Generalizable All Variables. Require Import Preamble. Require Import Categories_ch1_3. Require Import Functors_ch1_4. Require Import Isomorphisms_ch1_5. Require Import ProductCategories_ch1_6_1. Require Import InitialTerminal_ch2_2. Require Import Subcategories_ch7_1. Require Import NaturalTransformations_ch7_4. Require Import NaturalIsomorphisms_ch7_5. Require Import Coherence_ch7_8. Require Import BinoidalCategories. (* not in Awodey *) Class PreMonoidalCat `(bc:BinoidalCat(C:=C))(I:C) := { pmon_I := I ; pmon_bin := bc ; pmon_cat := C ; pmon_assoc : forall a b, (bin_second a >>>> bin_first b) <~~~> (bin_first b >>>> bin_second a) ; pmon_cancelr : (bin_first I) <~~~> functor_id C ; pmon_cancell : (bin_second I) <~~~> functor_id C ; 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)) ; 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)) (fun a => #(pmon_cancell a)) (fun a => #(pmon_cancelr a)) ; pmon_assoc_rr : forall a b, (bin_first (a⊗b)) <~~~> (bin_first a >>>> bin_first b) ; pmon_assoc_ll : forall a b, (bin_second (a⊗b)) <~~~> (bin_second b >>>> bin_second a) ; pmon_coherent_r : forall a c d:C, #(pmon_assoc_rr c d a) ~~ #(pmon_assoc a d c)⁻¹ ; pmon_coherent_l : forall a c d:C, #(pmon_assoc_ll c a d) ~~ #(pmon_assoc c d a) }. (* * Premonoidal categories actually have three associators (the "f" * indicates the position in which the operation is natural: * * assoc : (A ⋊ f) ⋉ C <-> A ⋊ (f ⋉ C) * assoc_rr : (f ⋉ B) ⋉ C <-> f ⋉ (B ⊗ C) * assoc_ll : (A ⋊ B) ⋊ f <-> (A ⊗ B) ⋊ f * * Fortunately, in a monoidal category these are all the same natural * isomorphism (and in any case -- monoidal or not -- the objects in * the left column are all the same and the objects in the right * column are all the same). This formalization assumes that is the * case even for premonoidal categories with non-central maps, in * order to keep the complexity manageable. I don't know much about * the consequences of having them and letting them be different; you * might need extra versions of the triangle/pentagon diagrams. *) Implicit Arguments pmon_cancell [ Ob Hom C bin_obj' bc I ]. Implicit Arguments pmon_cancelr [ Ob Hom C bin_obj' bc I ]. Implicit Arguments pmon_assoc [ Ob Hom C bin_obj' bc I ]. 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 mn) (a ⊗ b)) ~~ #((pmon_assoc mn a EI) b) >>> (a ⋊-) \ #((pmon_cancelr mn) 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 mn a b)) as xx. (* FIXME *) Admitted. Class PreMonoidalFunctor `(PM1:PreMonoidalCat(C:=C1)(I:=I1)) `(PM2:PreMonoidalCat(C:=C2)(I:=I2)) (fobj : C1 -> C2 ) := { mf_F :> Functor C1 C2 fobj ; mf_preserves_i : mf_F I1 ≅ I2 ; mf_preserves_first : forall a, bin_first a >>>> mf_F <~~~> mf_F >>>> bin_first (mf_F a) ; mf_preserves_second : forall a, bin_second a >>>> mf_F <~~~> mf_F >>>> bin_second (mf_F a) ; mf_preserves_center : forall `(f:a~>b), CentralMorphism f -> CentralMorphism (mf_F \ f) }. Coercion mf_F : PreMonoidalFunctor >-> Functor. (*******************************************************************************) (* Braided and Symmetric Categories *) Class BraidedCat `(mc:PreMonoidalCat) := { br_niso : forall a, bin_first a <~~~> bin_second a ; br_swap := fun a b => ni_iso (br_niso b) a ; triangleb : forall a:C, #(pmon_cancelr mc a) ~~ #(br_swap a (pmon_I(PreMonoidalCat:=mc))) >>> #(pmon_cancell mc a) ; hexagon1 : forall {a b c}, #(pmon_assoc mc _ _ _) >>> #(br_swap a _) >>> #(pmon_assoc mc _ _ _) ~~ #(br_swap _ _) ⋉ c >>> #(pmon_assoc mc _ _ _) >>> b ⋊ #(br_swap _ _) ; hexagon2 : forall {a b c}, #(pmon_assoc mc _ _ _)⁻¹ >>> #(br_swap _ c) >>> #(pmon_assoc mc _ _ _)⁻¹ ~~ a ⋊ #(br_swap _ _) >>> #(pmon_assoc mc _ _ _)⁻¹ >>> #(br_swap _ _) ⋉ b }. Class SymmetricCat `(bc:BraidedCat) := { symcat_swap : forall a b:C, #((br_swap(BraidedCat:=bc)) a b) ~~ #(br_swap _ _)⁻¹ }.