { bin_first := PreMonoidalFullSubcategory_first
+ ; bin_second := PreMonoidalFullSubcategory_second }.
+
+ Definition PreMonoidalFullSubcategory_assoc
+ : forall a b,
+ (PreMonoidalFullSubcategory_second a >>>> PreMonoidalFullSubcategory_first b) <~~~>
+ (PreMonoidalFullSubcategory_first b >>>> PreMonoidalFullSubcategory_second a).
+ Defined.
+
+ Definition PreMonoidalFullSubcategory_assoc_ll
+ : forall a b,
+ PreMonoidalFullSubcategory_second (a⊗b) <~~~>
+ PreMonoidalFullSubcategory_second b >>>> PreMonoidalFullSubcategory_second a.
+ intros.
+ Defined.
+
+ Definition PreMonoidalFullSubcategory_assoc_rr
+ : forall a b,
+ PreMonoidalFullSubcategory_first (a⊗b) <~~~>
+ PreMonoidalFullSubcategory_first a >>>> PreMonoidalFullSubcategory_first b.
+ intros.
+ Defined.
+
+ Definition PreMonoidalFullSubcategory_I := existT _ pmI Pobj_unit.
+
+ Definition PreMonoidalFullSubcategory_cancelr
+ : PreMonoidalFullSubcategory_first PreMonoidalFullSubcategory_I <~~~> functor_id _.
+ Defined.
+
+ Definition PreMonoidalFullSubcategory_cancell
+ : PreMonoidalFullSubcategory_second PreMonoidalFullSubcategory_I <~~~> functor_id _.
+ Defined.
+
+ Instance PreMonoidalFullSubcategory_PreMonoidal
+ : PreMonoidalCat PreMonoidalFullSubcategory_is_Binoidal PreMonoidalFullSubcategory_I :=
+ { pmon_assoc := PreMonoidalFullSubcategory_assoc
+ ; pmon_assoc_rr := PreMonoidalFullSubcategory_assoc_rr
+ ; pmon_assoc_ll := PreMonoidalFullSubcategory_assoc_ll
+ ; pmon_cancelr := PreMonoidalFullSubcategory_cancelr
+ ; pmon_cancell := PreMonoidalFullSubcategory_cancell
+ }.
+ Defined.
+End PreMonoidalFullSubcategory.
+*)
projects / coq-categories.git / blobdiff