Generalizable All Variables.
-Require Import Preamble.
+Require Import Notations.
Require Import Categories_ch1_3.
Require Import Functors_ch1_4.
Require Import Isomorphisms_ch1_5.
* might need extra versions of the triangle/pentagon diagrams.
*)
+Implicit Arguments pmon_I [ Ob Hom C bin_obj' bc I ].
Implicit Arguments pmon_cancell [ Ob Hom C bin_obj' bc I PreMonoidalCat ].
Implicit Arguments pmon_cancelr [ Ob Hom C bin_obj' bc I PreMonoidalCat ].
Implicit Arguments pmon_assoc [ Ob Hom C bin_obj' bc I PreMonoidalCat ].
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
`{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.
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 (fobj23 ○ fobj12) :=
+ 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 }.
Defined.
End PreMonoidalFunctorsCompose.
+Notation "a >>⊗>> b" := (PreMonoidalFunctorsCompose a b).
(*******************************************************************************)
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 a) ~~ #(br_swap a (pmon_I(PreMonoidalCat:=mc))) >>> #(pmon_cancell a)
+; triangleb : forall a:C, #(pmon_cancelr a) ~~ #(br_swap a (pmon_I mc)) >>> #(pmon_cancell a)
; hexagon1 : forall {a b c}, #(pmon_assoc _ _ _) >>> #(br_swap a _) >>> #(pmon_assoc _ _ _)
~~ #(br_swap _ _) ⋉ c >>> #(pmon_assoc _ _ _) >>> b ⋊ #(br_swap _ _)
; hexagon2 : forall {a b c}, #(pmon_assoc _ _ _)⁻¹ >>> #(br_swap _ c) >>> #(pmon_assoc _ _ _)⁻¹
End PreMonoidalWideSubcategory.
+Section IsoFullSubCategory.
+ Context `{C:Category}.
+ Context {Pobj}(S:FullSubcategory C Pobj).
+
+ Definition iso_full {a b:C}(i:a≅b)(pa:Pobj a)(pb:Pobj b) : (existT _ _ pa) ≅ (existT _ _ pb).
+ set (#i : existT Pobj a pa ~~{S}~~> existT Pobj b pb) as i1.
+ set (iso_backward i : existT Pobj b pb ~~{S}~~> existT Pobj a pa) as i2.
+ refine {| iso_forward := i1 ; iso_backward := i2 |}.
+ unfold i1; unfold i2; unfold hom; simpl.
+ apply iso_comp1.
+ unfold i1; unfold i2; unfold hom; simpl.
+ apply iso_comp2.
+ Defined.
+End IsoFullSubCategory.
(* a full subcategory inherits the premonoidal structure if it includes the unit object and is closed under object-pairing *)
-(*
Section PreMonoidalFullSubcategory.
Context `(pm:PreMonoidalCat(I:=pmI)).
Context {Pobj}(S:FullSubcategory pm Pobj).
+
Context (Pobj_unit:Pobj pmI).
Context (Pobj_closed:forall {a}{b}, Pobj a -> Pobj b -> Pobj (a⊗b)).
Implicit Arguments Pobj_closed [[a][b]].
{ bin_first := PreMonoidalFullSubcategory_first
; bin_second := PreMonoidalFullSubcategory_second }.
+ Definition central_full {a b}(f:a~~{S}~~>b)
+ : @CentralMorphism _ _ _ _ pm (projT1 a) (projT1 b) f -> CentralMorphism f.
+ intro cm.
+ apply Build_CentralMorphism; simpl.
+ intros.
+ destruct a as [a apf].
+ destruct b as [b bpf].
+ destruct c as [c cpf].
+ destruct d as [d dpf].
+ simpl.
+ apply cm.
+ intros.
+ destruct a as [a apf].
+ destruct b as [b bpf].
+ destruct c as [c cpf].
+ destruct d as [d dpf].
+ simpl.
+ apply cm.
+ Defined.
+
+ Notation "a ⊕ b" := (Pobj_closed a b).
Definition PreMonoidalFullSubcategory_assoc
: forall a b,
(PreMonoidalFullSubcategory_second a >>>> PreMonoidalFullSubcategory_first b) <~~~>
(PreMonoidalFullSubcategory_first b >>>> PreMonoidalFullSubcategory_second a).
- Defined.
+ intros.
+ refine {| ni_iso := (fun (c:S) => iso_full S (pmon_assoc(PreMonoidalCat:=pm) _ _ _)
+ ((projT2 a⊕projT2 c)⊕projT2 b)
+ (projT2 a⊕(projT2 c⊕projT2 b))) |}.
+ intros; simpl.
+ destruct a as [a apf].
+ destruct b as [b bpf].
+ destruct A as [A Apf].
+ destruct B as [B Bpf].
+ apply (ni_commutes (pmon_assoc(PreMonoidalCat:=pm) a b) f).
+ Defined.
Definition PreMonoidalFullSubcategory_assoc_ll
: forall a b,
PreMonoidalFullSubcategory_second (a⊗b) <~~~>
PreMonoidalFullSubcategory_second b >>>> PreMonoidalFullSubcategory_second a.
- intros.
- Defined.
+ intros.
+ refine {| ni_iso := (fun (c:S) => iso_full S (pmon_assoc_ll(PreMonoidalCat:=pm) _ _ _)
+ ((projT2 a⊕projT2 b)⊕projT2 c)
+ (projT2 a⊕(projT2 b⊕projT2 c))
+ ) |}.
+ intros; simpl.
+ destruct a as [a apf].
+ destruct b as [b bpf].
+ destruct A as [A Apf].
+ destruct B as [B Bpf].
+ apply (ni_commutes (pmon_assoc_ll(PreMonoidalCat:=pm) a b) f).
+ Defined.
Definition PreMonoidalFullSubcategory_assoc_rr
: forall a b,
PreMonoidalFullSubcategory_first (a⊗b) <~~~>
PreMonoidalFullSubcategory_first a >>>> PreMonoidalFullSubcategory_first b.
- intros.
- Defined.
+ intros.
+ refine {| ni_iso := (fun (c:S) => iso_full S (pmon_assoc_rr(PreMonoidalCat:=pm) _ _ _)
+ (projT2 c⊕(projT2 a⊕projT2 b))
+ ((projT2 c⊕projT2 a)⊕projT2 b)
+ ) |}.
+ intros; simpl.
+ destruct a as [a apf].
+ destruct b as [b bpf].
+ destruct A as [A Apf].
+ destruct B as [B Bpf].
+ apply (ni_commutes (pmon_assoc_rr(PreMonoidalCat:=pm) a b) f).
+ Defined.
Definition PreMonoidalFullSubcategory_I := existT _ pmI Pobj_unit.
+ Definition PreMonoidalFullSubcategory_cancelr_iso A
+ : (fun x : S => PreMonoidalFullSubcategory_bobj x (existT Pobj pmI Pobj_unit)) A ≅ (fun x : S => x) A.
+ destruct A.
+ apply (iso_full S).
+ apply pmon_cancelr.
+ Defined.
+
Definition PreMonoidalFullSubcategory_cancelr
: PreMonoidalFullSubcategory_first PreMonoidalFullSubcategory_I <~~~> functor_id _.
+ intros.
+ refine {| ni_iso := PreMonoidalFullSubcategory_cancelr_iso |}.
+ intros.
+ destruct A as [A Apf].
+ destruct B as [B Bpf].
+ simpl.
+ apply (ni_commutes (pmon_cancelr(PreMonoidalCat:=pm)) f).
+ Defined.
+
+ Definition PreMonoidalFullSubcategory_cancell_iso A
+ : (fun x : S => PreMonoidalFullSubcategory_bobj (existT Pobj pmI Pobj_unit) x) A ≅ (fun x : S => x) A.
+ destruct A.
+ apply (iso_full S).
+ apply pmon_cancell.
Defined.
Definition PreMonoidalFullSubcategory_cancell
: PreMonoidalFullSubcategory_second PreMonoidalFullSubcategory_I <~~~> functor_id _.
+ intros.
+ refine {| ni_iso := PreMonoidalFullSubcategory_cancell_iso |}.
+ intros.
+ destruct A as [A Apf].
+ destruct B as [B Bpf].
+ simpl.
+ apply (ni_commutes (pmon_cancell(PreMonoidalCat:=pm)) f).
Defined.
Instance PreMonoidalFullSubcategory_PreMonoidal
: PreMonoidalCat PreMonoidalFullSubcategory_is_Binoidal PreMonoidalFullSubcategory_I :=
- { pmon_assoc := PreMonoidalFullSubcategory_assoc
- ; pmon_assoc_rr := PreMonoidalFullSubcategory_assoc_rr
- ; pmon_assoc_ll := PreMonoidalFullSubcategory_assoc_ll
- ; pmon_cancelr := PreMonoidalFullSubcategory_cancelr
- ; pmon_cancell := PreMonoidalFullSubcategory_cancell
- }.
- Defined.
+ { pmon_assoc := PreMonoidalFullSubcategory_assoc
+ ; pmon_assoc_rr := PreMonoidalFullSubcategory_assoc_rr
+ ; pmon_assoc_ll := PreMonoidalFullSubcategory_assoc_ll
+ ; pmon_cancelr := PreMonoidalFullSubcategory_cancelr
+ ; pmon_cancell := PreMonoidalFullSubcategory_cancell
+ }.
+ apply Build_Pentagon.
+ intros.
+ destruct a as [a apf].
+ destruct b as [b bpf].
+ destruct c as [c cpf].
+ destruct d as [d dpf].
+ simpl.
+ apply (pmon_pentagon(PreMonoidalCat:=pm)).
+
+ apply Build_Triangle.
+ intros.
+ destruct a as [a apf].
+ destruct b as [b bpf].
+ simpl.
+ apply (pmon_triangle(PreMonoidalCat:=pm)).
+ simpl.
+ apply (pmon_triangle(PreMonoidalCat:=pm)).
+
+ intros.
+ destruct a as [a apf].
+ destruct c as [c cpf].
+ destruct d as [d dpf].
+ simpl.
+ apply (pmon_coherent_r(PreMonoidalCat:=pm)).
+
+ intros.
+ destruct a as [a apf].
+ destruct c as [c cpf].
+ destruct d as [d dpf].
+ simpl.
+ apply (pmon_coherent_l(PreMonoidalCat:=pm)).
+
+ intros.
+ destruct a as [a apf].
+ destruct b as [b bpf].
+ destruct c as [c cpf].
+ simpl.
+ apply central_full.
+ simpl.
+ apply (pmon_assoc_central(PreMonoidalCat:=pm)).
+
+ intros.
+ destruct a as [a apf].
+ simpl.
+ apply central_full.
+ simpl.
+ apply (pmon_cancelr_central(PreMonoidalCat:=pm)).
+
+ intros.
+ destruct a as [a apf].
+ simpl.
+ apply central_full.
+ simpl.
+ apply (pmon_cancell_central(PreMonoidalCat:=pm)).
+ Defined.
+
End PreMonoidalFullSubcategory.
-*)
+
projects / coq-categories.git / blobdiff