1 Generalizable All Variables.
2 Require Import Preamble.
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))
150 (fobj : C1 -> C2 ) :=
151 { mf_F :> Functor C1 C2 fobj
152 ; mf_i : I2 ≅ mf_F I1
153 ; mf_first : ∀ a, mf_F >>>> bin_first (mf_F a) <~~~> bin_first a >>>> mf_F
154 ; mf_second : ∀ a, mf_F >>>> bin_second (mf_F a) <~~~> bin_second a >>>> mf_F
155 ; mf_consistent : ∀ a b, #(mf_first a b) ~~ #(mf_second b a)
156 ; mf_center : forall `(f:a~>b), CentralMorphism f -> CentralMorphism (mf_F \ f)
157 ; mf_cancell : ∀ b, #(pmon_cancell _) ~~ #mf_i ⋉ _ >>> #(mf_first b I1) >>> mf_F \ #(pmon_cancell b)
158 ; mf_cancelr : ∀ a, #(pmon_cancelr _) ~~ _ ⋊ #mf_i >>> #(mf_second a I1) >>> mf_F \ #(pmon_cancelr a)
159 ; mf_assoc : ∀ a b c, #(pmon_assoc _ _ _) >>> _ ⋊ #(mf_second _ _) >>> #(mf_second _ _) ~~
160 #(mf_second _ _) ⋉ _ >>> #(mf_second _ _) >>> mf_F \ #(pmon_assoc a c b)
162 Coercion mf_F : PreMonoidalFunctor >-> Functor.
164 Section PreMonoidalFunctorsCompose.
166 `{PM1 :PreMonoidalCat(C:=C1)(I:=I1)}
167 `{PM2 :PreMonoidalCat(C:=C2)(I:=I2)}
169 (PMF12 :PreMonoidalFunctor PM1 PM2 fobj12)
170 `{PM3 :PreMonoidalCat(C:=C3)(I:=I3)}
172 (PMF23 :PreMonoidalFunctor PM2 PM3 fobj23).
174 Definition compose_mf := PMF12 >>>> PMF23.
176 Definition compose_mf_i : I3 ≅ PMF23 (PMF12 I1).
178 apply (mf_i(PreMonoidalFunctor:=PMF23)).
179 apply functors_preserve_isos.
180 apply (mf_i(PreMonoidalFunctor:=PMF12)).
183 Definition compose_mf_first a : compose_mf >>>> bin_first (compose_mf a) <~~~> bin_first a >>>> compose_mf.
184 set (mf_first(PreMonoidalFunctor:=PMF12) a) as mf_first12.
185 set (mf_first(PreMonoidalFunctor:=PMF23) (PMF12 a)) as mf_first23.
186 unfold functor_fobj in *; simpl in *.
189 apply (ni_associativity PMF12 PMF23 (- ⋉fobj23 (fobj12 a))).
191 apply (ni_respects PMF12 PMF12 (PMF23 >>>> - ⋉fobj23 (fobj12 a)) (- ⋉fobj12 a >>>> PMF23)).
198 apply (ni_associativity PMF12 (- ⋉fobj12 a) PMF23).
203 eapply (ni_associativity _ PMF12 PMF23).
205 apply ni_respects; [ idtac | apply ni_id ].
210 Definition compose_mf_second a : compose_mf >>>> bin_second (compose_mf a) <~~~> bin_second a >>>> compose_mf.
211 set (mf_second(PreMonoidalFunctor:=PMF12) a) as mf_second12.
212 set (mf_second(PreMonoidalFunctor:=PMF23) (PMF12 a)) as mf_second23.
213 unfold functor_fobj in *; simpl in *.
216 apply (ni_associativity PMF12 PMF23 (fobj23 (fobj12 a) ⋊-)).
218 apply (ni_respects PMF12 PMF12 (PMF23 >>>> fobj23 (fobj12 a) ⋊-) (fobj12 a ⋊- >>>> PMF23)).
225 apply (ni_associativity PMF12 (fobj12 a ⋊ -) PMF23).
230 eapply (ni_associativity (a ⋊-) PMF12 PMF23).
232 apply ni_respects; [ idtac | apply ni_id ].
237 Lemma compose_assoc_coherence a b c :
238 (#((pmon_assoc (compose_mf a) (fobj23 (fobj12 c))) (compose_mf b)) >>>
239 compose_mf a ⋊ #((compose_mf_second b) c)) >>>
240 #((compose_mf_second a) (b ⊗ c)) ~~
241 (#((compose_mf_second a) b) ⋉ fobj23 (fobj12 c) >>>
242 #((compose_mf_second (a ⊗ b)) c)) >>> compose_mf \ #((pmon_assoc a c) b).
244 set (mf_assoc a b c) as x.
245 set (mf_assoc (fobj12 a) (fobj12 b) (fobj12 c)) as x'.
246 unfold functor_fobj in *.
254 apply iso_shift_left' in x'.
256 unfold compose_mf_second; simpl.
257 unfold functor_fobj; simpl.
258 set (mf_second (fobj12 b)) as m.
259 assert (mf_second (fobj12 b)=m). reflexivity.
261 setoid_rewrite <- fmor_preserves_comp.
262 setoid_rewrite <- fmor_preserves_comp.
263 setoid_rewrite <- fmor_preserves_comp.
264 setoid_rewrite <- fmor_preserves_comp.
265 setoid_rewrite <- fmor_preserves_comp.
266 setoid_rewrite fmor_preserves_id.
267 setoid_rewrite fmor_preserves_id.
268 setoid_rewrite fmor_preserves_id.
269 setoid_rewrite right_identity.
270 setoid_rewrite left_identity.
271 setoid_rewrite left_identity.
272 setoid_rewrite left_identity.
274 set (mf_second (fobj12 (a ⊗ b))) as m''.
275 assert (mf_second (fobj12 (a ⊗ b))=m''). reflexivity.
277 unfold functor_fobj; simpl.
278 setoid_rewrite fmor_preserves_id.
279 setoid_rewrite fmor_preserves_id.
280 setoid_rewrite right_identity.
281 setoid_rewrite left_identity.
282 setoid_rewrite left_identity.
283 setoid_rewrite left_identity.
285 set (mf_second (fobj12 a)) as m'.
286 assert (mf_second (fobj12 a)=m'). reflexivity.
288 setoid_rewrite <- fmor_preserves_comp.
289 setoid_rewrite <- fmor_preserves_comp.
290 setoid_rewrite <- fmor_preserves_comp.
291 setoid_rewrite <- fmor_preserves_comp.
292 setoid_rewrite <- fmor_preserves_comp.
293 setoid_rewrite left_identity.
294 setoid_rewrite left_identity.
295 setoid_rewrite left_identity.
296 setoid_rewrite right_identity.
297 assert (fobj23 (fobj12 a) ⋊ PMF23 \ id (PMF12 (b ⊗ c)) ~~ id _).
300 setoid_rewrite left_identity.
301 assert ((id (fobj23 (fobj12 a) ⊗ fobj23 (fobj12 b)) ⋉ fobj23 (fobj12 c)) ~~ id _).
304 setoid_rewrite left_identity.
305 assert (id (fobj23 (fobj12 a ⊗ fobj12 b)) ⋉ fobj23 (fobj12 c) ~~ id _).
308 setoid_rewrite left_identity.
310 setoid_rewrite left_identity.
311 assert (id (fobj23 (fobj12 (a ⊗ b))) ⋉ fobj23 (fobj12 c) ~~ id _).
314 setoid_rewrite right_identity.
316 assert ((fobj23 (fobj12 a) ⋊ PMF23 \ id (PMF12 b)) ⋉ fobj23 (fobj12 c) ~~ id _).
319 setoid_rewrite left_identity.
321 unfold functor_comp in ni_commutes0; simpl in ni_commutes0.
322 unfold functor_comp in ni_commutes; simpl in ni_commutes.
323 unfold functor_comp in ni_commutes1; simpl in ni_commutes1.
326 unfold functor_fobj in *.
328 setoid_rewrite x in x'.
330 set (ni_commutes0 (a )
331 setoid_rewrite fmor_preserves_id.
333 eapply comp_respects.
335 eapply comp_respects.
336 eapply comp_respects.
339 eapply fmor_preserves_id.
340 setoid_rewrite (fmor_preserves_id PMF23).
345 Instance PreMonoidalFunctorsCompose : PreMonoidalFunctor PM1 PM3 (fobj23 ○ fobj12) :=
346 { mf_i := compose_mf_i
348 ; mf_first := compose_mf_first
349 ; mf_second := compose_mf_second }.
350 intros; unfold compose_mf_first; unfold compose_mf_second.
351 set (mf_first (PMF12 a)) as x in *.
352 set (mf_second (PMF12 b)) as y in *.
353 assert (x=mf_first (PMF12 a)). reflexivity.
354 assert (y=mf_second (PMF12 b)). reflexivity.
358 repeat setoid_rewrite left_identity.
359 repeat setoid_rewrite right_identity.
360 set (mf_consistent (PMF12 a) (PMF12 b)) as later.
361 apply comp_respects; try reflexivity.
363 unfold functor_fobj; simpl.
364 set (ni_commutes _ _ (id (fobj12 b))) as x.
365 unfold functor_comp in x.
367 unfold functor_fobj in x.
372 set (ni_commutes0 _ _ (id (fobj12 a))) as x'.
373 unfold functor_comp in x'.
375 unfold functor_fobj in x'.
376 etransitivity; [ idtac | apply x' ].
378 setoid_rewrite fmor_preserves_id.
379 setoid_rewrite fmor_preserves_id.
380 setoid_rewrite right_identity.
381 rewrite <- H in later.
382 rewrite <- H0 in later.
386 apply (mf_consistent a b).
395 unfold compose_mf_first; simpl.
396 set (mf_first (PMF12 b)) as m.
397 assert (mf_first (PMF12 b)=m). reflexivity.
400 unfold functor_fobj; simpl.
401 repeat setoid_rewrite <- fmor_preserves_comp.
402 repeat setoid_rewrite left_identity.
403 repeat setoid_rewrite right_identity.
405 set (mf_cancell b) as y.
406 set (mf_cancell (fobj12 b)) as y'.
407 unfold functor_fobj in *.
408 setoid_rewrite y in y'.
410 setoid_rewrite <- fmor_preserves_comp in y'.
411 setoid_rewrite <- fmor_preserves_comp in y'.
416 repeat setoid_rewrite <- associativity.
417 apply comp_respects; try reflexivity.
418 apply comp_respects; try reflexivity.
419 repeat setoid_rewrite associativity.
420 apply comp_respects; try reflexivity.
422 set (ni_commutes _ _ (id (fobj12 I1))) as x.
423 unfold functor_comp in x.
424 unfold functor_fobj in x.
428 setoid_rewrite fmor_preserves_id.
429 setoid_rewrite fmor_preserves_id.
430 setoid_rewrite right_identity.
435 unfold functor_comp in ni_commutes.
436 simpl in ni_commutes.
440 unfold compose_mf_second; simpl.
441 set (mf_second (PMF12 a)) as m.
442 assert (mf_second (PMF12 a)=m). reflexivity.
445 unfold functor_fobj; simpl.
446 repeat setoid_rewrite <- fmor_preserves_comp.
447 repeat setoid_rewrite left_identity.
448 repeat setoid_rewrite right_identity.
450 set (mf_cancelr a) as y.
451 set (mf_cancelr (fobj12 a)) as y'.
452 unfold functor_fobj in *.
453 setoid_rewrite y in y'.
455 setoid_rewrite <- fmor_preserves_comp in y'.
456 setoid_rewrite <- fmor_preserves_comp in y'.
461 repeat setoid_rewrite <- associativity.
462 apply comp_respects; try reflexivity.
463 apply comp_respects; try reflexivity.
464 repeat setoid_rewrite associativity.
465 apply comp_respects; try reflexivity.
467 set (ni_commutes _ _ (id (fobj12 I1))) as x.
468 unfold functor_comp in x.
469 unfold functor_fobj in x.
473 setoid_rewrite fmor_preserves_id.
474 setoid_rewrite fmor_preserves_id.
475 setoid_rewrite right_identity.
480 unfold functor_comp in ni_commutes.
481 simpl in ni_commutes.
484 apply compose_assoc_coherence.
487 End PreMonoidalFunctorsCompose.
490 (*******************************************************************************)
491 (* Braided and Symmetric Categories *)
493 Class BraidedCat `(mc:PreMonoidalCat) :=
494 { br_niso : forall a, bin_first a <~~~> bin_second a
495 ; br_swap := fun a b => ni_iso (br_niso b) a
496 ; triangleb : forall a:C, #(pmon_cancelr a) ~~ #(br_swap a (pmon_I(PreMonoidalCat:=mc))) >>> #(pmon_cancell a)
497 ; hexagon1 : forall {a b c}, #(pmon_assoc _ _ _) >>> #(br_swap a _) >>> #(pmon_assoc _ _ _)
498 ~~ #(br_swap _ _) ⋉ c >>> #(pmon_assoc _ _ _) >>> b ⋊ #(br_swap _ _)
499 ; hexagon2 : forall {a b c}, #(pmon_assoc _ _ _)⁻¹ >>> #(br_swap _ c) >>> #(pmon_assoc _ _ _)⁻¹
500 ~~ a ⋊ #(br_swap _ _) >>> #(pmon_assoc _ _ _)⁻¹ >>> #(br_swap _ _) ⋉ b
503 Class SymmetricCat `(bc:BraidedCat) :=
504 { symcat_swap : forall a b:C, #((br_swap(BraidedCat:=bc)) a b) ~~ #(br_swap _ _)⁻¹
508 (* a wide subcategory inherits the premonoidal structure if it includes all of the coherence maps *)
509 Section PreMonoidalWideSubcategory.
511 Context `(pm:PreMonoidalCat(I:=pmI)).
512 Context {Pmor}(S:WideSubcategory pm Pmor).
513 Context (Pmor_first : forall {a}{b}{c}{f}(pf:Pmor a b f), Pmor _ _ (f ⋉ c)).
514 Context (Pmor_second : forall {a}{b}{c}{f}(pf:Pmor a b f), Pmor _ _ (c ⋊ f)).
515 Context (Pmor_assoc : forall {a}{b}{c}, Pmor _ _ #(pmon_assoc a c b)).
516 Context (Pmor_unassoc: forall {a}{b}{c}, Pmor _ _ #(pmon_assoc a c b)⁻¹).
517 Context (Pmor_cancell: forall {a}, Pmor _ _ #(pmon_cancell a)).
518 Context (Pmor_uncancell: forall {a}, Pmor _ _ #(pmon_cancell a)⁻¹).
519 Context (Pmor_cancelr: forall {a}, Pmor _ _ #(pmon_cancelr a)).
520 Context (Pmor_uncancelr: forall {a}, Pmor _ _ #(pmon_cancelr a)⁻¹).
521 Implicit Arguments Pmor_first [[a][b][c][f]].
522 Implicit Arguments Pmor_second [[a][b][c][f]].
524 Definition PreMonoidalWideSubcategory_first_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y), (bin_obj' x a)~~{S}~~>(bin_obj' y a).
525 unfold hom; simpl; intros.
528 exists (bin_first(BinoidalCat:=pm) a \ x0).
529 apply Pmor_first; auto.
532 Definition PreMonoidalWideSubcategory_second_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y), (bin_obj' a x)~~{S}~~>(bin_obj' a y).
533 unfold hom; simpl; intros.
536 exists (bin_second(BinoidalCat:=pm) a \ x0).
537 apply Pmor_second; auto.
540 Instance PreMonoidalWideSubcategory_first (a:S) : Functor S S (fun x => bin_obj' x a) :=
541 { fmor := fun x y f => PreMonoidalWideSubcategory_first_fmor a f }.
542 unfold PreMonoidalWideSubcategory_first_fmor; intros; destruct f; destruct f'; simpl in *.
543 apply (fmor_respects (bin_first(BinoidalCat:=pm) a)); auto.
544 unfold PreMonoidalWideSubcategory_first_fmor; intros; simpl in *.
545 apply (fmor_preserves_id (bin_first(BinoidalCat:=pm) a)); auto.
546 unfold PreMonoidalWideSubcategory_first_fmor; intros; destruct f; destruct g; simpl in *.
547 apply (fmor_preserves_comp (bin_first(BinoidalCat:=pm) a)); auto.
550 Instance PreMonoidalWideSubcategory_second (a:S) : Functor S S (fun x => bin_obj' a x) :=
551 { fmor := fun x y f => PreMonoidalWideSubcategory_second_fmor a f }.
552 unfold PreMonoidalWideSubcategory_second_fmor; intros; destruct f; destruct f'; simpl in *.
553 apply (fmor_respects (bin_second(BinoidalCat:=pm) a)); auto.
554 unfold PreMonoidalWideSubcategory_second_fmor; intros; simpl in *.
555 apply (fmor_preserves_id (bin_second(BinoidalCat:=pm) a)); auto.
556 unfold PreMonoidalWideSubcategory_second_fmor; intros; destruct f; destruct g; simpl in *.
557 apply (fmor_preserves_comp (bin_second(BinoidalCat:=pm) a)); auto.
560 Instance PreMonoidalWideSubcategory_is_Binoidal : BinoidalCat S bin_obj' :=
561 { bin_first := PreMonoidalWideSubcategory_first
562 ; bin_second := PreMonoidalWideSubcategory_second }.
564 Definition PreMonoidalWideSubcategory_assoc_iso
565 : forall a b c, Isomorphic(C:=S) (bin_obj' (bin_obj' a b) c) (bin_obj' a (bin_obj' b c)).
567 refine {| iso_forward := existT _ _ (Pmor_assoc a b c) ; iso_backward := existT _ _ (Pmor_unassoc a b c) |}.
568 simpl; apply iso_comp1.
569 simpl; apply iso_comp2.
572 Definition PreMonoidalWideSubcategory_assoc
574 (PreMonoidalWideSubcategory_second a >>>> PreMonoidalWideSubcategory_first b) <~~~>
575 (PreMonoidalWideSubcategory_first b >>>> PreMonoidalWideSubcategory_second a).
577 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _ (PreMonoidalWideSubcategory_second a >>>>
578 PreMonoidalWideSubcategory_first b) (PreMonoidalWideSubcategory_first b >>>>
579 PreMonoidalWideSubcategory_second a) (fun c => PreMonoidalWideSubcategory_assoc_iso a c b)).
581 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
583 set (ni_commutes (pmon_assoc(PreMonoidalCat:=pm) a b) x) as q.
587 Definition PreMonoidalWideSubcategory_assoc_ll
589 PreMonoidalWideSubcategory_second (a⊗b) <~~~>
590 PreMonoidalWideSubcategory_second b >>>> PreMonoidalWideSubcategory_second a.
592 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
593 (PreMonoidalWideSubcategory_second (a⊗b))
594 (PreMonoidalWideSubcategory_second b >>>> PreMonoidalWideSubcategory_second a)
595 (fun c => PreMonoidalWideSubcategory_assoc_iso a b c)).
597 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
599 set (ni_commutes (pmon_assoc_ll(PreMonoidalCat:=pm) a b) x) as q.
600 unfold functor_comp in q; simpl in q.
601 set (pmon_coherent_l(PreMonoidalCat:=pm)) as q'.
602 setoid_rewrite q' in q.
606 Definition PreMonoidalWideSubcategory_assoc_rr
608 PreMonoidalWideSubcategory_first (a⊗b) <~~~>
609 PreMonoidalWideSubcategory_first a >>>> PreMonoidalWideSubcategory_first b.
612 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
613 (PreMonoidalWideSubcategory_first a >>>> PreMonoidalWideSubcategory_first b)
614 (PreMonoidalWideSubcategory_first (a⊗b))
615 (fun c => PreMonoidalWideSubcategory_assoc_iso c a b)).
617 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
619 set (ni_commutes (pmon_assoc_rr(PreMonoidalCat:=pm) a b) x) as q.
620 unfold functor_comp in q; simpl in q.
621 set (pmon_coherent_r(PreMonoidalCat:=pm)) as q'.
622 setoid_rewrite q' in q.
623 apply iso_shift_right' in q.
624 apply iso_shift_left.
626 setoid_rewrite iso_inv_inv in q.
627 setoid_rewrite associativity.
631 Definition PreMonoidalWideSubcategory_cancelr_iso : forall a, Isomorphic(C:=S) (bin_obj' a pmI) a.
633 refine {| iso_forward := existT _ _ (Pmor_cancelr a) ; iso_backward := existT _ _ (Pmor_uncancelr a) |}.
634 simpl; apply iso_comp1.
635 simpl; apply iso_comp2.
638 Definition PreMonoidalWideSubcategory_cancell_iso : forall a, Isomorphic(C:=S) (bin_obj' pmI a) a.
640 refine {| iso_forward := existT _ _ (Pmor_cancell a) ; iso_backward := existT _ _ (Pmor_uncancell a) |}.
641 simpl; apply iso_comp1.
642 simpl; apply iso_comp2.
645 Definition PreMonoidalWideSubcategory_cancelr : PreMonoidalWideSubcategory_first pmI <~~~> functor_id _.
646 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
647 (PreMonoidalWideSubcategory_first pmI) (functor_id _) PreMonoidalWideSubcategory_cancelr_iso).
649 unfold PreMonoidalWideSubcategory_first_fmor; simpl.
651 apply (ni_commutes (pmon_cancelr(PreMonoidalCat:=pm)) x).
654 Definition PreMonoidalWideSubcategory_cancell : PreMonoidalWideSubcategory_second pmI <~~~> functor_id _.
655 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
656 (PreMonoidalWideSubcategory_second pmI) (functor_id _) PreMonoidalWideSubcategory_cancell_iso).
658 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
660 apply (ni_commutes (pmon_cancell(PreMonoidalCat:=pm)) x).
663 Instance PreMonoidalWideSubcategory_PreMonoidal : PreMonoidalCat PreMonoidalWideSubcategory_is_Binoidal pmI :=
664 { pmon_assoc := PreMonoidalWideSubcategory_assoc
665 ; pmon_assoc_rr := PreMonoidalWideSubcategory_assoc_rr
666 ; pmon_assoc_ll := PreMonoidalWideSubcategory_assoc_ll
667 ; pmon_cancelr := PreMonoidalWideSubcategory_cancelr
668 ; pmon_cancell := PreMonoidalWideSubcategory_cancell
670 apply Build_Pentagon.
671 intros; unfold PreMonoidalWideSubcategory_assoc; simpl.
672 set (pmon_pentagon(PreMonoidalCat:=pm) a b c) as q.
675 apply Build_Triangle.
676 intros; unfold PreMonoidalWideSubcategory_assoc;
677 unfold PreMonoidalWideSubcategory_cancelr; unfold PreMonoidalWideSubcategory_cancell; simpl.
678 set (pmon_triangle(PreMonoidalCat:=pm) a b) as q.
683 set (pmon_triangle(PreMonoidalCat:=pm)) as q.
686 intros; simpl; reflexivity.
687 intros; simpl; reflexivity.
690 apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
691 apply (pmon_assoc_central(PreMonoidalCat:=pm) a b c).
692 apply (pmon_assoc_central(PreMonoidalCat:=pm) a b c).
695 apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
696 apply (pmon_cancelr_central(PreMonoidalCat:=pm) a).
697 apply (pmon_cancelr_central(PreMonoidalCat:=pm) a).
700 apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
701 apply (pmon_cancell_central(PreMonoidalCat:=pm) a).
702 apply (pmon_cancell_central(PreMonoidalCat:=pm) a).
705 End PreMonoidalWideSubcategory.
708 (* a full subcategory inherits the premonoidal structure if it includes the unit object and is closed under object-pairing *)
710 Section PreMonoidalFullSubcategory.
712 Context `(pm:PreMonoidalCat(I:=pmI)).
713 Context {Pobj}(S:FullSubcategory pm Pobj).
714 Context (Pobj_unit:Pobj pmI).
715 Context (Pobj_closed:forall {a}{b}, Pobj a -> Pobj b -> Pobj (a⊗b)).
716 Implicit Arguments Pobj_closed [[a][b]].
718 Definition PreMonoidalFullSubcategory_bobj (x y:S) :=
719 existT Pobj _ (Pobj_closed (projT2 x) (projT2 y)).
721 Definition PreMonoidalFullSubcategory_first_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y),
722 (PreMonoidalFullSubcategory_bobj x a)~~{S}~~>(PreMonoidalFullSubcategory_bobj y a).
723 unfold hom; simpl; intros.
724 destruct a as [a apf].
725 destruct x as [x xpf].
726 destruct y as [y ypf].
731 Definition PreMonoidalFullSubcategory_second_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y),
732 (PreMonoidalFullSubcategory_bobj a x)~~{S}~~>(PreMonoidalFullSubcategory_bobj a y).
733 unfold hom; simpl; intros.
734 destruct a as [a apf].
735 destruct x as [x xpf].
736 destruct y as [y ypf].
741 Instance PreMonoidalFullSubcategory_first (a:S)
742 : Functor S S (fun x => PreMonoidalFullSubcategory_bobj x a) :=
743 { fmor := fun x y f => PreMonoidalFullSubcategory_first_fmor a f }.
744 unfold PreMonoidalFullSubcategory_first_fmor; intros; destruct a; destruct a0; destruct b; simpl in *.
745 apply (fmor_respects (-⋉x)); auto.
746 unfold PreMonoidalFullSubcategory_first_fmor; intros; destruct a; destruct a0; simpl in *.
747 apply (fmor_preserves_id (-⋉x)); auto.
748 unfold PreMonoidalFullSubcategory_first_fmor; intros;
749 destruct a; destruct a0; destruct b; destruct c; simpl in *.
750 apply (fmor_preserves_comp (-⋉x)); auto.
753 Instance PreMonoidalFullSubcategory_second (a:S)
754 : Functor S S (fun x => PreMonoidalFullSubcategory_bobj a x) :=
755 { fmor := fun x y f => PreMonoidalFullSubcategory_second_fmor a f }.
756 unfold PreMonoidalFullSubcategory_second_fmor; intros; destruct a; destruct a0; destruct b; simpl in *.
757 apply (fmor_respects (x⋊-)); auto.
758 unfold PreMonoidalFullSubcategory_second_fmor; intros; destruct a; destruct a0; simpl in *.
759 apply (fmor_preserves_id (x⋊-)); auto.
760 unfold PreMonoidalFullSubcategory_second_fmor; intros;
761 destruct a; destruct a0; destruct b; destruct c; simpl in *.
762 apply (fmor_preserves_comp (x⋊-)); auto.
765 Instance PreMonoidalFullSubcategory_is_Binoidal : BinoidalCat S PreMonoidalFullSubcategory_bobj :=
766 { bin_first := PreMonoidalFullSubcategory_first
767 ; bin_second := PreMonoidalFullSubcategory_second }.
769 Definition PreMonoidalFullSubcategory_assoc
771 (PreMonoidalFullSubcategory_second a >>>> PreMonoidalFullSubcategory_first b) <~~~>
772 (PreMonoidalFullSubcategory_first b >>>> PreMonoidalFullSubcategory_second a).
775 Definition PreMonoidalFullSubcategory_assoc_ll
777 PreMonoidalFullSubcategory_second (a⊗b) <~~~>
778 PreMonoidalFullSubcategory_second b >>>> PreMonoidalFullSubcategory_second a.
782 Definition PreMonoidalFullSubcategory_assoc_rr
784 PreMonoidalFullSubcategory_first (a⊗b) <~~~>
785 PreMonoidalFullSubcategory_first a >>>> PreMonoidalFullSubcategory_first b.
789 Definition PreMonoidalFullSubcategory_I := existT _ pmI Pobj_unit.
791 Definition PreMonoidalFullSubcategory_cancelr
792 : PreMonoidalFullSubcategory_first PreMonoidalFullSubcategory_I <~~~> functor_id _.
795 Definition PreMonoidalFullSubcategory_cancell
796 : PreMonoidalFullSubcategory_second PreMonoidalFullSubcategory_I <~~~> functor_id _.
799 Instance PreMonoidalFullSubcategory_PreMonoidal
800 : PreMonoidalCat PreMonoidalFullSubcategory_is_Binoidal PreMonoidalFullSubcategory_I :=
801 { pmon_assoc := PreMonoidalFullSubcategory_assoc
802 ; pmon_assoc_rr := PreMonoidalFullSubcategory_assoc_rr
803 ; pmon_assoc_ll := PreMonoidalFullSubcategory_assoc_ll
804 ; pmon_cancelr := PreMonoidalFullSubcategory_cancelr
805 ; pmon_cancell := PreMonoidalFullSubcategory_cancell
808 End PreMonoidalFullSubcategory.