Class PreMonoidalFunctor
`(PM1:PreMonoidalCat(C:=C1)(I:=I1))
`(PM2:PreMonoidalCat(C:=C2)(I:=I2))
- (fobj : C1 -> C2 ) :=
+ (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
}.
Coercion mf_F : PreMonoidalFunctor >-> Functor.
-Definition PreMonoidalFunctorsCompose
+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)
- : PreMonoidalFunctor PM1 PM3 (fobj23 ○ fobj12).
- admit.
- Defined.
+ (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 *)
}.
-Section PreMonoidalSubCategory.
+(* 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}{Pmor}(S:SubCategory pm Pobj Pmor).
+ 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]].
- Context (Pmor_first: forall {a}{b}{c}{f}(pa:Pobj a)(pb:Pobj b)(pc:Pobj c)(pf:Pmor _ _ pa pb f),
- Pmor _ _ (Pobj_closed pa pc) (Pobj_closed pb pc) (f ⋉ c)).
- Context (Pmor_second: forall {a}{b}{c}{f}(pa:Pobj a)(pb:Pobj b)(pc:Pobj c)(pf:Pmor _ _ pa pb f),
- Pmor _ _ (Pobj_closed pc pa) (Pobj_closed pc pb) (c ⋊ f)).
- Context (Pmor_assoc: forall {a}{b}{c}(pa:Pobj a)(pb:Pobj b)(pc:Pobj c),
- Pmor _ _
- (Pobj_closed (Pobj_closed pa pb) pc)
- (Pobj_closed pa (Pobj_closed pb pc))
- #(pmon_assoc a c b)).
- Context (Pmor_unassoc: forall {a}{b}{c}(pa:Pobj a)(pb:Pobj b)(pc:Pobj c),
- Pmor _ _
- (Pobj_closed pa (Pobj_closed pb pc))
- (Pobj_closed (Pobj_closed pa pb) pc)
- #(pmon_assoc a c b)⁻¹).
- Context (Pmor_cancell: forall {a}(pa:Pobj a),
- Pmor _ _ (Pobj_closed Pobj_unit pa) pa
- #(pmon_cancell a)).
- Context (Pmor_uncancell: forall {a}(pa:Pobj a),
- Pmor _ _ pa (Pobj_closed Pobj_unit pa)
- #(pmon_cancell a)⁻¹).
- Context (Pmor_cancelr: forall {a}(pa:Pobj a),
- Pmor _ _ (Pobj_closed pa Pobj_unit) pa
- #(pmon_cancelr a)).
- Context (Pmor_uncancelr: forall {a}(pa:Pobj a),
- Pmor _ _ pa (Pobj_closed pa Pobj_unit)
- #(pmon_cancelr a)⁻¹).
- Implicit Arguments Pmor_first [[a][b][c][f]].
- Implicit Arguments Pmor_second [[a][b][c][f]].
- Definition PreMonoidalSubCategory_bobj (x y:S) :=
+ Definition PreMonoidalFullSubcategory_bobj (x y:S) :=
existT Pobj _ (Pobj_closed (projT2 x) (projT2 y)).
- Definition PreMonoidalSubCategory_first_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y),
- (PreMonoidalSubCategory_bobj x a)~~{S}~~>(PreMonoidalSubCategory_bobj y a).
+ 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 f.
destruct a as [a apf].
destruct x as [x xpf].
destruct y as [y ypf].
simpl in *.
- exists (x0 ⋉ a).
- apply Pmor_first; auto.
+ apply (f ⋉ a).
Defined.
- Definition PreMonoidalSubCategory_second_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y),
- (PreMonoidalSubCategory_bobj a x)~~{S}~~>(PreMonoidalSubCategory_bobj a y).
+ 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 f.
destruct a as [a apf].
destruct x as [x xpf].
destruct y as [y ypf].
simpl in *.
- exists (a ⋊ x0).
- apply Pmor_second; auto.
+ apply (a ⋊ f).
Defined.
- Instance PreMonoidalSubCategory_first (a:S)
- : Functor (S) (S) (fun x => PreMonoidalSubCategory_bobj x a) :=
- { fmor := fun x y f => PreMonoidalSubCategory_first_fmor a f }.
- unfold PreMonoidalSubCategory_first_fmor; intros; destruct a; destruct a0; destruct b; destruct f; destruct f'; simpl in *.
+ 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 PreMonoidalSubCategory_first_fmor; intros; destruct a; destruct a0; simpl in *.
+ unfold PreMonoidalFullSubcategory_first_fmor; intros; destruct a; destruct a0; simpl in *.
apply (fmor_preserves_id (-⋉x)); auto.
- unfold PreMonoidalSubCategory_first_fmor; intros;
- destruct a; destruct a0; destruct b; destruct c; destruct f; destruct g; simpl in *.
+ unfold PreMonoidalFullSubcategory_first_fmor; intros;
+ destruct a; destruct a0; destruct b; destruct c; simpl in *.
apply (fmor_preserves_comp (-⋉x)); auto.
Defined.
- Instance PreMonoidalSubCategory_second (a:S)
- : Functor (S) (S) (fun x => PreMonoidalSubCategory_bobj a x) :=
- { fmor := fun x y f => PreMonoidalSubCategory_second_fmor a f }.
- unfold PreMonoidalSubCategory_second_fmor; intros; destruct a; destruct a0; destruct b; destruct f; destruct f'; simpl in *.
+ 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 PreMonoidalSubCategory_second_fmor; intros; destruct a; destruct a0; simpl in *.
+ unfold PreMonoidalFullSubcategory_second_fmor; intros; destruct a; destruct a0; simpl in *.
apply (fmor_preserves_id (x⋊-)); auto.
- unfold PreMonoidalSubCategory_second_fmor; intros;
- destruct a; destruct a0; destruct b; destruct c; destruct f; destruct g; simpl in *.
+ unfold PreMonoidalFullSubcategory_second_fmor; intros;
+ destruct a; destruct a0; destruct b; destruct c; simpl in *.
apply (fmor_preserves_comp (x⋊-)); auto.
Defined.
- Instance PreMonoidalSubCategory_is_Binoidal : BinoidalCat S PreMonoidalSubCategory_bobj :=
- { bin_first := PreMonoidalSubCategory_first
- ; bin_second := PreMonoidalSubCategory_second }.
+ Instance PreMonoidalFullSubcategory_is_Binoidal : BinoidalCat S PreMonoidalFullSubcategory_bobj :=
+ { bin_first := PreMonoidalFullSubcategory_first
+ ; bin_second := PreMonoidalFullSubcategory_second }.
- Definition PreMonoidalSubCategory_assoc
+ Definition PreMonoidalFullSubcategory_assoc
: forall a b,
- (PreMonoidalSubCategory_second a >>>> PreMonoidalSubCategory_first b) <~~~>
- (PreMonoidalSubCategory_first b >>>> PreMonoidalSubCategory_second a).
- admit.
+ (PreMonoidalFullSubcategory_second a >>>> PreMonoidalFullSubcategory_first b) <~~~>
+ (PreMonoidalFullSubcategory_first b >>>> PreMonoidalFullSubcategory_second a).
Defined.
- Definition PreMonoidalSubCategory_assoc_ll
+ Definition PreMonoidalFullSubcategory_assoc_ll
: forall a b,
- PreMonoidalSubCategory_second (a⊗b) <~~~>
- PreMonoidalSubCategory_second b >>>> PreMonoidalSubCategory_second a.
+ PreMonoidalFullSubcategory_second (a⊗b) <~~~>
+ PreMonoidalFullSubcategory_second b >>>> PreMonoidalFullSubcategory_second a.
intros.
- admit.
Defined.
- Definition PreMonoidalSubCategory_assoc_rr
+ Definition PreMonoidalFullSubcategory_assoc_rr
: forall a b,
- PreMonoidalSubCategory_first (a⊗b) <~~~>
- PreMonoidalSubCategory_first a >>>> PreMonoidalSubCategory_first b.
+ PreMonoidalFullSubcategory_first (a⊗b) <~~~>
+ PreMonoidalFullSubcategory_first a >>>> PreMonoidalFullSubcategory_first b.
intros.
- admit.
Defined.
- Definition PreMonoidalSubCategory_I := existT _ pmI (Pobj_unit).
+ Definition PreMonoidalFullSubcategory_I := existT _ pmI Pobj_unit.
- Definition PreMonoidalSubCategory_cancelr : PreMonoidalSubCategory_first PreMonoidalSubCategory_I <~~~> functor_id _.
- admit.
+ Definition PreMonoidalFullSubcategory_cancelr
+ : PreMonoidalFullSubcategory_first PreMonoidalFullSubcategory_I <~~~> functor_id _.
Defined.
- Definition PreMonoidalSubCategory_cancell : PreMonoidalSubCategory_second PreMonoidalSubCategory_I <~~~> functor_id _.
- admit.
+ Definition PreMonoidalFullSubcategory_cancell
+ : PreMonoidalFullSubcategory_second PreMonoidalFullSubcategory_I <~~~> functor_id _.
Defined.
- Instance PreMonoidalSubCategory_PreMonoidal : PreMonoidalCat PreMonoidalSubCategory_is_Binoidal PreMonoidalSubCategory_I :=
- { pmon_assoc := PreMonoidalSubCategory_assoc
- ; pmon_assoc_rr := PreMonoidalSubCategory_assoc_rr
- ; pmon_assoc_ll := PreMonoidalSubCategory_assoc_ll
- ; pmon_cancelr := PreMonoidalSubCategory_cancelr
- ; pmon_cancell := PreMonoidalSubCategory_cancell
+ 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
}.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
Defined.
-
-End PreMonoidalSubCategory.
+End PreMonoidalFullSubcategory.
+*)
\ No newline at end of file
projects / coq-categories.git / blobdiff