include "sets/sets.ma".
-ndefinition binary_morph_setoid : setoid → setoid → setoid → setoid.
-#S1; #S2; #T; @ (binary_morphism S1 S2 T); @;
-##[ #f; #g; napply (∀x,y. f x y = g x y);
-##| #f; #x; #y; napply #;
-##| #f; #g; #H; #x; #y; napply ((H x y)^-1);
-##| #f; #g; #h; #H1; #H2; #x; #y; napply (trans … (H1 …) (H2 …)); ##]
-nqed.
-
-ndefinition unary_morph_setoid : setoid → setoid → setoid.
-#S1; #S2; @ (unary_morphism S1 S2); @;
-##[ #f; #g; napply (∀x. f x = g x);
-##| #f; #x; napply #;
-##| #f; #g; #H; #x; napply ((H x)^-1);
-##| #f; #g; #h; #H1; #H2; #x; napply (trans … (H1 …) (H2 …)); ##]
-nqed.
-
nrecord category : Type[2] ≝
{ objs:> Type[1];
arrows: objs → objs → setoid;
id: ∀o:objs. arrows o o;
- comp: ∀o1,o2,o3. binary_morphism (arrows o1 o2) (arrows o2 o3) (arrows o1 o3);
- comp_assoc: ∀o1,o2,o3,o4. ∀a12,a23,a34.
- comp o1 o3 o4 (comp o1 o2 o3 a12 a23) a34 = comp o1 o2 o4 a12 (comp o2 o3 o4 a23 a34);
- id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o1) a = a;
- id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o2) = a
+ comp: ∀o1,o2,o3. binary_morphism (arrows o2 o3) (arrows o1 o2) (arrows o1 o3);
+ comp_assoc: ∀o1,o2,o3,o4. ∀a34,a23,a12.
+ comp o1 o3 o4 a34 (comp o1 o2 o3 a23 a12) = comp o1 o2 o4 (comp o2 o3 o4 a34 a23) a12;
+ id_neutral_left: ∀o1,o2. ∀a: arrows o1 o2. comp ??? (id o2) a = a;
+ id_neutral_right: ∀o1,o2. ∀a: arrows o1 o2. comp ??? a (id o1) = a
}.
notation "hvbox(A break ⇒ B)" right associative with precedence 50 for @{ 'arrows $A $B }.
##[ napply setoid;
##| napply unary_morph_setoid;
##| #o; @ (λx.x); #a; #b; #H; napply H;
-##| #o1; #o2; #o3; @;
- ##[ #f; #g; @(λx.g (f x)); #a; #b; #H; napply (.= (††H)); napply #;
- ##| #f; #g; #f'; #g'; #H1; #H2; nwhd; #x; napply (.= (H2 (f x)));
- napply (.= (†(H1 x))); napply #; ##]
+##| napply comp_binary_morphisms; (*CSC: why not ∘?*)
##| #o1; #o2; #o3; #o4; #f; #g; #h; nwhd; #x; napply #;
##|##6,7: #o1; #o2; #f; nwhd; #x; napply #; ##]
nqed.