X-Git-Url: http://git.megacz.com/?p=coq-categories.git;a=blobdiff_plain;f=src%2FPreMonoidalCategories.v;h=2764abde0e733059e9cec15cdbabf35fa6d5007a;hp=e09f8abb775a3e4d5b2bd3ea04875ebf5196767f;hb=fd14c25703d15bd78088c67ff3d417d435b6b136;hpb=27ffdd2265eb1c15acc62970f49d25a07bcadb05 diff --git a/src/PreMonoidalCategories.v b/src/PreMonoidalCategories.v index e09f8ab..2764abd 100644 --- a/src/PreMonoidalCategories.v +++ b/src/PreMonoidalCategories.v @@ -1,8 +1,9 @@ Generalizable All Variables. -Require Import Preamble. +Require Import Notations. Require Import Categories_ch1_3. Require Import Functors_ch1_4. Require Import Isomorphisms_ch1_5. +Require Import EpicMonic_ch2_1. Require Import InitialTerminal_ch2_2. Require Import Subcategories_ch7_1. Require Import NaturalTransformations_ch7_4. @@ -47,28 +48,109 @@ Class PreMonoidalCat `(bc:BinoidalCat(C:=C))(I:C) := * might need extra versions of the triangle/pentagon diagrams. *) +Implicit Arguments pmon_I [ 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 (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. +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 +`(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 @@ -76,22 +158,305 @@ Class PreMonoidalFunctor ; 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) +; 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. -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) + {PMFF12:Functor C1 C2 fobj12 } + (PMF12 :PreMonoidalFunctor PM1 PM2 PMFF12) `{PM3 :PreMonoidalCat(C:=C3)(I:=I3)} {fobj23:C2 -> C3 } - (PMF23 :PreMonoidalFunctor PM2 PM3 fobj23) - : PreMonoidalFunctor PM1 PM3 (fobj23 ○ fobj12). - admit. - Defined. + {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. +Notation "a >>⊗>> b" := (PreMonoidalFunctorsCompose a b). + (*******************************************************************************) (* Braided and Symmetric Categories *) @@ -99,7 +464,7 @@ Definition PreMonoidalFunctorsCompose 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) +; triangleb : forall a:C, #(pmon_cancelr a) ~~ #(br_swap a (pmon_I 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 @@ -111,144 +476,514 @@ Class SymmetricCat `(bc:BraidedCat) := }. -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 {Pobj}{Pmor}(S:SubCategory pm Pobj Pmor). + 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. + +Section IsoFullSubCategory. + Context `{C:Category}. + Context {Pobj}(S:FullSubcategory C Pobj). + + Definition iso_full {a b:C}(i:a≅b)(pa:Pobj a)(pb:Pobj b) : (existT _ _ pa) ≅ (existT _ _ pb). + set (#i : existT Pobj a pa ~~{S}~~> existT Pobj b pb) as i1. + set (iso_backward i : existT Pobj b pb ~~{S}~~> existT Pobj a pa) as i2. + refine {| iso_forward := i1 ; iso_backward := i2 |}. + unfold i1; unfold i2; unfold hom; simpl. + apply iso_comp1. + unfold i1; unfold i2; unfold hom; simpl. + apply iso_comp2. + Defined. +End IsoFullSubCategory. + +(* 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]]. - 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 central_full {a b}(f:a~~{S}~~>b) + : @CentralMorphism _ _ _ _ pm (projT1 a) (projT1 b) f -> CentralMorphism f. + intro cm. + apply Build_CentralMorphism; simpl. + intros. + destruct a as [a apf]. + destruct b as [b bpf]. + destruct c as [c cpf]. + destruct d as [d dpf]. + simpl. + apply cm. + intros. + destruct a as [a apf]. + destruct b as [b bpf]. + destruct c as [c cpf]. + destruct d as [d dpf]. + simpl. + apply cm. + Defined. - Definition PreMonoidalSubCategory_assoc + Notation "a ⊕ b" := (Pobj_closed a b). + Definition PreMonoidalFullSubcategory_assoc : forall a b, - (PreMonoidalSubCategory_second a >>>> PreMonoidalSubCategory_first b) <~~~> - (PreMonoidalSubCategory_first b >>>> PreMonoidalSubCategory_second a). - admit. - Defined. + (PreMonoidalFullSubcategory_second a >>>> PreMonoidalFullSubcategory_first b) <~~~> + (PreMonoidalFullSubcategory_first b >>>> PreMonoidalFullSubcategory_second a). + intros. + refine {| ni_iso := (fun (c:S) => iso_full S (pmon_assoc(PreMonoidalCat:=pm) _ _ _) + ((projT2 a⊕projT2 c)⊕projT2 b) + (projT2 a⊕(projT2 c⊕projT2 b))) |}. + intros; simpl. + destruct a as [a apf]. + destruct b as [b bpf]. + destruct A as [A Apf]. + destruct B as [B Bpf]. + apply (ni_commutes (pmon_assoc(PreMonoidalCat:=pm) a b) f). + Defined. + + Definition PreMonoidalFullSubcategory_assoc_ll + : forall a b, + PreMonoidalFullSubcategory_second (a⊗b) <~~~> + PreMonoidalFullSubcategory_second b >>>> PreMonoidalFullSubcategory_second a. + intros. + refine {| ni_iso := (fun (c:S) => iso_full S (pmon_assoc_ll(PreMonoidalCat:=pm) _ _ _) + ((projT2 a⊕projT2 b)⊕projT2 c) + (projT2 a⊕(projT2 b⊕projT2 c)) + ) |}. + intros; simpl. + destruct a as [a apf]. + destruct b as [b bpf]. + destruct A as [A Apf]. + destruct B as [B Bpf]. + apply (ni_commutes (pmon_assoc_ll(PreMonoidalCat:=pm) a b) f). + Defined. - Definition PreMonoidalSubCategory_assoc_ll + Definition PreMonoidalFullSubcategory_assoc_rr : forall a b, - PreMonoidalSubCategory_second (a⊗b) <~~~> - PreMonoidalSubCategory_second b >>>> PreMonoidalSubCategory_second a. + PreMonoidalFullSubcategory_first (a⊗b) <~~~> + PreMonoidalFullSubcategory_first a >>>> PreMonoidalFullSubcategory_first b. + intros. + refine {| ni_iso := (fun (c:S) => iso_full S (pmon_assoc_rr(PreMonoidalCat:=pm) _ _ _) + (projT2 c⊕(projT2 a⊕projT2 b)) + ((projT2 c⊕projT2 a)⊕projT2 b) + ) |}. + intros; simpl. + destruct a as [a apf]. + destruct b as [b bpf]. + destruct A as [A Apf]. + destruct B as [B Bpf]. + apply (ni_commutes (pmon_assoc_rr(PreMonoidalCat:=pm) a b) f). + Defined. + + Definition PreMonoidalFullSubcategory_I := existT _ pmI Pobj_unit. + + Definition PreMonoidalFullSubcategory_cancelr_iso A + : (fun x : S => PreMonoidalFullSubcategory_bobj x (existT Pobj pmI Pobj_unit)) A ≅ (fun x : S => x) A. + destruct A. + apply (iso_full S). + apply pmon_cancelr. + Defined. + + Definition PreMonoidalFullSubcategory_cancelr + : PreMonoidalFullSubcategory_first PreMonoidalFullSubcategory_I <~~~> functor_id _. intros. - admit. + refine {| ni_iso := PreMonoidalFullSubcategory_cancelr_iso |}. + intros. + destruct A as [A Apf]. + destruct B as [B Bpf]. + simpl. + apply (ni_commutes (pmon_cancelr(PreMonoidalCat:=pm)) f). Defined. - Definition PreMonoidalSubCategory_assoc_rr - : forall a b, - PreMonoidalSubCategory_first (a⊗b) <~~~> - PreMonoidalSubCategory_first a >>>> PreMonoidalSubCategory_first b. + Definition PreMonoidalFullSubcategory_cancell_iso A + : (fun x : S => PreMonoidalFullSubcategory_bobj (existT Pobj pmI Pobj_unit) x) A ≅ (fun x : S => x) A. + destruct A. + apply (iso_full S). + apply pmon_cancell. + Defined. + + Definition PreMonoidalFullSubcategory_cancell + : PreMonoidalFullSubcategory_second PreMonoidalFullSubcategory_I <~~~> functor_id _. + intros. + refine {| ni_iso := PreMonoidalFullSubcategory_cancell_iso |}. intros. - admit. + destruct A as [A Apf]. + destruct B as [B Bpf]. + simpl. + apply (ni_commutes (pmon_cancell(PreMonoidalCat:=pm)) f). Defined. - Definition PreMonoidalSubCategory_I := existT _ pmI (Pobj_unit). + 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 + }. + apply Build_Pentagon. + intros. + destruct a as [a apf]. + destruct b as [b bpf]. + destruct c as [c cpf]. + destruct d as [d dpf]. + simpl. + apply (pmon_pentagon(PreMonoidalCat:=pm)). + + apply Build_Triangle. + intros. + destruct a as [a apf]. + destruct b as [b bpf]. + simpl. + apply (pmon_triangle(PreMonoidalCat:=pm)). + simpl. + apply (pmon_triangle(PreMonoidalCat:=pm)). + + intros. + destruct a as [a apf]. + destruct c as [c cpf]. + destruct d as [d dpf]. + simpl. + apply (pmon_coherent_r(PreMonoidalCat:=pm)). + + intros. + destruct a as [a apf]. + destruct c as [c cpf]. + destruct d as [d dpf]. + simpl. + apply (pmon_coherent_l(PreMonoidalCat:=pm)). + + intros. + destruct a as [a apf]. + destruct b as [b bpf]. + destruct c as [c cpf]. + simpl. + apply central_full. + simpl. + apply (pmon_assoc_central(PreMonoidalCat:=pm)). + + intros. + destruct a as [a apf]. + simpl. + apply central_full. + simpl. + apply (pmon_cancelr_central(PreMonoidalCat:=pm)). - Definition PreMonoidalSubCategory_cancelr : PreMonoidalSubCategory_first PreMonoidalSubCategory_I <~~~> functor_id _. - admit. + intros. + destruct a as [a apf]. + simpl. + apply central_full. + simpl. + apply (pmon_cancell_central(PreMonoidalCat:=pm)). Defined. - Definition PreMonoidalSubCategory_cancell : PreMonoidalSubCategory_second PreMonoidalSubCategory_I <~~~> functor_id _. - admit. + Instance inclusion_first : ∀a : S, + FullSubcategoryInclusionFunctor S >>>> + - ⋉(FullSubcategoryInclusionFunctor S) a <~~~> + - ⋉a >>>> FullSubcategoryInclusionFunctor S + := { ni_iso := fun A => iso_id ((projT1 A)⊗(projT1 a)) }. + intros; simpl. + symmetry. + setoid_rewrite right_identity. + setoid_rewrite left_identity. + destruct A. + destruct B. + destruct a. + simpl. + reflexivity. 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 - }. - admit. - admit. - admit. - admit. - admit. - admit. - admit. - Defined. - -End PreMonoidalSubCategory. + Instance inclusion_second : ∀a : S, + FullSubcategoryInclusionFunctor S >>>> + (FullSubcategoryInclusionFunctor S) a ⋊- <~~~> + a ⋊- >>>> FullSubcategoryInclusionFunctor S + := { ni_iso := fun A => iso_id ((projT1 a)⊗(projT1 A)) }. + intros; simpl. + symmetry. + setoid_rewrite right_identity. + setoid_rewrite left_identity. + destruct A. + destruct B. + destruct a. + simpl. + reflexivity. + Defined. + + (* Curiously, the inclusion functor for a PREmonoidal category isn't necessarily premonoidal (it might fail to preserve + * the center. But in the monoidal case we're okay *) + Instance PreMonoidalFullSubcategoryInclusionFunctor_PreMonoidal (mc:CommutativeCat pm) + : PreMonoidalFunctor PreMonoidalFullSubcategory_PreMonoidal pm (FullSubcategoryInclusionFunctor S) := + { mf_i := iso_id _ + ; mf_first := inclusion_first + ; mf_first := inclusion_second + }. + intros; destruct a; destruct b; reflexivity. + intros; destruct a; destruct b; simpl in *. + apply mc. + intros; destruct b; simpl. + setoid_rewrite right_identity. + setoid_rewrite fmor_preserves_id. + setoid_rewrite left_identity. + reflexivity. + intros; destruct a; simpl. + setoid_rewrite right_identity. + setoid_rewrite fmor_preserves_id. + setoid_rewrite left_identity. + reflexivity. + intros; destruct a; destruct b; destruct c; simpl. + setoid_rewrite right_identity. + setoid_rewrite fmor_preserves_id. + setoid_rewrite left_identity. + setoid_rewrite right_identity. + reflexivity. + Defined. + +End PreMonoidalFullSubcategory. +