X-Git-Url: http://git.megacz.com/?p=coq-categories.git;a=blobdiff_plain;f=src%2FPreMonoidalCategories.v;h=fdda1a5584bd878597215ac5efc07f1a143ff120;hp=4ab56d2dde51792e57c1453886d7a7c2d911efa7;hb=9acb9b7b4ed12543e54c39c82d7b0f34d04d0207;hpb=e928451c4c45cdbdd975bbfb229e8cc2616b8194 diff --git a/src/PreMonoidalCategories.v b/src/PreMonoidalCategories.v index 4ab56d2..fdda1a5 100644 --- a/src/PreMonoidalCategories.v +++ b/src/PreMonoidalCategories.v @@ -3,6 +3,7 @@ Require Import Preamble. 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. @@ -53,22 +54,102 @@ 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 @@ -76,8 +157,8 @@ 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. @@ -86,10 +167,12 @@ Section PreMonoidalFunctorsCompose. `{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). + {PMFF23:Functor C2 C3 fobj23 } + (PMF23 :PreMonoidalFunctor PM2 PM3 PMFF23). Definition compose_mf := PMF12 >>>> PMF23. @@ -108,8 +191,7 @@ Section PreMonoidalFunctorsCompose. eapply ni_comp. apply (ni_associativity PMF12 PMF23 (- ⋉fobj23 (fobj12 a))). eapply ni_comp. - apply (ni_respects PMF12 PMF12 (PMF23 >>>> - ⋉fobj23 (fobj12 a)) (- ⋉fobj12 a >>>> PMF23)). - apply ni_id. + apply (ni_respects1 PMF12 (PMF23 >>>> - ⋉fobj23 (fobj12 a)) (- ⋉fobj12 a >>>> PMF23)). apply mf_first23. clear mf_first23. @@ -122,7 +204,7 @@ Section PreMonoidalFunctorsCompose. eapply ni_inv. eapply (ni_associativity _ PMF12 PMF23). - apply ni_respects; [ idtac | apply ni_id ]. + apply ni_respects2. apply ni_inv. apply mf_first12. Defined. @@ -135,8 +217,7 @@ Section PreMonoidalFunctorsCompose. eapply ni_comp. apply (ni_associativity PMF12 PMF23 (fobj23 (fobj12 a) ⋊-)). eapply ni_comp. - apply (ni_respects PMF12 PMF12 (PMF23 >>>> fobj23 (fobj12 a) ⋊-) (fobj12 a ⋊- >>>> PMF23)). - apply ni_id. + apply (ni_respects1 PMF12 (PMF23 >>>> fobj23 (fobj12 a) ⋊-) (fobj12 a ⋊- >>>> PMF23)). apply mf_second23. clear mf_second23. @@ -149,124 +230,128 @@ Section PreMonoidalFunctorsCompose. eapply ni_inv. eapply (ni_associativity (a ⋊-) PMF12 PMF23). - apply ni_respects; [ idtac | apply ni_id ]. + apply ni_respects2. apply ni_inv. apply mf_second12. Defined. - Lemma compose_assoc_coherence a b c : - (#((pmon_assoc (compose_mf a) (fobj23 (fobj12 c))) (compose_mf b)) >>> - compose_mf a ⋊ #((compose_mf_second b) c)) >>> + (* 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) ⋉ fobj23 (fobj12 c) >>> - #((compose_mf_second (a ⊗ b)) c)) >>> compose_mf \ #((pmon_assoc a c) b). -(* - set (mf_assoc a b c) as x. - set (mf_assoc (fobj12 a) (fobj12 b) (fobj12 c)) as x'. - unfold functor_fobj in *. - simpl in *. - etransitivity. - etransitivity. - etransitivity. - Focus 3. - apply x'. - - apply iso_shift_left' in x'. - + (#((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_second (fobj12 b)) as m. - assert (mf_second (fobj12 b)=m). reflexivity. - destruct m; simpl. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite right_identity. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - - set (mf_second (fobj12 (a ⊗ b))) as m''. - assert (mf_second (fobj12 (a ⊗ b))=m''). reflexivity. - destruct m''; simpl. - unfold functor_fobj; simpl. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite right_identity. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - - set (mf_second (fobj12 a)) as m'. - assert (mf_second (fobj12 a)=m'). reflexivity. + + set (mf_first (fobj12 c)) as m'. + assert (mf_first (fobj12 c)=m'). reflexivity. destruct m'; simpl. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite <- fmor_preserves_comp. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - setoid_rewrite left_identity. - setoid_rewrite right_identity. - assert (fobj23 (fobj12 a) ⋊ PMF23 \ id (PMF12 (b ⊗ c)) ~~ id _). - (* *) - setoid_rewrite H2. - setoid_rewrite left_identity. - assert ((id (fobj23 (fobj12 a) ⊗ fobj23 (fobj12 b)) ⋉ fobj23 (fobj12 c)) ~~ id _). - (* *) - setoid_rewrite H3. - setoid_rewrite left_identity. - assert (id (fobj23 (fobj12 a ⊗ fobj12 b)) ⋉ fobj23 (fobj12 c) ~~ id _). - (* *) - setoid_rewrite H4. - setoid_rewrite left_identity. - clear H4. - setoid_rewrite left_identity. - assert (id (fobj23 (fobj12 (a ⊗ b))) ⋉ fobj23 (fobj12 c) ~~ id _). - (* *) - setoid_rewrite H4. - setoid_rewrite right_identity. - clear H4. - assert ((fobj23 (fobj12 a) ⋊ PMF23 \ id (PMF12 b)) ⋉ fobj23 (fobj12 c) ~~ id _). - (* *) - setoid_rewrite H4. - setoid_rewrite left_identity. - clear H4. - unfold functor_comp in ni_commutes0; simpl in ni_commutes0. - unfold functor_comp in ni_commutes; simpl in ni_commutes. - unfold functor_comp in ni_commutes1; simpl in ni_commutes1. + set (mf_second (fobj12 a)) as m. + assert (mf_second (fobj12 a)=m). reflexivity. + destruct m; simpl. - unfold functor_fobj in *. - simpl in *. - setoid_rewrite x in x'. - rewrite H1. - set (ni_commutes0 (a ) - setoid_rewrite fmor_preserves_id. + Implicit Arguments id [[Ob][Hom][Category][a]]. + idtac. + + symmetry. etransitivity. - eapply comp_respects. + 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. - eapply comp_respects. - eapply comp_respects. - apply - Focus 2. - eapply fmor_preserves_id. - setoid_rewrite (fmor_preserves_id PMF23). -*) - admit. - Qed. - Instance PreMonoidalFunctorsCompose : PreMonoidalFunctor PM1 PM3 (fobj23 ○ fobj12) := + 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_F := compose_mf ; mf_first := compose_mf_first ; mf_second := compose_mf_second }. + intros; unfold compose_mf_first; unfold compose_mf_second. set (mf_first (PMF12 a)) as x in *. set (mf_second (PMF12 b)) as y in *. @@ -279,31 +364,12 @@ Section PreMonoidalFunctorsCompose. repeat setoid_rewrite right_identity. set (mf_consistent (PMF12 a) (PMF12 b)) as later. apply comp_respects; try reflexivity. - unfold functor_comp. - unfold functor_fobj; simpl. - set (ni_commutes _ _ (id (fobj12 b))) as x. - unfold functor_comp in x. - simpl in x. - unfold functor_fobj in x. - symmetry in x. - etransitivity. - apply x. - clear x. - set (ni_commutes0 _ _ (id (fobj12 a))) as x'. - unfold functor_comp in x'. - simpl in x'. - unfold functor_fobj in x'. - etransitivity; [ idtac | apply x' ]. - clear x'. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite right_identity. rewrite <- H in later. rewrite <- H0 in later. simpl in later. apply later. apply fmor_respects. - apply (mf_consistent a b). + apply mf_consistent. intros. simpl. @@ -339,22 +405,13 @@ Section PreMonoidalFunctorsCompose. repeat setoid_rewrite associativity. apply comp_respects; try reflexivity. - set (ni_commutes _ _ (id (fobj12 I1))) as x. + set (ni_commutes _ _ #mf_i) as x. unfold functor_comp in x. unfold functor_fobj in x. simpl in x. - setoid_rewrite <- x. - clear x. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite right_identity. - rewrite H. simpl. - clear H. - unfold functor_comp in ni_commutes. - simpl in ni_commutes. - apply ni_commutes. + apply x. intros. unfold compose_mf_second; simpl. @@ -384,24 +441,16 @@ Section PreMonoidalFunctorsCompose. repeat setoid_rewrite associativity. apply comp_respects; try reflexivity. - set (ni_commutes _ _ (id (fobj12 I1))) as x. + set (ni_commutes _ _ #mf_i) as x. unfold functor_comp in x. unfold functor_fobj in x. simpl in x. - setoid_rewrite <- x. - clear x. - setoid_rewrite fmor_preserves_id. - setoid_rewrite fmor_preserves_id. - setoid_rewrite right_identity. - rewrite H. simpl. - clear H. - unfold functor_comp in ni_commutes. - simpl in ni_commutes. - apply ni_commutes. + apply x. + + apply mf_associativity_comp. - apply compose_assoc_coherence. Defined. End PreMonoidalFunctorsCompose. @@ -726,4 +775,4 @@ Section PreMonoidalFullSubcategory. }. Defined. End PreMonoidalFullSubcategory. -*) \ No newline at end of file +*)