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_first _ _) >>> #(mf_second _ _) ~~
160 #(mf_second _ _) ⋉ _ >>> #(mf_first _ _) >>> 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_respects1 PMF12 (PMF23 >>>> - ⋉fobj23 (fobj12 a)) (- ⋉fobj12 a >>>> PMF23)).
197 apply (ni_associativity PMF12 (- ⋉fobj12 a) PMF23).
202 eapply (ni_associativity _ PMF12 PMF23).
209 Definition compose_mf_second a : compose_mf >>>> bin_second (compose_mf a) <~~~> bin_second a >>>> compose_mf.
210 set (mf_second(PreMonoidalFunctor:=PMF12) a) as mf_second12.
211 set (mf_second(PreMonoidalFunctor:=PMF23) (PMF12 a)) as mf_second23.
212 unfold functor_fobj in *; simpl in *.
215 apply (ni_associativity PMF12 PMF23 (fobj23 (fobj12 a) ⋊-)).
217 apply (ni_respects1 PMF12 (PMF23 >>>> fobj23 (fobj12 a) ⋊-) (fobj12 a ⋊- >>>> PMF23)).
223 apply (ni_associativity PMF12 (fobj12 a ⋊ -) PMF23).
228 eapply (ni_associativity (a ⋊-) PMF12 PMF23).
235 (* this proof is really gross; I will write a better one some other day *)
236 Lemma mf_associativity_comp :
238 (#((pmon_assoc (compose_mf a) (compose_mf c)) (fobj23 (fobj12 b))) >>>
239 compose_mf a ⋊ #((compose_mf_first c) b)) >>>
240 #((compose_mf_second a) (b ⊗ c)) ~~
241 (#((compose_mf_second a) b) ⋉ compose_mf c >>>
242 #((compose_mf_first c) (a ⊗ b))) >>> compose_mf \ #((pmon_assoc a c) b).
244 unfold compose_mf_second; simpl.
245 unfold compose_mf_first; simpl.
246 unfold functor_comp; simpl.
248 unfold functor_fobj; simpl.
250 set (mf_first (fobj12 c)) as m'.
251 assert (mf_first (fobj12 c)=m'). reflexivity.
254 set (mf_second (fobj12 a)) as m.
255 assert (mf_second (fobj12 a)=m). reflexivity.
258 Implicit Arguments id [[Ob][Hom][Category][a]].
263 repeat setoid_rewrite <- fmor_preserves_comp.
264 repeat setoid_rewrite fmor_preserves_id.
265 repeat setoid_rewrite left_identity.
266 repeat setoid_rewrite right_identity.
270 repeat setoid_rewrite <- fmor_preserves_comp.
271 repeat setoid_rewrite fmor_preserves_id.
272 repeat setoid_rewrite left_identity.
273 repeat setoid_rewrite right_identity.
276 assert ( (#((pmon_assoc (fobj23 (fobj12 a)) (fobj23 (fobj12 c)))
277 (fobj23 (fobj12 b))) >>>
280 (#(ni_iso (fobj12 b)) >>> ( (PMF23 \ #((mf_first c) b) ))))) >>>
282 (#(ni_iso0 (fobj12 (b ⊗ c))) >>>
283 ((PMF23 \ #((mf_second a) (b ⊗ c)))))) ~~
285 (#(ni_iso0 (fobj12 b)) >>> ( (PMF23 \ #((mf_second a) b) ))))
286 ⋉ fobj23 (fobj12 c) >>>
288 (#(ni_iso (fobj12 (a ⊗ b))) >>>
289 ( (PMF23 \ #((mf_first c) (a ⊗ b))))))) >>>
290 PMF23 \ (PMF12 \ #((pmon_assoc a c) b))
293 repeat setoid_rewrite associativity.
294 setoid_rewrite (fmor_preserves_comp PMF23).
295 unfold functor_comp in *.
296 unfold functor_fobj in *.
298 rename ni_commutes into ni_commutes7.
299 set (mf_assoc(PreMonoidalFunctor:=PMF12)) as q.
300 set (ni_commutes7 _ _ (#((mf_second a) b))) as q'.
302 repeat setoid_rewrite associativity.
304 setoid_rewrite <- (fmor_preserves_comp (-⋉ fobj23 (fobj12 c))).
305 repeat setoid_rewrite <- associativity.
306 setoid_rewrite juggle1.
307 setoid_rewrite <- q'.
308 repeat setoid_rewrite associativity.
309 setoid_rewrite fmor_preserves_comp.
311 unfold functor_fobj in *.
313 repeat setoid_rewrite <- associativity.
316 repeat setoid_rewrite <- fmor_preserves_comp.
317 repeat setoid_rewrite <- associativity.
318 apply comp_respects; try reflexivity.
320 set (mf_assoc(PreMonoidalFunctor:=PMF23) (fobj12 a) (fobj12 b) (fobj12 c)) as q.
321 unfold functor_fobj in *.
327 repeat setoid_rewrite <- associativity.
328 repeat setoid_rewrite <- associativity in q.
331 unfold functor_fobj; simpl.
333 repeat setoid_rewrite associativity.
334 apply comp_respects; try reflexivity.
335 apply comp_respects; try reflexivity.
338 repeat setoid_rewrite associativity.
339 repeat setoid_rewrite associativity in H1.
340 repeat setoid_rewrite <- fmor_preserves_comp in H1.
341 repeat setoid_rewrite associativity in H1.
344 Implicit Arguments id [[Ob][Hom][Category]].
346 (* this proof is really gross; I will write a better one some other day *)
347 Instance PreMonoidalFunctorsCompose : PreMonoidalFunctor PM1 PM3 (fobj23 ○ fobj12) :=
348 { mf_i := compose_mf_i
350 ; mf_first := compose_mf_first
351 ; mf_second := compose_mf_second }.
353 intros; unfold compose_mf_first; unfold compose_mf_second.
354 set (mf_first (PMF12 a)) as x in *.
355 set (mf_second (PMF12 b)) as y in *.
356 assert (x=mf_first (PMF12 a)). reflexivity.
357 assert (y=mf_second (PMF12 b)). reflexivity.
361 repeat setoid_rewrite left_identity.
362 repeat setoid_rewrite right_identity.
363 set (mf_consistent (PMF12 a) (PMF12 b)) as later.
364 apply comp_respects; try reflexivity.
365 rewrite <- H in later.
366 rewrite <- H0 in later.
379 unfold compose_mf_first; simpl.
380 set (mf_first (PMF12 b)) as m.
381 assert (mf_first (PMF12 b)=m). reflexivity.
384 unfold functor_fobj; simpl.
385 repeat setoid_rewrite <- fmor_preserves_comp.
386 repeat setoid_rewrite left_identity.
387 repeat setoid_rewrite right_identity.
389 set (mf_cancell b) as y.
390 set (mf_cancell (fobj12 b)) as y'.
391 unfold functor_fobj in *.
392 setoid_rewrite y in y'.
394 setoid_rewrite <- fmor_preserves_comp in y'.
395 setoid_rewrite <- fmor_preserves_comp in y'.
400 repeat setoid_rewrite <- associativity.
401 apply comp_respects; try reflexivity.
402 apply comp_respects; try reflexivity.
403 repeat setoid_rewrite associativity.
404 apply comp_respects; try reflexivity.
406 set (ni_commutes _ _ #mf_i) as x.
407 unfold functor_comp in x.
408 unfold functor_fobj in x.
415 unfold compose_mf_second; simpl.
416 set (mf_second (PMF12 a)) as m.
417 assert (mf_second (PMF12 a)=m). reflexivity.
420 unfold functor_fobj; simpl.
421 repeat setoid_rewrite <- fmor_preserves_comp.
422 repeat setoid_rewrite left_identity.
423 repeat setoid_rewrite right_identity.
425 set (mf_cancelr a) as y.
426 set (mf_cancelr (fobj12 a)) as y'.
427 unfold functor_fobj in *.
428 setoid_rewrite y in y'.
430 setoid_rewrite <- fmor_preserves_comp in y'.
431 setoid_rewrite <- fmor_preserves_comp in y'.
436 repeat setoid_rewrite <- associativity.
437 apply comp_respects; try reflexivity.
438 apply comp_respects; try reflexivity.
439 repeat setoid_rewrite associativity.
440 apply comp_respects; try reflexivity.
442 set (ni_commutes _ _ #mf_i) as x.
443 unfold functor_comp in x.
444 unfold functor_fobj in x.
450 apply mf_associativity_comp.
454 End PreMonoidalFunctorsCompose.
457 (*******************************************************************************)
458 (* Braided and Symmetric Categories *)
460 Class BraidedCat `(mc:PreMonoidalCat) :=
461 { br_niso : forall a, bin_first a <~~~> bin_second a
462 ; br_swap := fun a b => ni_iso (br_niso b) a
463 ; triangleb : forall a:C, #(pmon_cancelr a) ~~ #(br_swap a (pmon_I(PreMonoidalCat:=mc))) >>> #(pmon_cancell a)
464 ; hexagon1 : forall {a b c}, #(pmon_assoc _ _ _) >>> #(br_swap a _) >>> #(pmon_assoc _ _ _)
465 ~~ #(br_swap _ _) ⋉ c >>> #(pmon_assoc _ _ _) >>> b ⋊ #(br_swap _ _)
466 ; hexagon2 : forall {a b c}, #(pmon_assoc _ _ _)⁻¹ >>> #(br_swap _ c) >>> #(pmon_assoc _ _ _)⁻¹
467 ~~ a ⋊ #(br_swap _ _) >>> #(pmon_assoc _ _ _)⁻¹ >>> #(br_swap _ _) ⋉ b
470 Class SymmetricCat `(bc:BraidedCat) :=
471 { symcat_swap : forall a b:C, #((br_swap(BraidedCat:=bc)) a b) ~~ #(br_swap _ _)⁻¹
475 (* a wide subcategory inherits the premonoidal structure if it includes all of the coherence maps *)
476 Section PreMonoidalWideSubcategory.
478 Context `(pm:PreMonoidalCat(I:=pmI)).
479 Context {Pmor}(S:WideSubcategory pm Pmor).
480 Context (Pmor_first : forall {a}{b}{c}{f}(pf:Pmor a b f), Pmor _ _ (f ⋉ c)).
481 Context (Pmor_second : forall {a}{b}{c}{f}(pf:Pmor a b f), Pmor _ _ (c ⋊ f)).
482 Context (Pmor_assoc : forall {a}{b}{c}, Pmor _ _ #(pmon_assoc a c b)).
483 Context (Pmor_unassoc: forall {a}{b}{c}, Pmor _ _ #(pmon_assoc a c b)⁻¹).
484 Context (Pmor_cancell: forall {a}, Pmor _ _ #(pmon_cancell a)).
485 Context (Pmor_uncancell: forall {a}, Pmor _ _ #(pmon_cancell a)⁻¹).
486 Context (Pmor_cancelr: forall {a}, Pmor _ _ #(pmon_cancelr a)).
487 Context (Pmor_uncancelr: forall {a}, Pmor _ _ #(pmon_cancelr a)⁻¹).
488 Implicit Arguments Pmor_first [[a][b][c][f]].
489 Implicit Arguments Pmor_second [[a][b][c][f]].
491 Definition PreMonoidalWideSubcategory_first_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y), (bin_obj' x a)~~{S}~~>(bin_obj' y a).
492 unfold hom; simpl; intros.
495 exists (bin_first(BinoidalCat:=pm) a \ x0).
496 apply Pmor_first; auto.
499 Definition PreMonoidalWideSubcategory_second_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y), (bin_obj' a x)~~{S}~~>(bin_obj' a y).
500 unfold hom; simpl; intros.
503 exists (bin_second(BinoidalCat:=pm) a \ x0).
504 apply Pmor_second; auto.
507 Instance PreMonoidalWideSubcategory_first (a:S) : Functor S S (fun x => bin_obj' x a) :=
508 { fmor := fun x y f => PreMonoidalWideSubcategory_first_fmor a f }.
509 unfold PreMonoidalWideSubcategory_first_fmor; intros; destruct f; destruct f'; simpl in *.
510 apply (fmor_respects (bin_first(BinoidalCat:=pm) a)); auto.
511 unfold PreMonoidalWideSubcategory_first_fmor; intros; simpl in *.
512 apply (fmor_preserves_id (bin_first(BinoidalCat:=pm) a)); auto.
513 unfold PreMonoidalWideSubcategory_first_fmor; intros; destruct f; destruct g; simpl in *.
514 apply (fmor_preserves_comp (bin_first(BinoidalCat:=pm) a)); auto.
517 Instance PreMonoidalWideSubcategory_second (a:S) : Functor S S (fun x => bin_obj' a x) :=
518 { fmor := fun x y f => PreMonoidalWideSubcategory_second_fmor a f }.
519 unfold PreMonoidalWideSubcategory_second_fmor; intros; destruct f; destruct f'; simpl in *.
520 apply (fmor_respects (bin_second(BinoidalCat:=pm) a)); auto.
521 unfold PreMonoidalWideSubcategory_second_fmor; intros; simpl in *.
522 apply (fmor_preserves_id (bin_second(BinoidalCat:=pm) a)); auto.
523 unfold PreMonoidalWideSubcategory_second_fmor; intros; destruct f; destruct g; simpl in *.
524 apply (fmor_preserves_comp (bin_second(BinoidalCat:=pm) a)); auto.
527 Instance PreMonoidalWideSubcategory_is_Binoidal : BinoidalCat S bin_obj' :=
528 { bin_first := PreMonoidalWideSubcategory_first
529 ; bin_second := PreMonoidalWideSubcategory_second }.
531 Definition PreMonoidalWideSubcategory_assoc_iso
532 : forall a b c, Isomorphic(C:=S) (bin_obj' (bin_obj' a b) c) (bin_obj' a (bin_obj' b c)).
534 refine {| iso_forward := existT _ _ (Pmor_assoc a b c) ; iso_backward := existT _ _ (Pmor_unassoc a b c) |}.
535 simpl; apply iso_comp1.
536 simpl; apply iso_comp2.
539 Definition PreMonoidalWideSubcategory_assoc
541 (PreMonoidalWideSubcategory_second a >>>> PreMonoidalWideSubcategory_first b) <~~~>
542 (PreMonoidalWideSubcategory_first b >>>> PreMonoidalWideSubcategory_second a).
544 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _ (PreMonoidalWideSubcategory_second a >>>>
545 PreMonoidalWideSubcategory_first b) (PreMonoidalWideSubcategory_first b >>>>
546 PreMonoidalWideSubcategory_second a) (fun c => PreMonoidalWideSubcategory_assoc_iso a c b)).
548 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
550 set (ni_commutes (pmon_assoc(PreMonoidalCat:=pm) a b) x) as q.
554 Definition PreMonoidalWideSubcategory_assoc_ll
556 PreMonoidalWideSubcategory_second (a⊗b) <~~~>
557 PreMonoidalWideSubcategory_second b >>>> PreMonoidalWideSubcategory_second a.
559 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
560 (PreMonoidalWideSubcategory_second (a⊗b))
561 (PreMonoidalWideSubcategory_second b >>>> PreMonoidalWideSubcategory_second a)
562 (fun c => PreMonoidalWideSubcategory_assoc_iso a b c)).
564 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
566 set (ni_commutes (pmon_assoc_ll(PreMonoidalCat:=pm) a b) x) as q.
567 unfold functor_comp in q; simpl in q.
568 set (pmon_coherent_l(PreMonoidalCat:=pm)) as q'.
569 setoid_rewrite q' in q.
573 Definition PreMonoidalWideSubcategory_assoc_rr
575 PreMonoidalWideSubcategory_first (a⊗b) <~~~>
576 PreMonoidalWideSubcategory_first a >>>> PreMonoidalWideSubcategory_first b.
579 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
580 (PreMonoidalWideSubcategory_first a >>>> PreMonoidalWideSubcategory_first b)
581 (PreMonoidalWideSubcategory_first (a⊗b))
582 (fun c => PreMonoidalWideSubcategory_assoc_iso c a b)).
584 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
586 set (ni_commutes (pmon_assoc_rr(PreMonoidalCat:=pm) a b) x) as q.
587 unfold functor_comp in q; simpl in q.
588 set (pmon_coherent_r(PreMonoidalCat:=pm)) as q'.
589 setoid_rewrite q' in q.
590 apply iso_shift_right' in q.
591 apply iso_shift_left.
593 setoid_rewrite iso_inv_inv in q.
594 setoid_rewrite associativity.
598 Definition PreMonoidalWideSubcategory_cancelr_iso : forall a, Isomorphic(C:=S) (bin_obj' a pmI) a.
600 refine {| iso_forward := existT _ _ (Pmor_cancelr a) ; iso_backward := existT _ _ (Pmor_uncancelr a) |}.
601 simpl; apply iso_comp1.
602 simpl; apply iso_comp2.
605 Definition PreMonoidalWideSubcategory_cancell_iso : forall a, Isomorphic(C:=S) (bin_obj' pmI a) a.
607 refine {| iso_forward := existT _ _ (Pmor_cancell a) ; iso_backward := existT _ _ (Pmor_uncancell a) |}.
608 simpl; apply iso_comp1.
609 simpl; apply iso_comp2.
612 Definition PreMonoidalWideSubcategory_cancelr : PreMonoidalWideSubcategory_first pmI <~~~> functor_id _.
613 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
614 (PreMonoidalWideSubcategory_first pmI) (functor_id _) PreMonoidalWideSubcategory_cancelr_iso).
616 unfold PreMonoidalWideSubcategory_first_fmor; simpl.
618 apply (ni_commutes (pmon_cancelr(PreMonoidalCat:=pm)) x).
621 Definition PreMonoidalWideSubcategory_cancell : PreMonoidalWideSubcategory_second pmI <~~~> functor_id _.
622 apply (@Build_NaturalIsomorphism _ _ _ _ _ _ _ _
623 (PreMonoidalWideSubcategory_second pmI) (functor_id _) PreMonoidalWideSubcategory_cancell_iso).
625 unfold PreMonoidalWideSubcategory_second_fmor; simpl.
627 apply (ni_commutes (pmon_cancell(PreMonoidalCat:=pm)) x).
630 Instance PreMonoidalWideSubcategory_PreMonoidal : PreMonoidalCat PreMonoidalWideSubcategory_is_Binoidal pmI :=
631 { pmon_assoc := PreMonoidalWideSubcategory_assoc
632 ; pmon_assoc_rr := PreMonoidalWideSubcategory_assoc_rr
633 ; pmon_assoc_ll := PreMonoidalWideSubcategory_assoc_ll
634 ; pmon_cancelr := PreMonoidalWideSubcategory_cancelr
635 ; pmon_cancell := PreMonoidalWideSubcategory_cancell
637 apply Build_Pentagon.
638 intros; unfold PreMonoidalWideSubcategory_assoc; simpl.
639 set (pmon_pentagon(PreMonoidalCat:=pm) a b c) as q.
642 apply Build_Triangle.
643 intros; unfold PreMonoidalWideSubcategory_assoc;
644 unfold PreMonoidalWideSubcategory_cancelr; unfold PreMonoidalWideSubcategory_cancell; simpl.
645 set (pmon_triangle(PreMonoidalCat:=pm) a b) as q.
650 set (pmon_triangle(PreMonoidalCat:=pm)) as q.
653 intros; simpl; reflexivity.
654 intros; simpl; reflexivity.
657 apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
658 apply (pmon_assoc_central(PreMonoidalCat:=pm) a b c).
659 apply (pmon_assoc_central(PreMonoidalCat:=pm) a b c).
662 apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
663 apply (pmon_cancelr_central(PreMonoidalCat:=pm) a).
664 apply (pmon_cancelr_central(PreMonoidalCat:=pm) a).
667 apply Build_CentralMorphism; intros; simpl; destruct g; simpl.
668 apply (pmon_cancell_central(PreMonoidalCat:=pm) a).
669 apply (pmon_cancell_central(PreMonoidalCat:=pm) a).
672 End PreMonoidalWideSubcategory.
675 (* a full subcategory inherits the premonoidal structure if it includes the unit object and is closed under object-pairing *)
677 Section PreMonoidalFullSubcategory.
679 Context `(pm:PreMonoidalCat(I:=pmI)).
680 Context {Pobj}(S:FullSubcategory pm Pobj).
681 Context (Pobj_unit:Pobj pmI).
682 Context (Pobj_closed:forall {a}{b}, Pobj a -> Pobj b -> Pobj (a⊗b)).
683 Implicit Arguments Pobj_closed [[a][b]].
685 Definition PreMonoidalFullSubcategory_bobj (x y:S) :=
686 existT Pobj _ (Pobj_closed (projT2 x) (projT2 y)).
688 Definition PreMonoidalFullSubcategory_first_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y),
689 (PreMonoidalFullSubcategory_bobj x a)~~{S}~~>(PreMonoidalFullSubcategory_bobj y a).
690 unfold hom; simpl; intros.
691 destruct a as [a apf].
692 destruct x as [x xpf].
693 destruct y as [y ypf].
698 Definition PreMonoidalFullSubcategory_second_fmor (a:S) : forall {x}{y}(f:x~~{S}~~>y),
699 (PreMonoidalFullSubcategory_bobj a x)~~{S}~~>(PreMonoidalFullSubcategory_bobj a y).
700 unfold hom; simpl; intros.
701 destruct a as [a apf].
702 destruct x as [x xpf].
703 destruct y as [y ypf].
708 Instance PreMonoidalFullSubcategory_first (a:S)
709 : Functor S S (fun x => PreMonoidalFullSubcategory_bobj x a) :=
710 { fmor := fun x y f => PreMonoidalFullSubcategory_first_fmor a f }.
711 unfold PreMonoidalFullSubcategory_first_fmor; intros; destruct a; destruct a0; destruct b; simpl in *.
712 apply (fmor_respects (-⋉x)); auto.
713 unfold PreMonoidalFullSubcategory_first_fmor; intros; destruct a; destruct a0; simpl in *.
714 apply (fmor_preserves_id (-⋉x)); auto.
715 unfold PreMonoidalFullSubcategory_first_fmor; intros;
716 destruct a; destruct a0; destruct b; destruct c; simpl in *.
717 apply (fmor_preserves_comp (-⋉x)); auto.
720 Instance PreMonoidalFullSubcategory_second (a:S)
721 : Functor S S (fun x => PreMonoidalFullSubcategory_bobj a x) :=
722 { fmor := fun x y f => PreMonoidalFullSubcategory_second_fmor a f }.
723 unfold PreMonoidalFullSubcategory_second_fmor; intros; destruct a; destruct a0; destruct b; simpl in *.
724 apply (fmor_respects (x⋊-)); auto.
725 unfold PreMonoidalFullSubcategory_second_fmor; intros; destruct a; destruct a0; simpl in *.
726 apply (fmor_preserves_id (x⋊-)); auto.
727 unfold PreMonoidalFullSubcategory_second_fmor; intros;
728 destruct a; destruct a0; destruct b; destruct c; simpl in *.
729 apply (fmor_preserves_comp (x⋊-)); auto.
732 Instance PreMonoidalFullSubcategory_is_Binoidal : BinoidalCat S PreMonoidalFullSubcategory_bobj :=
733 { bin_first := PreMonoidalFullSubcategory_first
734 ; bin_second := PreMonoidalFullSubcategory_second }.
736 Definition PreMonoidalFullSubcategory_assoc
738 (PreMonoidalFullSubcategory_second a >>>> PreMonoidalFullSubcategory_first b) <~~~>
739 (PreMonoidalFullSubcategory_first b >>>> PreMonoidalFullSubcategory_second a).
742 Definition PreMonoidalFullSubcategory_assoc_ll
744 PreMonoidalFullSubcategory_second (a⊗b) <~~~>
745 PreMonoidalFullSubcategory_second b >>>> PreMonoidalFullSubcategory_second a.
749 Definition PreMonoidalFullSubcategory_assoc_rr
751 PreMonoidalFullSubcategory_first (a⊗b) <~~~>
752 PreMonoidalFullSubcategory_first a >>>> PreMonoidalFullSubcategory_first b.
756 Definition PreMonoidalFullSubcategory_I := existT _ pmI Pobj_unit.
758 Definition PreMonoidalFullSubcategory_cancelr
759 : PreMonoidalFullSubcategory_first PreMonoidalFullSubcategory_I <~~~> functor_id _.
762 Definition PreMonoidalFullSubcategory_cancell
763 : PreMonoidalFullSubcategory_second PreMonoidalFullSubcategory_I <~~~> functor_id _.
766 Instance PreMonoidalFullSubcategory_PreMonoidal
767 : PreMonoidalCat PreMonoidalFullSubcategory_is_Binoidal PreMonoidalFullSubcategory_I :=
768 { pmon_assoc := PreMonoidalFullSubcategory_assoc
769 ; pmon_assoc_rr := PreMonoidalFullSubcategory_assoc_rr
770 ; pmon_assoc_ll := PreMonoidalFullSubcategory_assoc_ll
771 ; pmon_cancelr := PreMonoidalFullSubcategory_cancelr
772 ; pmon_cancell := PreMonoidalFullSubcategory_cancell
775 End PreMonoidalFullSubcategory.