Generalizable All Variables. Require Import Preamble. Require Import Categories_ch1_3. Require Import Functors_ch1_4. Require Import Isomorphisms_ch1_5. Require Import InitialTerminal_ch2_2. Require Import Subcategories_ch7_1. Require Import NaturalTransformations_ch7_4. Require Import NaturalIsomorphisms_ch7_5. Require Import Coherence_ch7_8. Require Import BinoidalCategories. (* 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_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" * 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 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 (a ⊗ b)) ~~ #((pmon_assoc a EI) b) >>> (a ⋊-) \ #(pmon_cancelr 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 a b)) as xx. (* FIXME *) Admitted. Class PreMonoidalFunctor `(PM1:PreMonoidalCat(C:=C1)(I:=I1)) `(PM2:PreMonoidalCat(C:=C2)(I:=I2)) (fobj : C1 -> C2 ) := { mf_F :> Functor C1 C2 fobj ; 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_second _ _) >>> #(mf_second _ _) ~~ #(mf_second _ _) ⋉ _ >>> #(mf_second _ _) >>> 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 } (PMF12 :PreMonoidalFunctor PM1 PM2 fobj12) `{PM3 :PreMonoidalCat(C:=C3)(I:=I3)} {fobj23:C2 -> C3 } (PMF23 :PreMonoidalFunctor PM2 PM3 fobj23). 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_respects PMF12 PMF12 (PMF23 >>>> - ⋉fobj23 (fobj12 a)) (- ⋉fobj12 a >>>> PMF23)). apply ni_id. 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_respects; [ idtac | apply ni_id ]. 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_respects PMF12 PMF12 (PMF23 >>>> fobj23 (fobj12 a) ⋊-) (fobj12 a ⋊- >>>> PMF23)). apply ni_id. 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_respects; [ idtac | apply ni_id ]. apply ni_inv. apply mf_second12. Defined. Lemma compose_assoc_coherence a b c : (#((pmon_assoc (compose_mf a) (fobj23 (fobj12 c))) (compose_mf b)) >>> compose_mf a ⋊ #((compose_mf_second b) c)) >>> #((compose_mf_second a) (b ⊗ c)) ~~ (#((compose_mf_second a) b) ⋉ fobj23 (fobj12 c) >>> #((compose_mf_second (a ⊗ b)) c)) >>> compose_mf \ #((pmon_assoc a c) b). (* set (mf_assoc a b c) as x. set (mf_assoc (fobj12 a) (fobj12 b) (fobj12 c)) as x'. unfold functor_fobj in *. simpl in *. etransitivity. etransitivity. etransitivity. Focus 3. apply x'. apply iso_shift_left' in x'. unfold compose_mf_second; simpl. unfold functor_fobj; simpl. set (mf_second (fobj12 b)) as m. assert (mf_second (fobj12 b)=m). reflexivity. destruct m; simpl. setoid_rewrite <- fmor_preserves_comp. setoid_rewrite <- fmor_preserves_comp. setoid_rewrite <- fmor_preserves_comp. setoid_rewrite <- fmor_preserves_comp. setoid_rewrite <- fmor_preserves_comp. setoid_rewrite fmor_preserves_id. setoid_rewrite fmor_preserves_id. setoid_rewrite fmor_preserves_id. setoid_rewrite right_identity. setoid_rewrite left_identity. setoid_rewrite left_identity. setoid_rewrite left_identity. set (mf_second (fobj12 (a ⊗ b))) as m''. assert (mf_second (fobj12 (a ⊗ b))=m''). reflexivity. destruct m''; simpl. unfold functor_fobj; simpl. setoid_rewrite fmor_preserves_id. setoid_rewrite fmor_preserves_id. setoid_rewrite right_identity. setoid_rewrite left_identity. setoid_rewrite left_identity. setoid_rewrite left_identity. set (mf_second (fobj12 a)) as m'. assert (mf_second (fobj12 a)=m'). reflexivity. destruct m'; simpl. setoid_rewrite <- fmor_preserves_comp. setoid_rewrite <- fmor_preserves_comp. setoid_rewrite <- fmor_preserves_comp. setoid_rewrite <- fmor_preserves_comp. setoid_rewrite <- fmor_preserves_comp. setoid_rewrite left_identity. setoid_rewrite left_identity. setoid_rewrite left_identity. setoid_rewrite right_identity. assert (fobj23 (fobj12 a) ⋊ PMF23 \ id (PMF12 (b ⊗ c)) ~~ id _). (* *) setoid_rewrite H2. setoid_rewrite left_identity. assert ((id (fobj23 (fobj12 a) ⊗ fobj23 (fobj12 b)) ⋉ fobj23 (fobj12 c)) ~~ id _). (* *) setoid_rewrite H3. setoid_rewrite left_identity. assert (id (fobj23 (fobj12 a ⊗ fobj12 b)) ⋉ fobj23 (fobj12 c) ~~ id _). (* *) setoid_rewrite H4. setoid_rewrite left_identity. clear H4. setoid_rewrite left_identity. assert (id (fobj23 (fobj12 (a ⊗ b))) ⋉ fobj23 (fobj12 c) ~~ id _). (* *) setoid_rewrite H4. setoid_rewrite right_identity. clear H4. assert ((fobj23 (fobj12 a) ⋊ PMF23 \ id (PMF12 b)) ⋉ fobj23 (fobj12 c) ~~ id _). (* *) setoid_rewrite H4. setoid_rewrite left_identity. clear H4. unfold functor_comp in ni_commutes0; simpl in ni_commutes0. unfold functor_comp in ni_commutes; simpl in ni_commutes. unfold functor_comp in ni_commutes1; simpl in ni_commutes1. unfold functor_fobj in *. simpl in *. setoid_rewrite x in x'. rewrite H1. set (ni_commutes0 (a ) setoid_rewrite fmor_preserves_id. etransitivity. eapply comp_respects. reflexivity. eapply comp_respects. eapply comp_respects. apply Focus 2. eapply fmor_preserves_id. setoid_rewrite (fmor_preserves_id PMF23). *) admit. Qed. Instance PreMonoidalFunctorsCompose : PreMonoidalFunctor PM1 PM3 (fobj23 ○ fobj12) := { mf_i := compose_mf_i ; mf_F := compose_mf ; 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. unfold functor_comp. unfold functor_fobj; simpl. set (ni_commutes _ _ (id (fobj12 b))) as x. unfold functor_comp in x. simpl in x. unfold functor_fobj in x. symmetry in x. etransitivity. apply x. clear x. set (ni_commutes0 _ _ (id (fobj12 a))) as x'. unfold functor_comp in x'. simpl in x'. unfold functor_fobj in x'. etransitivity; [ idtac | apply x' ]. clear x'. setoid_rewrite fmor_preserves_id. setoid_rewrite fmor_preserves_id. setoid_rewrite right_identity. rewrite <- H in later. rewrite <- H0 in later. simpl in later. apply later. apply fmor_respects. apply (mf_consistent a b). 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 _ _ (id (fobj12 I1))) as x. unfold functor_comp in x. unfold functor_fobj in x. simpl in x. setoid_rewrite <- x. clear x. setoid_rewrite fmor_preserves_id. setoid_rewrite fmor_preserves_id. setoid_rewrite right_identity. rewrite H. simpl. clear H. unfold functor_comp in ni_commutes. simpl in ni_commutes. apply ni_commutes. 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 _ _ (id (fobj12 I1))) as x. unfold functor_comp in x. unfold functor_fobj in x. simpl in x. setoid_rewrite <- x. clear x. setoid_rewrite fmor_preserves_id. setoid_rewrite fmor_preserves_id. setoid_rewrite right_identity. rewrite H. simpl. clear H. unfold functor_comp in ni_commutes. simpl in ni_commutes. apply ni_commutes. apply compose_assoc_coherence. 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 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. *)