:: Yoneda Embedding
:: by Miros{\l}aw Wojciechowski
::
:: Received June 12, 1997
:: Copyright (c) 1997-2021 Association of Mizar Users

definition

let A be Category;

func EnsHom A -> Category equals :: YONEDA_1:def 1
Ens (Hom A);

correctness
coherence
Ens (Hom A) is Category;
;

end;

:: deftheorem defines EnsHom YONEDA_1:def 1 :
for A being Category holds EnsHom A = Ens (Hom A);

theorem Th1: :: YONEDA_1:1

for A being Category
for f, g being Function
for m1, m2 being Morphism of (EnsHom A) st cod m1 = dom m2 & [[(dom m1),(cod m1)],f] = m1 & [[(dom m2),(cod m2)],g] = m2 holds
[[(dom m1),(cod m2)],(g * f)] = m2 (*) m1

proof end;

definition

let A be Category;
let a be Object of A;

func <|a,?> -> Functor of A, EnsHom A equals :: YONEDA_1:def 2
hom?- a;

coherence
hom?- a is Functor of A, EnsHom A by ENS_1:49;

end;

:: deftheorem defines <| YONEDA_1:def 2 :
for A being Category
for a being Object of A holds <|a,?> = hom?- a;

theorem Th2: :: YONEDA_1:2

for A being Category
for f being Morphism of A holds <|(cod f),?> is_naturally_transformable_to <|(dom f),?>

proof end;

definition

let A be Category;
let f be Morphism of A;

func <|f,?> -> natural_transformation of <|(cod f),?>,<|(dom f),?> means :Def3: :: YONEDA_1:def 3
for o being Object of A holds it . o = [[(Hom ((cod f),o)),(Hom ((dom f),o))],(hom (f,(id o)))];

existence
ex b₁ being natural_transformation of <|(cod f),?>,<|(dom f),?> st
for o being Object of A holds b₁ . o = [[(Hom ((cod f),o)),(Hom ((dom f),o))],(hom (f,(id o)))]

proof end;

uniqueness
for b₁, b₂ being natural_transformation of <|(cod f),?>,<|(dom f),?> st ( for o being Object of A holds b₁ . o = [[(Hom ((cod f),o)),(Hom ((dom f),o))],(hom (f,(id o)))] ) & ( for o being Object of A holds b₂ . o = [[(Hom ((cod f),o)),(Hom ((dom f),o))],(hom (f,(id o)))] ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def3 defines <| YONEDA_1:def 3 :
for A being Category
for f being Morphism of A
for b₃ being natural_transformation of <|(cod f),?>,<|(dom f),?> holds
( b₃ = <|f,?> iff for o being Object of A holds b₃ . o = [[(Hom ((cod f),o)),(Hom ((dom f),o))],(hom (f,(id o)))] );

theorem Th3: :: YONEDA_1:3

for A being Category
for f being Element of the carrier' of A holds [[<|(cod f),?>,<|(dom f),?>],<|f,?>] is Element of the carrier' of (Functors (A,(EnsHom A)))

proof end;

definition

let A be Category;

func Yoneda A -> Contravariant_Functor of A, Functors (A,(EnsHom A)) means :Def4: :: YONEDA_1:def 4
for f being Morphism of A holds it . f = [[<|(cod f),?>,<|(dom f),?>],<|f,?>];

existence
ex b₁ being Contravariant_Functor of A, Functors (A,(EnsHom A)) st
for f being Morphism of A holds b₁ . f = [[<|(cod f),?>,<|(dom f),?>],<|f,?>]

proof end;

uniqueness
for b₁, b₂ being Contravariant_Functor of A, Functors (A,(EnsHom A)) st ( for f being Morphism of A holds b₁ . f = [[<|(cod f),?>,<|(dom f),?>],<|f,?>] ) & ( for f being Morphism of A holds b₂ . f = [[<|(cod f),?>,<|(dom f),?>],<|f,?>] ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines Yoneda YONEDA_1:def 4 :
for A being Category
for b₂ being Contravariant_Functor of A, Functors (A,(EnsHom A)) holds
( b₂ = Yoneda A iff for f being Morphism of A holds b₂ . f = [[<|(cod f),?>,<|(dom f),?>],<|f,?>] );

definition

let A, B be Category;
let F be Contravariant_Functor of A,B;
let c be Object of A;

func F . c -> Object of B equals :: YONEDA_1:def 5
(Obj F) . c;

correctness
coherence
(Obj F) . c is Object of B;
;

end;

:: deftheorem defines . YONEDA_1:def 5 :
for A, B being Category
for F being Contravariant_Functor of A,B
for c being Object of A holds F . c = (Obj F) . c;

theorem :: YONEDA_1:4

for A being Category
for F being Functor of A, Functors (A,(EnsHom A)) st Obj F is one-to-one & F is faithful holds
F is one-to-one

proof end;

definition

let C, D be Category;
let T be Contravariant_Functor of C,D;

attr T is faithful means :: YONEDA_1:def 6
for c, c9 being Object of C st Hom (c,c9) <> {} holds
for f1, f2 being Morphism of c,c9 st T . f1 = T . f2 holds
f1 = f2;

end;

:: deftheorem defines faithful YONEDA_1:def 6 :
for C, D being Category
for T being Contravariant_Functor of C,D holds
( T is faithful iff for c, c9 being Object of C st Hom (c,c9) <> {} holds
for f1, f2 being Morphism of c,c9 st T . f1 = T . f2 holds
f1 = f2 );

theorem Th5: :: YONEDA_1:5

for A being Category
for F being Contravariant_Functor of A, Functors (A,(EnsHom A)) st Obj F is one-to-one & F is faithful holds
F is one-to-one

proof end;

registration

let A be Category;

cluster Yoneda A -> faithful ;

coherence
Yoneda A is faithful

proof end;

end;

registration

let A be Category;

cluster Yoneda A -> one-to-one ;

coherence
Yoneda A is one-to-one

proof end;

end;

definition

let C, D be Category;
let T be Contravariant_Functor of C,D;

attr T is full means :: YONEDA_1:def 7
for c, c9 being Object of C st Hom ((T . c9),(T . c)) <> {} holds
for g being Morphism of T . c9,T . c holds
( Hom (c,c9) <> {} & ex f being Morphism of c,c9 st g = T . f );

end;

:: deftheorem defines full YONEDA_1:def 7 :
for C, D being Category
for T being Contravariant_Functor of C,D holds
( T is full iff for c, c9 being Object of C st Hom ((T . c9),(T . c)) <> {} holds
for g being Morphism of T . c9,T . c holds
( Hom (c,c9) <> {} & ex f being Morphism of c,c9 st g = T . f ) );

registration

let A be Category;

cluster Yoneda A -> full ;

coherence
Yoneda A is full

proof end;

end;