+(*********************************************************************************************************************************)
+(* SemiCategory: *)
+(* *)
+(* Same as a category, but without identity maps. See *)
+(* *)
+(* http://ncatlab.org/nlab/show/semicategory *)
+(* *)
+(*********************************************************************************************************************************)
+
+Generalizable All Variables.
+Require Import Preamble.
+Require Import General.
+
+Class SemiCategory (Ob:Type)(Hom:Ob->Ob->Type) :=
+{ semi_hom := Hom
+; semi_ob := Ob
+; semi_comp : forall {a}{b}{c}, Hom a b -> Hom b c -> Hom a c
+; semi_eqv : forall a b, (Hom a b) -> (Hom a b) -> Prop
+; semi_eqv_equivalence : forall a b, Equivalence (semi_eqv a b)
+; semi_comp_respects : forall a b c, Proper (semi_eqv a b ==> semi_eqv b c ==> semi_eqv a c) (@semi_comp _ _ _)
+; semi_associativity :
+ forall `(f:Hom a b)`(g:Hom b c)`(h:Hom c d), semi_eqv _ _ (semi_comp (semi_comp f g) h) (semi_comp f (semi_comp g h))
+}.
+Coercion semi_ob : SemiCategory >-> Sortclass.
+
+Notation "a ~-> b" := (@semi_hom _ _ _ a b) (at level 70).
+Notation "f ~-~ g" := (@semi_eqv _ _ _ _ _ f g) (at level 54).
+Notation "f >>->> g" := (@semi_comp _ _ _ _ _ _ f g) (at level 45).
+
+Add Parametric Relation (Ob:Type)(Hom:Ob->Ob->Type)(C:SemiCategory Ob Hom)(a b:Ob) : (semi_hom a b) (semi_eqv a b)
+ reflexivity proved by (@Equivalence_Reflexive _ _ (semi_eqv_equivalence a b))
+ symmetry proved by (@Equivalence_Symmetric _ _ (semi_eqv_equivalence a b))
+ transitivity proved by (@Equivalence_Transitive _ _ (semi_eqv_equivalence a b))
+ as parametric_relation_semi_eqv.
+ Add Parametric Morphism `(c:SemiCategory Ob Hom)(a b c:Ob) : (@semi_comp _ _ _ a b c)
+ with signature (semi_eqv _ _ ==> semi_eqv _ _ ==> semi_eqv _ _) as parametric_morphism_semi_comp.
+ intros.
+ apply semi_comp_respects; auto.
+ Defined.
+
+Class SemiFunctor
+`( c1 : SemiCategory )
+`( c2 : SemiCategory )
+( fobj : c1 -> c2 ) :=
+{ semifunctor_fobj := fobj
+; semi_fmor : forall {a b}, (a~->b) -> (fobj a)~->(fobj b)
+; semi_fmor_respects : forall a b (f f':a~->b), (f ~-~ f') -> (semi_fmor f ~-~ semi_fmor f')
+; semi_fmor_preserves_comp : forall `(f:a~->b)`(g:b~->c), (semi_fmor f) >>->> (semi_fmor g) ~-~ semi_fmor (f >>->> g)
+}.
+Implicit Arguments semi_fmor [[Ob][Hom][c1][Ob0][Hom0][c2][fobj][a][b]].
+
+ (* register "fmor" so we can rewrite through it *)
+ Implicit Arguments semi_fmor [ Ob Hom Ob0 Hom0 c1 c2 fobj a b ].
+ Implicit Arguments semi_fmor_respects [ Ob Hom Ob0 Hom0 c1 c2 fobj a b ].
+ Implicit Arguments semi_fmor_preserves_comp [ Ob Hom Ob0 Hom0 c1 c2 fobj a b c ].
+ Notation "F \- f" := (semi_fmor F f) (at level 20) : category_scope.
+ Add Parametric Morphism `(C1:SemiCategory)`(C2:SemiCategory)
+ (Fobj:C1->C2)
+ (F:SemiFunctor C1 C2 Fobj)
+ (a b:C1)
+ : (@semi_fmor _ _ C1 _ _ C2 Fobj F a b)
+ with signature ((@semi_eqv C1 _ C1 a b) ==> (@semi_eqv C2 _ C2 (Fobj a) (Fobj b))) as parametric_morphism_semi_fmor'.
+ intros; apply (@semi_fmor_respects _ _ C1 _ _ C2 Fobj F a b x y); auto.
+ Defined.
+ Coercion semifunctor_fobj : SemiFunctor >-> Funclass.
+
+Definition semifunctor_comp
+ `(C1:SemiCategory)
+ `(C2:SemiCategory)
+ `(C3:SemiCategory)
+ `(F:@SemiFunctor _ _ C1 _ _ C2 Fobj)`(G:@SemiFunctor _ _ C2 _ _ C3 Gobj) : SemiFunctor C1 C3 (Gobj ○ Fobj).
+ intros. apply (Build_SemiFunctor _ _ _ _ _ _ _ (fun a b m => semi_fmor G (semi_fmor F m))).
+ intros.
+ setoid_rewrite H.
+ reflexivity.
+ intros.
+ setoid_rewrite semi_fmor_preserves_comp; auto.
+ setoid_rewrite semi_fmor_preserves_comp; auto.
+ reflexivity.
+ Defined.
+Notation "f >>>–>>> g" := (@semifunctor_comp _ _ _ _ _ _ _ _ _ _ f _ g) (at level 20) : category_scope.
+
+Class IsomorphicSemiCategories `(C:SemiCategory)`(D:SemiCategory) :=
+{ isc_f_obj : C -> D
+; isc_g_obj : D -> C
+; isc_f : SemiFunctor C D isc_f_obj
+; isc_g : SemiFunctor D C isc_g_obj
+; isc_forward : forall a b (f:a~->b), semi_fmor isc_f (semi_fmor isc_g f) ~-~ f
+}.
+; isc_backward : isc_g >>>> isc_f ~~~~ functor_id D
+}.
+
+