1 Generalizable All Variables.
2 Require Import Notations.
3 Require Import Categories_ch1_3.
4 Require Import Functors_ch1_4.
5 Require Import Isomorphisms_ch1_5.
6 Require Import EpicMonic_ch2_1.
7 Require Import InitialTerminal_ch2_2.
8 Require Import Subcategories_ch7_1.
9 Require Import NaturalTransformations_ch7_4.
10 Require Import NaturalIsomorphisms_ch7_5.
11 Require Import Coherence_ch7_8.
12 Require Import BinoidalCategories.
15 Class PreMonoidalCat `(bc:BinoidalCat(C:=C))(I:C) :=
19 ; pmon_assoc : forall a b, (bin_second a >>>> bin_first b) <~~~> (bin_first b >>>> bin_second a)
20 ; pmon_cancelr : (bin_first I) <~~~> functor_id C
21 ; pmon_cancell : (bin_second I) <~~~> functor_id C
22 ; 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))
23 ; 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))
24 (fun a => #(pmon_cancell a)) (fun a => #(pmon_cancelr a))
25 ; pmon_assoc_rr : forall a b, (bin_first (a⊗b)) <~~~> (bin_first a >>>> bin_first b)
26 ; pmon_assoc_ll : forall a b, (bin_second (a⊗b)) <~~~> (bin_second b >>>> bin_second a)
27 ; pmon_coherent_r : forall a c d:C, #(pmon_assoc_rr c d a) ~~ #(pmon_assoc a d c)⁻¹
28 ; pmon_coherent_l : forall a c d:C, #(pmon_assoc_ll c a d) ~~ #(pmon_assoc c d a)
29 ; pmon_assoc_central : forall a b c, CentralMorphism #(pmon_assoc a b c)
30 ; pmon_cancelr_central : forall a , CentralMorphism #(pmon_cancelr a)
31 ; pmon_cancell_central : forall a , CentralMorphism #(pmon_cancell a)
34 * Premonoidal categories actually have three associators (the "f"
35 * indicates the position in which the operation is natural:
37 * assoc : (A ⋊ f) ⋉ C <-> A ⋊ (f ⋉ C)
38 * assoc_rr : (f ⋉ B) ⋉ C <-> f ⋉ (B ⊗ C)
39 * assoc_ll : (A ⋊ B) ⋊ f <-> (A ⊗ B) ⋊ f
41 * Fortunately, in a monoidal category these are all the same natural
42 * isomorphism (and in any case -- monoidal or not -- the objects in
43 * the left column are all the same and the objects in the right
44 * column are all the same). This formalization assumes that is the
45 * case even for premonoidal categories with non-central maps, in
46 * order to keep the complexity manageable. I don't know much about
47 * the consequences of having them and letting them be different; you
48 * might need extra versions of the triangle/pentagon diagrams.
51 Implicit Arguments pmon_cancell [ Ob Hom C bin_obj' bc I PreMonoidalCat ].
52 Implicit Arguments pmon_cancelr [ Ob Hom C bin_obj' bc I PreMonoidalCat ].
53 Implicit Arguments pmon_assoc [ Ob Hom C bin_obj' bc I PreMonoidalCat ].
54 Coercion pmon_bin : PreMonoidalCat >-> BinoidalCat.
56 (* this turns out to be Exercise VII.1.1 from Mac Lane's CWM *)
57 Lemma MacLane_ex_VII_1_1 `{mn:PreMonoidalCat(I:=EI)} b a
59 let α := fun a b c => #((pmon_assoc a c) b)
60 in α a b EI >>> _ ⋊ #(pmon_cancelr _) ~~ #(pmon_cancelr _).
64 (* following Mac Lane's hint, we aim for (λ >>> α >>> λ×1)~~(λ >>> λ) *)
65 set (epic _ (iso_epic (pmon_cancelr ((a⊗b)⊗EI)))) as q.
69 (* next, we show that the hint goal above is implied by the bottom-left 1/5th of the big whiteboard diagram *)
70 set (ni_commutes pmon_cancelr (α a b EI)) as q.
71 setoid_rewrite <- associativity.
74 setoid_rewrite associativity.
76 set (ni_commutes pmon_cancelr (a ⋊ #(pmon_cancelr b))) as q.
81 set (ni_commutes pmon_cancelr (#(pmon_cancelr (a⊗b)))) as q.
86 setoid_rewrite <- associativity.
87 apply comp_respects; try reflexivity.
89 (* now we carry out the proof in the whiteboard diagram, starting from the pentagon diagram *)
92 assert (α (a⊗b) EI EI >>> α _ _ _ >>> (_ ⋊ (_ ⋊ #(pmon_cancell _))) ~~ #(pmon_cancelr _) ⋉ _ >>> α _ _ _).
93 set (pmon_triangle (a⊗b) EI) as tria.
98 setoid_rewrite associativity.
99 apply comp_respects; try reflexivity.
100 set (ni_commutes (pmon_assoc_ll a b) #(pmon_cancell EI)) as x.
102 setoid_rewrite pmon_coherent_l in x.
106 assert (((#((pmon_assoc a EI) b) ⋉ EI >>> #((pmon_assoc a EI) (b ⊗ EI))) >>>
107 a ⋊ #((pmon_assoc b EI) EI)) >>> a ⋊ (b ⋊ #(pmon_cancell EI))
108 ~~ α _ _ _ ⋉ _ >>> (_ ⋊ #(pmon_cancelr _)) ⋉ _ >>> α _ _ _).
111 repeat setoid_rewrite associativity.
112 apply comp_respects; try reflexivity.
114 set (ni_commutes (pmon_assoc a EI) (#(pmon_cancelr b) )) as x.
116 setoid_rewrite <- associativity.
121 setoid_rewrite associativity.
122 apply comp_respects; try reflexivity.
123 setoid_rewrite (fmor_preserves_comp (a⋊-)).
124 apply (fmor_respects (a⋊-)).
126 set (pmon_triangle b EI) as tria.
131 set (pmon_pentagon a b EI EI) as penta. unfold pmon_pentagon in penta. simpl in penta.
133 set (@comp_respects _ _ _ _ _ _ _ _ penta (a ⋊ (b ⋊ #(pmon_cancell EI))) (a ⋊ (b ⋊ #(pmon_cancell EI)))) as qq.
135 setoid_rewrite H in qq.
137 setoid_rewrite H0 in qq.
141 apply (monic _ (iso_monic ((pmon_assoc a EI) b))).
147 Class PreMonoidalFunctor
148 `(PM1 : PreMonoidalCat(C:=C1)(I:=I1))
149 `(PM2 : PreMonoidalCat(C:=C2)(I:=I2))
151 (F : Functor C1 C2 fobj ) :=
153 ; mf_i : I2 ≅ mf_F I1
154 ; mf_first : ∀ a, mf_F >>>> bin_first (mf_F a) <~~~> bin_first a >>>> mf_F
155 ; mf_second : ∀ a, mf_F >>>> bin_second (mf_F a) <~~~> bin_second a >>>> mf_F
156 ; mf_consistent : ∀ a b, #(mf_first a b) ~~ #(mf_second b a)
157 ; mf_center : forall `(f:a~>b), CentralMorphism f -> CentralMorphism (mf_F \ f)
158 ; mf_cancell : ∀ b, #(pmon_cancell _) ~~ #mf_i ⋉ _ >>> #(mf_first b I1) >>> mf_F \ #(pmon_cancell b)
159 ; mf_cancelr : ∀ a, #(pmon_cancelr _) ~~ _ ⋊ #mf_i >>> #(mf_second a I1) >>> mf_F \ #(pmon_cancelr a)
160 ; mf_assoc : ∀ a b c, #(pmon_assoc _ _ _) >>> _ ⋊ #(mf_first _ _) >>> #(mf_second _ _) ~~
161 #(mf_second _ _) ⋉ _ >>> #(mf_first _ _) >>> mf_F \ #(pmon_assoc a c b)
163 Coercion mf_F : PreMonoidalFunctor >-> Functor.
165 Section PreMonoidalFunctorsCompose.
167 `{PM1 :PreMonoidalCat(C:=C1)(I:=I1)}
168 `{PM2 :PreMonoidalCat(C:=C2)(I:=I2)}
170 {PMFF12:Functor C1 C2 fobj12 }
171 (PMF12 :PreMonoidalFunctor PM1 PM2 PMFF12)
172 `{PM3 :PreMonoidalCat(C:=C3)(I:=I3)}
174 {PMFF23:Functor C2 C3 fobj23 }
175 (PMF23 :PreMonoidalFunctor PM2 PM3 PMFF23).
177 Definition compose_mf := PMF12 >>>> PMF23.
179 Definition compose_mf_i : I3 ≅ PMF23 (PMF12 I1).
181 apply (mf_i(PreMonoidalFunctor:=PMF23)).
182 apply functors_preserve_isos.
183 apply (mf_i(PreMonoidalFunctor:=PMF12)).
186 Definition compose_mf_first a : compose_mf >>>> bin_first (compose_mf a) <~~~> bin_first a >>>> compose_mf.
187 set (mf_first(PreMonoidalFunctor:=PMF12) a) as mf_first12.
188 set (mf_first(PreMonoidalFunctor:=PMF23) (PMF12 a)) as mf_first23.
189 unfold functor_fobj in *; simpl in *.
192 apply (ni_associativity PMF12 PMF23 (- ⋉fobj23 (fobj12 a))).
194 apply (ni_respects1 PMF12 (PMF23 >>>> - ⋉fobj23 (fobj12 a)) (- ⋉fobj12 a >>>> PMF23)).
200 apply (ni_associativity PMF12 (- ⋉fobj12 a) PMF23).
205 eapply (ni_associativity _ PMF12 PMF23).
212 Definition compose_mf_second a : compose_mf >>>> bin_second (compose_mf a) <~~~> bin_second a >>>> compose_mf.
213 set (mf_second(PreMonoidalFunctor:=PMF12) a) as mf_second12.
214 set (mf_second(PreMonoidalFunctor:=PMF23) (PMF12 a)) as mf_second23.
215 unfold functor_fobj in *; simpl in *.
218 apply (ni_associativity PMF12 PMF23 (fobj23 (fobj12 a) ⋊-)).
220 apply (ni_respects1 PMF12 (PMF23 >>>> fobj23 (fobj12 a) ⋊-) (fobj12 a ⋊- >>>> PMF23)).
226 apply (ni_associativity PMF12 (fobj12 a ⋊ -) PMF23).
231 eapply (ni_associativity (a ⋊-) PMF12 PMF23).
238 (* this proof is really gross; I will write a better one some other day *)
239 Lemma mf_associativity_comp :
241 (#((pmon_assoc (compose_mf a) (compose_mf c)) (fobj23 (fobj12 b))) >>>
242 compose_mf a ⋊ #((compose_mf_first c) b)) >>>
243 #((compose_mf_second a) (b ⊗ c)) ~~
244 (#((compose_mf_second a) b) ⋉ compose_mf c >>>
245 #((compose_mf_first c) (a ⊗ b))) >>> compose_mf \ #((pmon_assoc a c) b).
247 unfold compose_mf_second; simpl.
248 unfold compose_mf_first; simpl.
249 unfold functor_comp; simpl.
251 unfold functor_fobj; simpl.
253 set (mf_first (fobj12 c)) as m'.
254 assert (mf_first (fobj12 c)=m'). reflexivity.
257 set (mf_second (fobj12 a)) as m.
258 assert (mf_second (fobj12 a)=m). reflexivity.
261 Implicit Arguments id [[Ob][Hom][Category][a]].
266 repeat setoid_rewrite <- fmor_preserves_comp.
267 repeat setoid_rewrite fmor_preserves_id.
268 repeat setoid_rewrite left_identity.
269 repeat setoid_rewrite right_identity.
273 repeat setoid_rewrite <- fmor_preserves_comp.
274 repeat setoid_rewrite fmor_preserves_id.
275 repeat setoid_rewrite left_identity.
276 repeat setoid_rewrite right_identity.
279 assert ( (#((pmon_assoc (fobj23 (fobj12 a)) (fobj23 (fobj12 c)))
280 (fobj23 (fobj12 b))) >>>
283 (#(ni_iso (fobj12 b)) >>> ( (PMF23 \ #((mf_first c) b) ))))) >>>
285 (#(ni_iso0 (fobj12 (b ⊗ c))) >>>
286 ((PMF23 \ #((mf_second a) (b ⊗ c)))))) ~~
288 (#(ni_iso0 (fobj12 b)) >>> ( (PMF23 \ #((mf_second a) b) ))))
289 ⋉ fobj23 (fobj12 c) >>>
291 (#(ni_iso (fobj12 (a ⊗ b))) >>>
292 ( (PMF23 \ #((mf_first c) (a ⊗ b))))))) >>>
293 PMF23 \ (PMF12 \ #((pmon_assoc a c) b))
296 repeat setoid_rewrite associativity.
297 setoid_rewrite (fmor_preserves_comp PMF23).
298 unfold functor_comp in *.
299 unfold functor_fobj in *.
301 rename ni_commutes into ni_commutes7.
302 set (mf_assoc(PreMonoidalFunctor:=PMF12)) as q.
303 set (ni_commutes7 _ _ (#((mf_second a) b))) as q'.
305 repeat setoid_rewrite associativity.
307 setoid_rewrite <- (fmor_preserves_comp (-⋉ fobj23 (fobj12 c))).
308 repeat setoid_rewrite <- associativity.
309 setoid_rewrite juggle1.
310 setoid_rewrite <- q'.
311 repeat setoid_rewrite associativity.
312 setoid_rewrite fmor_preserves_comp.
314 unfold functor_fobj in *.
316 repeat setoid_rewrite <- associativity.
319 repeat setoid_rewrite <- fmor_preserves_comp.
320 repeat setoid_rewrite <- associativity.
321 apply comp_respects; try reflexivity.
323 set (mf_assoc(PreMonoidalFunctor:=PMF23) (fobj12 a) (fobj12 b) (fobj12 c)) as q.
324 unfold functor_fobj in *.
330 repeat setoid_rewrite <- associativity.
331 repeat setoid_rewrite <- associativity in q.
334 unfold functor_fobj; simpl.
336 repeat setoid_rewrite associativity.
337 apply comp_respects; try reflexivity.
338 apply comp_respects; try reflexivity.
341 repeat setoid_rewrite associativity.
342 repeat setoid_rewrite associativity in H1.
343 repeat setoid_rewrite <- fmor_preserves_comp in H1.
344 repeat setoid_rewrite associativity in H1.
347 Implicit Arguments id [[Ob][Hom][Category]].
349 (* this proof is really gross; I will write a better one some other day *)
350 Instance PreMonoidalFunctorsCompose : PreMonoidalFunctor PM1 PM3 compose_mf :=
351 { mf_i := compose_mf_i
352 ; mf_first := compose_mf_first
353 ; mf_second := compose_mf_second }.
355 intros; unfold compose_mf_first; unfold compose_mf_second.
356 set (mf_first (PMF12 a)) as x in *.
357 set (mf_second (PMF12 b)) as y in *.
358 assert (x=mf_first (PMF12 a)). reflexivity.
359 assert (y=mf_second (PMF12 b)). reflexivity.
363 repeat setoid_rewrite left_identity.
364 repeat setoid_rewrite right_identity.
365 set (mf_consistent (PMF12 a) (PMF12 b)) as later.
366 apply comp_respects; try reflexivity.
367 rewrite <- H in later.
368 rewrite <- H0 in later.
381 unfold compose_mf_first; simpl.
382 set (mf_first (PMF12 b)) as m.
383 assert (mf_first (PMF12 b)=m). reflexivity.
386 unfold functor_fobj; simpl.
387 repeat setoid_rewrite <- fmor_preserves_comp.
388 repeat setoid_rewrite left_identity.
389 repeat setoid_rewrite right_identity.
391 set (mf_cancell b) as y.
392 set (mf_cancell (fobj12 b)) as y'.
393 unfold functor_fobj in *.
394 setoid_rewrite y in y'.
396 setoid_rewrite <- fmor_preserves_comp in y'.
397 setoid_rewrite <- fmor_preserves_comp in y'.
402 repeat setoid_rewrite <- associativity.
403 apply comp_respects; try reflexivity.
404 apply comp_respects; try reflexivity.
405 repeat setoid_rewrite associativity.
406 apply comp_respects; try reflexivity.
408 set (ni_commutes _ _ #mf_i) as x.
409 unfold functor_comp in x.
410 unfold functor_fobj in x.
417 unfold compose_mf_second; simpl.
418 set (mf_second (PMF12 a)) as m.
419 assert (mf_second (PMF12 a)=m). reflexivity.
422 unfold functor_fobj; simpl.
423 repeat setoid_rewrite <- fmor_preserves_comp.
424 repeat setoid_rewrite left_identity.
425 repeat setoid_rewrite right_identity.
427 set (mf_cancelr a) as y.
428 set (mf_cancelr (fobj12 a)) as y'.
429 unfold functor_fobj in *.
430 setoid_rewrite y in y'.
432 setoid_rewrite <- fmor_preserves_comp in y'.
433 setoid_rewrite <- fmor_preserves_comp in y'.
438 repeat setoid_rewrite <- associativity.
439 apply comp_respects; try reflexivity.
440 apply comp_respects; try reflexivity.
441 repeat setoid_rewrite associativity.
442 apply comp_respects; try reflexivity.
444 set (ni_commutes _ _ #mf_i) as x.
445 unfold functor_comp in x.
446 unfold functor_fobj in x.
452 apply mf_associativity_comp.
456 End PreMonoidalFunctorsCompose.
459 (*******************************************************************************)
460 (* Braided and Symmetric Categories *)
462 Class BraidedCat `(mc:PreMonoidalCat) :=
463 { br_niso : forall a, bin_first a <~~~> bin_second a
464 ; br_swap := fun a b => ni_iso (br_niso b) a
465 ; triangleb : forall a:C, #(pmon_cancelr a) ~~ #(br_swap a (pmon_I(PreMonoidalCat:=mc))) >>> #(pmon_cancell a)
466 ; hexagon1 : forall {a b c}, #(pmon_assoc _ _ _) >>> #(br_swap a _) >>> #(pmon_assoc _ _ _)
467 ~~ #(br_swap _ _) ⋉ c >>> #(pmon_assoc _ _ _) >>> b ⋊ #(br_swap _ _)
468 ; hexagon2 : forall {a b c}, #(pmon_assoc _ _ _)⁻¹ >>> #(br_swap _ c) >>> #(pmon_assoc _ _ _)⁻¹
469 ~~ a ⋊ #(br_swap _ _) >>> #(pmon_assoc _ _ _)⁻¹ >>> #(br_swap _ _) ⋉ b
472 Class SymmetricCat `(bc:BraidedCat) :=
473 { symcat_swap : forall a b:C, #((br_swap(BraidedCat:=bc)) a b) ~~ #(br_swap _ _)⁻¹
477 (* a wide subcategory inherits the premonoidal structure if it includes all of the coherence maps *)
478 Section PreMonoidalWideSubcategory.
480 Context `(pm:PreMonoidalCat(I:=pmI)).
481 Context {Pmor}(S:WideSubcategory pm Pmor).
482 Context (Pmor_first : forall {a}{b}{c}{f}(pf:Pmor a b f), Pmor _ _ (f ⋉ c)).
483 Context (Pmor_second : forall {a}{b}{c}{f}(pf:Pmor a b f), Pmor _ _ (c ⋊ f)).
484 Context (Pmor_assoc : forall {a}{b}{c}, Pmor _ _ #(pmon_assoc a c b)).
485 Context (Pmor_unassoc: forall {a}{b}{c}, Pmor _ _ #(pmon_assoc a c b)⁻¹).
486 Context (Pmor_cancell: forall {a}, Pmor _ _ #(pmon_cancell a)).
487 Context (Pmor_uncancell: forall {a}, Pmor _ _ #(pmon_cancell a)⁻¹).
488 Context (Pmor_cancelr: forall {a}, Pmor _ _ #(pmon_cancelr a)).
489 Context (Pmor_uncancelr: forall {a}, Pmor _ _ #(pmon_cancelr a)⁻¹).
490 Implicit Arguments Pmor_first [[a][b][c][f]].
491 Implicit Arguments Pmor_second [[a][b][c][f]].
493 Definition PreMonoidalWideSubcategory_first_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y), (bin_obj' x a)~~{S}~~>(bin_obj' y a).
494 unfold hom; simpl; intros.
497 exists (bin_first(BinoidalCat:=pm) a \ x0).
498 apply Pmor_first; auto.
501 Definition PreMonoidalWideSubcategory_second_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y), (bin_obj' a x)~~{S}~~>(bin_obj' a y).
502 unfold hom; simpl; intros.
505 exists (bin_second(BinoidalCat:=pm) a \ x0).
506 apply Pmor_second; auto.
509 Instance PreMonoidalWideSubcategory_first (a:S) : Functor S S (fun x => bin_obj' x a) :=
510 { fmor := fun x y f => PreMonoidalWideSubcategory_first_fmor a f }.
511 unfold PreMonoidalWideSubcategory_first_fmor; intros; destruct f; destruct f'; simpl in *.
512 apply (fmor_respects (bin_first(BinoidalCat:=pm) a)); auto.
513 unfold PreMonoidalWideSubcategory_first_fmor; intros; simpl in *.
514 apply (fmor_preserves_id (bin_first(BinoidalCat:=pm) a)); auto.
515 unfold PreMonoidalWideSubcategory_first_fmor; intros; destruct f; destruct g; simpl in *.
516 apply (fmor_preserves_comp (bin_first(BinoidalCat:=pm) a)); auto.
519 Instance PreMonoidalWideSubcategory_second (a:S) : Functor S S (fun x => bin_obj' a x) :=
520 { fmor := fun x y f => PreMonoidalWideSubcategory_second_fmor a f }.
521 unfold PreMonoidalWideSubcategory_second_fmor; intros; destruct f; destruct f'; simpl in *.
522 apply (fmor_respects (bin_second(BinoidalCat:=pm) a)); auto.
523 unfold PreMonoidalWideSubcategory_second_fmor; intros; simpl in *.
524 apply (fmor_preserves_id (bin_second(BinoidalCat:=pm) a)); auto.
525 unfold PreMonoidalWideSubcategory_second_fmor; intros; destruct f; destruct g; simpl in *.
526 apply (fmor_preserves_comp (bin_second(BinoidalCat:=pm) a)); auto.
529 Instance PreMonoidalWideSubcategory_is_Binoidal : BinoidalCat S bin_obj' :=
530 { bin_first := PreMonoidalWideSubcategory_first
531 ; bin_second := PreMonoidalWideSubcategory_second }.
533 Definition PreMonoidalWideSubcategory_assoc_iso
534 : forall a b c, Isomorphic(C:=S) (bin_obj' (bin_obj' a b) c) (bin_obj' a (bin_obj' b c)).
536 refine {| iso_forward := existT _ _ (Pmor_assoc a b c) ; iso_backward := existT _ _ (Pmor_unassoc a b c) |}.
537 simpl; apply iso_comp1.
538 simpl; apply iso_comp2.
541 Definition PreMonoidalWideSubcategory_assoc
543 (PreMonoidalWideSubcategory_second a >>>> PreMonoidalWideSubcategory_first b) <~~~>
544 (PreMonoidalWideSubcategory_first b >>>> PreMonoidalWideSubcategory_second a).
546 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _ (PreMonoidalWideSubcategory_second a >>>>
547 PreMonoidalWideSubcategory_first b) (PreMonoidalWideSubcategory_first b >>>>
548 PreMonoidalWideSubcategory_second a) (fun c => PreMonoidalWideSubcategory_assoc_iso a c b)).
550 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
552 set (ni_commutes (pmon_assoc(PreMonoidalCat:=pm) a b) x) as q.
556 Definition PreMonoidalWideSubcategory_assoc_ll
558 PreMonoidalWideSubcategory_second (a⊗b) <~~~>
559 PreMonoidalWideSubcategory_second b >>>> PreMonoidalWideSubcategory_second a.
561 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
562 (PreMonoidalWideSubcategory_second (a⊗b))
563 (PreMonoidalWideSubcategory_second b >>>> PreMonoidalWideSubcategory_second a)
564 (fun c => PreMonoidalWideSubcategory_assoc_iso a b c)).
566 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
568 set (ni_commutes (pmon_assoc_ll(PreMonoidalCat:=pm) a b) x) as q.
569 unfold functor_comp in q; simpl in q.
570 set (pmon_coherent_l(PreMonoidalCat:=pm)) as q'.
571 setoid_rewrite q' in q.
575 Definition PreMonoidalWideSubcategory_assoc_rr
577 PreMonoidalWideSubcategory_first (a⊗b) <~~~>
578 PreMonoidalWideSubcategory_first a >>>> PreMonoidalWideSubcategory_first b.
581 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
582 (PreMonoidalWideSubcategory_first a >>>> PreMonoidalWideSubcategory_first b)
583 (PreMonoidalWideSubcategory_first (a⊗b))
584 (fun c => PreMonoidalWideSubcategory_assoc_iso c a b)).
586 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
588 set (ni_commutes (pmon_assoc_rr(PreMonoidalCat:=pm) a b) x) as q.
589 unfold functor_comp in q; simpl in q.
590 set (pmon_coherent_r(PreMonoidalCat:=pm)) as q'.
591 setoid_rewrite q' in q.
592 apply iso_shift_right' in q.
593 apply iso_shift_left.
595 setoid_rewrite iso_inv_inv in q.
596 setoid_rewrite associativity.
600 Definition PreMonoidalWideSubcategory_cancelr_iso : forall a, Isomorphic(C:=S) (bin_obj' a pmI) a.
602 refine {| iso_forward := existT _ _ (Pmor_cancelr a) ; iso_backward := existT _ _ (Pmor_uncancelr a) |}.
603 simpl; apply iso_comp1.
604 simpl; apply iso_comp2.
607 Definition PreMonoidalWideSubcategory_cancell_iso : forall a, Isomorphic(C:=S) (bin_obj' pmI a) a.
609 refine {| iso_forward := existT _ _ (Pmor_cancell a) ; iso_backward := existT _ _ (Pmor_uncancell a) |}.
610 simpl; apply iso_comp1.
611 simpl; apply iso_comp2.
614 Definition PreMonoidalWideSubcategory_cancelr : PreMonoidalWideSubcategory_first pmI <~~~> functor_id _.
615 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
616 (PreMonoidalWideSubcategory_first pmI) (functor_id _) PreMonoidalWideSubcategory_cancelr_iso).
618 unfold PreMonoidalWideSubcategory_first_fmor; simpl.
620 apply (ni_commutes (pmon_cancelr(PreMonoidalCat:=pm)) x).
623 Definition PreMonoidalWideSubcategory_cancell : PreMonoidalWideSubcategory_second pmI <~~~> functor_id _.
624 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
625 (PreMonoidalWideSubcategory_second pmI) (functor_id _) PreMonoidalWideSubcategory_cancell_iso).
627 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
629 apply (ni_commutes (pmon_cancell(PreMonoidalCat:=pm)) x).
632 Instance PreMonoidalWideSubcategory_PreMonoidal : PreMonoidalCat PreMonoidalWideSubcategory_is_Binoidal pmI :=
633 { pmon_assoc := PreMonoidalWideSubcategory_assoc
634 ; pmon_assoc_rr := PreMonoidalWideSubcategory_assoc_rr
635 ; pmon_assoc_ll := PreMonoidalWideSubcategory_assoc_ll
636 ; pmon_cancelr := PreMonoidalWideSubcategory_cancelr
637 ; pmon_cancell := PreMonoidalWideSubcategory_cancell
639 apply Build_Pentagon.
640 intros; unfold PreMonoidalWideSubcategory_assoc; simpl.
641 set (pmon_pentagon(PreMonoidalCat:=pm) a b c) as q.
644 apply Build_Triangle.
645 intros; unfold PreMonoidalWideSubcategory_assoc;
646 unfold PreMonoidalWideSubcategory_cancelr; unfold PreMonoidalWideSubcategory_cancell; simpl.
647 set (pmon_triangle(PreMonoidalCat:=pm) a b) as q.
652 set (pmon_triangle(PreMonoidalCat:=pm)) as q.
655 intros; simpl; reflexivity.
656 intros; simpl; reflexivity.
659 apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
660 apply (pmon_assoc_central(PreMonoidalCat:=pm) a b c).
661 apply (pmon_assoc_central(PreMonoidalCat:=pm) a b c).
664 apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
665 apply (pmon_cancelr_central(PreMonoidalCat:=pm) a).
666 apply (pmon_cancelr_central(PreMonoidalCat:=pm) a).
669 apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
670 apply (pmon_cancell_central(PreMonoidalCat:=pm) a).
671 apply (pmon_cancell_central(PreMonoidalCat:=pm) a).
674 End PreMonoidalWideSubcategory.
677 (* a full subcategory inherits the premonoidal structure if it includes the unit object and is closed under object-pairing *)
679 Section PreMonoidalFullSubcategory.
681 Context `(pm:PreMonoidalCat(I:=pmI)).
682 Context {Pobj}(S:FullSubcategory pm Pobj).
683 Context (Pobj_unit:Pobj pmI).
684 Context (Pobj_closed:forall {a}{b}, Pobj a -> Pobj b -> Pobj (a⊗b)).
685 Implicit Arguments Pobj_closed [[a][b]].
687 Definition PreMonoidalFullSubcategory_bobj (x y:S) :=
688 existT Pobj _ (Pobj_closed (projT2 x) (projT2 y)).
690 Definition PreMonoidalFullSubcategory_first_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y),
691 (PreMonoidalFullSubcategory_bobj x a)~~{S}~~>(PreMonoidalFullSubcategory_bobj y a).
692 unfold hom; simpl; intros.
693 destruct a as [a apf].
694 destruct x as [x xpf].
695 destruct y as [y ypf].
700 Definition PreMonoidalFullSubcategory_second_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y),
701 (PreMonoidalFullSubcategory_bobj a x)~~{S}~~>(PreMonoidalFullSubcategory_bobj a y).
702 unfold hom; simpl; intros.
703 destruct a as [a apf].
704 destruct x as [x xpf].
705 destruct y as [y ypf].
710 Instance PreMonoidalFullSubcategory_first (a:S)
711 : Functor S S (fun x => PreMonoidalFullSubcategory_bobj x a) :=
712 { fmor := fun x y f => PreMonoidalFullSubcategory_first_fmor a f }.
713 unfold PreMonoidalFullSubcategory_first_fmor; intros; destruct a; destruct a0; destruct b; simpl in *.
714 apply (fmor_respects (-⋉x)); auto.
715 unfold PreMonoidalFullSubcategory_first_fmor; intros; destruct a; destruct a0; simpl in *.
716 apply (fmor_preserves_id (-⋉x)); auto.
717 unfold PreMonoidalFullSubcategory_first_fmor; intros;
718 destruct a; destruct a0; destruct b; destruct c; simpl in *.
719 apply (fmor_preserves_comp (-⋉x)); auto.
722 Instance PreMonoidalFullSubcategory_second (a:S)
723 : Functor S S (fun x => PreMonoidalFullSubcategory_bobj a x) :=
724 { fmor := fun x y f => PreMonoidalFullSubcategory_second_fmor a f }.
725 unfold PreMonoidalFullSubcategory_second_fmor; intros; destruct a; destruct a0; destruct b; simpl in *.
726 apply (fmor_respects (x⋊-)); auto.
727 unfold PreMonoidalFullSubcategory_second_fmor; intros; destruct a; destruct a0; simpl in *.
728 apply (fmor_preserves_id (x⋊-)); auto.
729 unfold PreMonoidalFullSubcategory_second_fmor; intros;
730 destruct a; destruct a0; destruct b; destruct c; simpl in *.
731 apply (fmor_preserves_comp (x⋊-)); auto.
734 Instance PreMonoidalFullSubcategory_is_Binoidal : BinoidalCat S PreMonoidalFullSubcategory_bobj :=
735 { bin_first := PreMonoidalFullSubcategory_first
736 ; bin_second := PreMonoidalFullSubcategory_second }.
738 Definition PreMonoidalFullSubcategory_assoc
740 (PreMonoidalFullSubcategory_second a >>>> PreMonoidalFullSubcategory_first b) <~~~>
741 (PreMonoidalFullSubcategory_first b >>>> PreMonoidalFullSubcategory_second a).
744 Definition PreMonoidalFullSubcategory_assoc_ll
746 PreMonoidalFullSubcategory_second (a⊗b) <~~~>
747 PreMonoidalFullSubcategory_second b >>>> PreMonoidalFullSubcategory_second a.
751 Definition PreMonoidalFullSubcategory_assoc_rr
753 PreMonoidalFullSubcategory_first (a⊗b) <~~~>
754 PreMonoidalFullSubcategory_first a >>>> PreMonoidalFullSubcategory_first b.
758 Definition PreMonoidalFullSubcategory_I := existT _ pmI Pobj_unit.
760 Definition PreMonoidalFullSubcategory_cancelr
761 : PreMonoidalFullSubcategory_first PreMonoidalFullSubcategory_I <~~~> functor_id _.
764 Definition PreMonoidalFullSubcategory_cancell
765 : PreMonoidalFullSubcategory_second PreMonoidalFullSubcategory_I <~~~> functor_id _.
768 Instance PreMonoidalFullSubcategory_PreMonoidal
769 : PreMonoidalCat PreMonoidalFullSubcategory_is_Binoidal PreMonoidalFullSubcategory_I :=
770 { pmon_assoc := PreMonoidalFullSubcategory_assoc
771 ; pmon_assoc_rr := PreMonoidalFullSubcategory_assoc_rr
772 ; pmon_assoc_ll := PreMonoidalFullSubcategory_assoc_ll
773 ; pmon_cancelr := PreMonoidalFullSubcategory_cancelr
774 ; pmon_cancell := PreMonoidalFullSubcategory_cancell
777 End PreMonoidalFullSubcategory.