X-Git-Url: http://git.megacz.com/?p=coq-categories.git;a=blobdiff_plain;f=src%2FSubcategories_ch7_1.v;h=25391a12eda6a431ed4bac1e99f4d432314bc903;hp=e6b8aa4361d776657419b2a9f8317b468ebedddb;hb=20641452e40570b4bfc9429ca57b0cffca6eccfb;hpb=a0b31d2cc2b6cf7184efe4ff01ad682749f779ad

diff --git a/src/Subcategories_ch7_1.v b/src/Subcategories_ch7_1.v
index e6b8aa4..25391a1 100644
--- a/src/Subcategories_ch7_1.v
+++ b/src/Subcategories_ch7_1.v
@@ -3,7 +3,7 @@
 (****************************************************************************)
 
 Generalizable All Variables.
-Require Import Preamble.
+Require Import Notations.
 Require Import Categories_ch1_3.
 Require Import Functors_ch1_4.
 Require Import Isomorphisms_ch1_5.
@@ -11,38 +11,62 @@ Require Import OppositeCategories_ch1_6_2.
 Require Import NaturalTransformations_ch7_4.
 Require Import NaturalIsomorphisms_ch7_5.
 
-(* Any morphism-predicate which is closed under composition and
- * passing to identity morphisms (of either the domain or codomain)
- *
- * We could recycle the "predicate on morphisms" to determine the
- * "predicate on objects", but this causes technical difficulties with
- * Coq *)
-Class SubCategory `(C:Category Ob Hom)(Pobj:Ob->Type)(Pmor:forall a b:Ob, Pobj a -> Pobj b -> (a~>b) ->Type) : Type :=
-{ sc_id_included    : forall (a:Ob)(pa:Pobj a), Pmor _ _ pa pa (id a)
-; sc_comp_included  : forall (a b c:Ob)(pa:Pobj a)(pb:Pobj b)(pc:Pobj c) f g,
-                        (Pmor _ _ pa pb f) -> (Pmor _ _ pb pc g) -> (Pmor _ _ pa pc (f>>>g))
+(*
+ * See the README for an explanation of why there is "WideSubcategory"
+ * and "FullSubcategory" but no "Subcategory"
+ *)
+
+(* a full subcategory requires nothing more than a predicate on objects *)
+Class FullSubcategory `(C:Category)(Pobj:C->Type) := { }.
+
+(* the category construction for full subcategories is simpler: *)
+Instance FullSubCategoriesAreCategories `(fsc:@FullSubcategory Ob Hom C Pobj)
+  : Category (sigT Pobj) (fun dom ran => (projT1 dom)~~{C}~~>(projT1 ran)) :=
+{ id   := fun t         => id (projT1 t)
+; eqv  := fun a b f g   => eqv _ _ f g
+; comp := fun a b c f g => f >>> g
 }.
+  intros; apply Build_Equivalence. unfold Reflexive.
+     intros; reflexivity.
+     unfold Symmetric; intros; simpl; symmetry; auto.
+     unfold Transitive; intros; simpl. transitivity y; auto.
+  intros; unfold Proper. unfold respectful. intros. simpl. apply comp_respects. auto. auto.
+  intros; simpl. apply left_identity.
+  intros; simpl. apply right_identity.
+  intros; simpl. apply associativity.
+  Defined.
+Coercion FullSubCategoriesAreCategories : FullSubcategory >-> Category.
 
-(* every category is a subcategory of itself! *)
+(* every category is a subcategory of itself *)
+(*
 Instance IdentitySubCategory `(C:Category Ob Hom) : SubCategory C (fun _ => True) (fun _ _ _ _ _ => True).
   intros; apply Build_SubCategory.
   intros; auto.
   intros; auto.
   Defined.
-
-(* a full subcategory requires nothing more than a predicate on objects *)
-Definition FullSubcategory `(C:Category)(Pobj:C->Type) : SubCategory C Pobj (fun _ _ _ _ _ => True).
-  apply Build_SubCategory; intros; auto.
+*)
+(* the inclusion operation from a subcategory to its host is a functor *)
+Instance FullSubcategoryInclusionFunctor `(SP:@FullSubcategory Ob Hom C Pobj)
+  : Functor SP C (fun x => projT1 x) :=
+  { fmor := fun dom ran f => f }.
+  intros. unfold eqv in H. simpl in H. auto.
+  intros. simpl. reflexivity.
+  intros. simpl. reflexivity.
   Defined.
 
-Section SubCategoriesAreCategories.
-  (* Any such predicate determines a category *)
-  Instance SubCategoriesAreCategories `(C:Category Ob Hom)`(SP:@SubCategory _ _ C Pobj Pmor)
-    : Category (sigT Pobj) (fun dom ran => sigT (fun f => Pmor (projT1 dom) (projT1 ran) (projT2 dom) (projT2 ran) f)) :=
-  { id   := fun t         => existT (fun f => Pmor _ _ _ _ f) (id (projT1 t)) (sc_id_included _ (projT2 t))
+(* a wide subcategory includes all objects, so it requires nothing more than a predicate on each hom-set *)
+Class WideSubcategory `(C:Category Ob Hom)(Pmor:forall a b:Ob, (a~>b) ->Type) : Type :=
+{ wsc_id_included    : forall (a:Ob), Pmor a a (id a)
+; wsc_comp_included  : forall (a b c:Ob) f g, (Pmor a b f) -> (Pmor b c g) -> (Pmor a c (f>>>g))
+}.
+
+(* the category construction for full subcategories is simpler: *)
+Instance WideSubCategoriesAreCategories `{C:Category(Ob:=Ob)}{Pmor}(wsc:WideSubcategory C Pmor)
+  : Category Ob (fun x y => sigT (Pmor x y)) :=
+  { id   := fun t         => existT _ (id t) (@wsc_id_included _ _ _ _ wsc t)
   ; eqv  := fun a b f g   => eqv _ _ (projT1 f) (projT1 g)
-  ; comp := fun a b c f g => existT (fun f => Pmor _ _ _ _ f) (projT1 f >>> projT1 g)
-                                    (sc_comp_included _ _ _ (projT2 a) (projT2 b) (projT2 c) _ _ (projT2 f) (projT2 g))
+  ; comp := fun a b c f g => existT (Pmor a c) (projT1 f >>> projT1 g)
+                                    (@wsc_comp_included _ _ _ _ wsc _ _ _ _ _ (projT2 f) (projT2 g))
   }.
   intros; apply Build_Equivalence. unfold Reflexive.
      intros; reflexivity.
@@ -53,55 +77,90 @@ Section SubCategoriesAreCategories.
   intros; simpl. apply right_identity.
   intros; simpl. apply associativity.
   Defined.
-End SubCategoriesAreCategories.
-Coercion SubCategoriesAreCategories : SubCategory >-> Category.
+Coercion WideSubCategoriesAreCategories : WideSubcategory >-> Category.
 
-(* the inclusion operation from a subcategory to its host is a functor *)
-Instance InclusionFunctor `(C:Category Ob Hom)`(SP:@SubCategory _ _ C Pobj Pmor)
-  : Functor SP C (fun x => projT1 x) :=
-  { fmor := fun dom ran f => projT1 f }.
-  intros. unfold eqv in H. simpl in H. auto.
-  intros. simpl. reflexivity.
-  intros. simpl. reflexivity.
-  Defined.
+(* the full image of a functor is a full subcategory *)
+Section FullImage.
 
-Definition FullImage `(F:Functor(c1:=C)(c2:=D)(fobj:=Fobj)) := FullSubcategory D (fun d => { c:C & (Fobj c)=d }).
+  Context `(F:Functor(c1:=C)(c2:=D)).
 
-(* any functor may be restricted to its image *)
-Section RestrictToImage.
-  Context `(F:Functor(c1:=C)(c2:=D)(fobj:=Fobj)).
-  Definition RestrictToImage_fobj : C -> FullImage F.
-    intros.
-    exists (F X).
-    exists X.
-    reflexivity.
-    Defined.
-  Definition RestrictToImage_fmor a b (f:a~>b) : (RestrictToImage_fobj a)~~{FullImage F}~~>(RestrictToImage_fobj b).
-    exists (F \ f); auto.
+  Instance FullImage : Category C (fun x y => (F x)~~{D}~~>(F y)) :=
+  { id    := fun t         => id (F t)
+  ; eqv   := fun x y   f g => eqv(Category:=D)  _ _ f g
+  ; comp  := fun x y z f g => comp(Category:=D) _ _ _ f g
+  }.
+  intros; apply Build_Equivalence. unfold Reflexive.
+     intros; reflexivity.
+     unfold Symmetric; intros; simpl; symmetry; auto.
+     unfold Transitive; intros; simpl. transitivity y; auto.
+  intros; unfold Proper. unfold respectful. intros. simpl. apply comp_respects. auto. auto.
+  intros; simpl. apply left_identity.
+  intros; simpl. apply right_identity.
+  intros; simpl. apply associativity.
+  Defined.
+
+  Instance FullImage_InclusionFunctor : Functor FullImage D (fun x => F x) :=
+    { fmor := fun x y f => f }.
+    intros; auto.
+    intros; simpl; reflexivity.
+    intros; simpl; reflexivity.
     Defined.
-  Instance RestrictToImage : Functor C (FullImage F) RestrictToImage_fobj :=
-    { fmor := fun a b f => RestrictToImage_fmor a b f }.
+
+  Instance RestrictToImage : Functor C FullImage (fun x => x) :=
+    { fmor := fun a b f => F \ f }.
     intros; simpl; apply fmor_respects; auto.
     intros; simpl; apply fmor_preserves_id; auto.
     intros; simpl; apply fmor_preserves_comp; auto.
     Defined.
-  Lemma RestrictToImage_splits : F ≃ (RestrictToImage >>>> InclusionFunctor _ _).
-    exists (fun A => iso_id (F A)).
-    intros; simpl.
+
+  Lemma RestrictToImage_splits : F ~~~~ (RestrictToImage >>>> FullImage_InclusionFunctor).
+    unfold EqualFunctors; simpl; intros; apply heq_morphisms_intro.
+    apply fmor_respects.
+    auto.
+    Defined.
+
+  Lemma RestrictToImage_splits_niso : F ≃ (RestrictToImage >>>> FullImage_InclusionFunctor).
+    unfold IsomorphicFunctors.
+    exists (fun A => iso_id (fobj A)).
+    intros.
+    simpl.
     setoid_rewrite left_identity.
     setoid_rewrite right_identity.
     reflexivity.
     Qed.
-End RestrictToImage.
 
-Instance func_opSubcat `(c1:Category Ob1 Hom1)`(c2:Category Ob Hom)`(SP:@SubCategory _ _ c2 Pobj Pmor)
+End FullImage.
+
+(* any functor may be restricted to a subcategory of its domain *)
+Section RestrictDomain.
+
+  Context `{C:Category}.
+  Context `{D:Category}.
+  Context `(F:!Functor C D fobj).
+  Context  {Pmor}(S:WideSubcategory C Pmor).
+
+  Instance RestrictDomain : Functor S D fobj :=
+    { fmor  := fun a b f => F \ (projT1 f) }.
+    intros; destruct f; destruct f'; simpl in *.
+      apply fmor_respects; auto.
+    intros. simpl. apply fmor_preserves_id.
+    intros; simpl; destruct f; destruct g; simpl in *.
+      apply fmor_preserves_comp.
+    Defined.
+
+End RestrictDomain.
+
+(*
+Instance func_opSubcat `(c1:Category)`(c2:Category)`(SP:@SubCategory _ _ c2 Pobj Pmor)
   {fobj}(F:Functor c1⁽ºᑭ⁾ SP fobj) : Functor c1 SP⁽ºᑭ⁾ fobj :=
   { fmor                := fun a b f => fmor F f }.
   intros. apply (@fmor_respects _ _ _ _ _ _ _ F _ _ f f' H).
   intros. apply (@fmor_preserves_id _ _ _ _ _ _ _ F a).
   intros. apply (@fmor_preserves_comp _ _ _ _ _ _ _ F _ _ g _ f).
   Defined.
+*)
 
+(*
 (* if a functor's range falls within a subcategory, then it is already a functor into that subcategory *)
 Section FunctorWithRangeInSubCategory.
   Context `(Cat1:Category O1 Hom1).
@@ -160,6 +219,7 @@ Section FunctorWithRangeInSubCategory.
       *)
   End Opposite.
 End FunctorWithRangeInSubCategory.
+*)
 
 
 (* Definition 7.1: faithful functors *)
@@ -178,6 +238,8 @@ Definition EssentiallySurjectiveOnObjects `(F:Functor(c1:=C1)(c2:=C2)(fobj:=Fobj
   forall o2:C2, { o1:C1 & (F o1) ≅ o2 }.
 
 (* Definition 7.1: (essentially) injective on objects *)
+(* TODO *)
+
 Class ConservativeFunctor `(F:Functor(c1:=C1)(c2:=C2)(fobj:=Fobj)) :=
   { cf_reflect_iso  : forall (a b:C1),  (F a) ≅ (F b) -> a ≅ b
   ; cf_reflect_iso1 : forall a b (i:(F a) ≅ (F b)), F \ #(cf_reflect_iso a b i) ~~ #i
@@ -190,7 +252,7 @@ Definition WeaklyMonic
     `{D:Category}
      {Fobj}
      (F:@Functor _ _ C _ _ D Fobj) := forall
-    `{E:Category}
+     Eob EHom (E:@Category Eob EHom)
     `{G :@Functor _ _ E _ _ C Gobj'}
     `{H :@Functor _ _ E _ _ C Hobj'},
     G >>>> F ≃ H >>>> F
@@ -211,20 +273,10 @@ Section FullFaithfulFunctor_section.
     auto.
     Qed.
 
-  Definition ff_functor_section_fobj (a:FullImage F) : C1 := projT1 (projT2 a).
-
-  Definition ff_functor_section_fmor {a b:FullImage F} (f:a~~{FullImage F}~~>b)
-    : (ff_functor_section_fobj a)~~{C1}~~>(ff_functor_section_fobj b).
-    destruct a as [ a1 [ a2 a3 ] ].
-    subst.
-    unfold ff_functor_section_fobj.
-    simpl.
-    destruct b as [ b1 [ b2 b3 ] ].
-    subst.
-    unfold ff_functor_section_fobj.
-    simpl.
-    apply (@ff_invert _ _ _ _ _ _ _ _ F_full).
-    apply f.
+  Definition ff_functor_section_fmor {a b:FullImage F} (f:a~~{FullImage F}~~>b) : a~~{C1}~~>b.
+    set (@ff_invert _ _ _ _ _ _ _ _ F_full _ _ f) as f'.
+    destruct f'.
+    apply x.
     Defined.
 
   Lemma ff_functor_section_respectful {a2 b2 c2 : C1}
@@ -247,116 +299,67 @@ Section FullFaithfulFunctor_section.
     reflexivity.
     Qed.
 
-  Instance ff_functor_section_functor : Functor (FullImage F) C1 ff_functor_section_fobj :=
+  Instance ff_functor_section_functor : Functor (FullImage F) C1 (fun x => x) :=
     { fmor := fun a b f => ff_functor_section_fmor f }.
-    abstract (intros;
-      destruct a; destruct b; destruct s; destruct s0; destruct f; destruct f'; simpl in *;
-      subst; simpl; set (F_full x1 x2 x3) as ff1; set (F_full x1 x2 x4) as ff2; destruct ff1; destruct ff2;
-      apply F_faithful;
-      etransitivity; [ apply e | idtac ];
-      symmetry;
-      etransitivity; [ apply e0 | idtac ];
-      symmetry; auto).
-    abstract (intros;
-      simpl;
-      destruct a as [ a1 [ a2 a3 ] ];
-      subst;
-      simpl;
-      apply ff_functor_section_id_preserved).
-    abstract (intros;
-      destruct a as [ a1 [ a2 a3 ] ];
-      destruct b as [ b1 [ b2 b3 ] ];
-      destruct c as [ c1 [ c2 c3 ] ];
-      subst;
-      simpl in *;
-      destruct f;
-      destruct g;
-      simpl in *;
-      apply ff_functor_section_respectful).
+    intros.
+      unfold ff_functor_section_fmor; simpl.
+      destruct (F_full a b f).
+      destruct (F_full a b f').
+      apply F_faithful.
+      setoid_rewrite e0.
+      setoid_rewrite e.
+      auto.
+    intros; simpl; subst.
+      apply ff_functor_section_id_preserved.
+    intros; simpl in *.
+      apply ff_functor_section_respectful.
       Defined.
-
+(*
   Lemma ff_functor_section_splits_helper (a2 b2:C1)(f:existT (fun d : C2, {c : C1 & Fobj c = d}) (Fobj a2)
         (existT (fun c : C1, Fobj c = Fobj a2) a2 (eq_refl _)) ~~{ 
       FullImage F
       }~~> existT (fun d : C2, {c : C1 & Fobj c = d}) 
              (Fobj b2) (existT (fun c : C1, Fobj c = Fobj b2) b2 (eq_refl _)))
-     : F \ (let (x1, _) := F_full a2 b2 (let (x1, _) := f in x1) in x1) ~~ projT1 f.
-    destruct f.
+     : F \ (let (x1, _) := F_full a2 b2 f in x1) ~~ f.
     simpl.
-    set (F_full a2 b2 x) as qq.
+    set (F_full a2 b2 f) as qq.
     destruct qq.
     apply e.
     Qed.
-
+*)
   Lemma ff_functor_section_splits : (ff_functor_section_functor >>>> RestrictToImage F) ~~~~ functor_id _.
-    unfold EqualFunctors.
-    intros.
-    simpl.
-    destruct a as [ a1 [ a2 a3 ] ].
-    destruct b as [ b1 [ b2 b3 ] ].
-    subst.
-    simpl in *.
-    inversion f; subst.
-    inversion f'; subst.
-    simpl in *.
-    apply heq_morphisms_intro.
-    simpl.
-    etransitivity; [ idtac | apply H ].
-    clear H.
-    clear f'.
-    apply ff_functor_section_splits_helper.
+    unfold EqualFunctors; intros; simpl.
+    unfold ff_functor_section_fmor; simpl.
+    destruct (F_full a b f).
+    idtac.
+    apply (@heq_morphisms_intro _ _ (FullImage F) a b).
+    unfold eqv; simpl.
+    setoid_rewrite e.
+    apply H.
     Qed.
 
-  Definition ff_functor_section_splits_niso_helper a : ((ff_functor_section_functor >>>> RestrictToImage F) a ≅ (functor_id (FullImage F)) a).
-    intros; simpl.
-    unfold functor_fobj.
-    unfold ff_functor_section_fobj.
-    unfold RestrictToImage_fobj.
-    destruct a as [ a1 [ a2 a3 ] ].
-    simpl.
-    subst.
-    unfold functor_fobj.
-    apply iso_id.
-    Defined.
-
   Lemma ff_functor_section_splits_niso : (ff_functor_section_functor >>>> RestrictToImage F) ≃ functor_id _.
     intros; simpl.
-    exists ff_functor_section_splits_niso_helper.
-    intros.
-    simpl in *.
-    destruct A as [ a1 [ a2 a3 ] ].
-    destruct B as [ b1 [ b2 b3 ] ].
-    simpl.
-    destruct f; subst.
-    simpl.
+    exists iso_id; intros.
+    symmetry.
+    unfold functor_comp; simpl.
     setoid_rewrite left_identity.
     setoid_rewrite right_identity.
-    set (F_full a2 b2 x) as qr.
-    destruct qr.
-    symmetry; auto.
+    unfold ff_functor_section_fmor.
+    destruct (F_full A B f).
+    auto.
     Qed.
 
-  Definition ff_functor_section_splits_niso_helper' a
-    : ((RestrictToImage F >>>> ff_functor_section_functor) a ≅ (functor_id _) a).
-    intros; simpl.
-    unfold functor_fobj.
-    unfold ff_functor_section_fobj.
-    unfold RestrictToImage_fobj.
-    simpl.
-    apply iso_id.
-    Defined.
-
   Lemma ff_functor_section_splits_niso' : (RestrictToImage F >>>> ff_functor_section_functor) ≃ functor_id _.
     intros; simpl.
-    exists ff_functor_section_splits_niso_helper'.
-    intros.
-    simpl in *.
+    exists iso_id; intros.
+    symmetry.
+    unfold functor_comp; simpl.
     setoid_rewrite left_identity.
     setoid_rewrite right_identity.
-    set (F_full _ _ (F \ f)) as qr.
-    destruct qr.
-    apply F_faithful in e.
-    symmetry.
+    unfold ff_functor_section_fmor.
+    destruct (F_full A B (F \ f)).
+    apply F_faithful.
     auto.
     Qed.
 
@@ -393,13 +396,13 @@ Section FullFaithfulFunctor_section.
     apply if_right_identity.
     eapply if_inv.
     apply (if_comp(F2:=G>>>>(RestrictToImage F >>>> ff_functor_section_functor))).
-    apply (@if_respects _ _ _ _ _ _ _ _ _ _ G _ G _ (functor_id C1) _ (RestrictToImage F >>>> ff_functor_section_functor)).
+    apply (if_respects G G (functor_id C1) (RestrictToImage F >>>> ff_functor_section_functor)).
     apply if_id.
     apply if_inv.
     apply ff_functor_section_splits_niso'.
     apply if_inv.
     apply (if_comp(F2:=H0>>>>(RestrictToImage F >>>> ff_functor_section_functor))).
-    apply (@if_respects _ _ _ _ _ _ _ _ _ _ H0 _ H0 _ (functor_id C1) _ (RestrictToImage F >>>> ff_functor_section_functor)).
+    apply (if_respects H0 H0 (functor_id C1) (RestrictToImage F >>>> ff_functor_section_functor)).
     apply if_id.
     apply if_inv.
     apply ff_functor_section_splits_niso'.
@@ -407,7 +410,7 @@ Section FullFaithfulFunctor_section.
       ((H0 >>>> (RestrictToImage F >>>> ff_functor_section_functor))
         ≃ ((H0 >>>> RestrictToImage F) >>>> ff_functor_section_functor)).
     apply if_inv.
-    apply if_associativity.
+    apply (if_associativity H0 (RestrictToImage F) ff_functor_section_functor).
     apply (if_comp H2).
     clear H2.
     apply if_inv.
@@ -415,15 +418,20 @@ Section FullFaithfulFunctor_section.
       ((G >>>> (RestrictToImage F >>>> ff_functor_section_functor))
         ≃ ((G >>>> RestrictToImage F) >>>> ff_functor_section_functor)).
     apply if_inv.
-    apply if_associativity.
+    apply (if_associativity G (RestrictToImage F) ff_functor_section_functor).
     apply (if_comp H2).
     clear H2.
-    apply if_respects.
+    apply (if_respects (G >>>> RestrictToImage F) (H0 >>>> RestrictToImage F)
+      ff_functor_section_functor ff_functor_section_functor).
     apply if_fullimage.
     apply H1.
-    apply if_id.
+    simpl.
+    exists (ni_id _).
+    intros.
+    simpl.
+    setoid_rewrite left_identity.
+    setoid_rewrite right_identity.
+    reflexivity.
     Qed.
 
-  Opaque ff_functor_section_splits_niso_helper.
-
 End FullFaithfulFunctor_section.