:: Subcategories and Products of Categories
:: by Czes{\l}aw Byli\'nski
::
:: Received May 31, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

theorem :: CAT_2:1
canceled;

theorem :: CAT_2:2
canceled;

theorem :: CAT_2:3
canceled;

theorem :: CAT_2:4
canceled;

:: Auxiliary theorems on Functors
definition
let B, C be Category;
let c be Object of C;
func B --> c -> Functor of B,C equals :: CAT_2:def 1
the carrier' of B --> (id c);
coherence
the carrier' of B --> (id c) is Functor of B,C
proof end;
end;

:: deftheorem defines --> CAT_2:def 1 :
for B, C being Category
for c being Object of C holds B --> c = the carrier' of B --> (id c);

theorem :: CAT_2:5
for B, C being Category
for c being Object of C
for b being Object of B holds (Obj (B --> c)) . b = c
proof end;

definition
let C, D be Category;
func Funct (C,D) -> set means :Def2: :: CAT_2:def 2
for x being set holds
( x in it iff x is Functor of C,D );
existence
ex b1 being set st
for x being set holds
( x in b1 iff x is Functor of C,D )
proof end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff x is Functor of C,D ) ) & ( for x being set holds
( x in b2 iff x is Functor of C,D ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines Funct CAT_2:def 2 :
for C, D being Category
for b3 being set holds
( b3 = Funct (C,D) iff for x being set holds
( x in b3 iff x is Functor of C,D ) );

registration
let C, D be Category;
cluster Funct (C,D) -> non empty ;
coherence
not Funct (C,D) is empty
proof end;
end;

definition
let C, D be Category;
mode FUNCTOR-DOMAIN of C,D -> non empty set means :Def3: :: CAT_2:def 3
for x being Element of it holds x is Functor of C,D;
existence
ex b1 being non empty set st
for x being Element of b1 holds x is Functor of C,D
proof end;
end;

:: deftheorem Def3 defines FUNCTOR-DOMAIN CAT_2:def 3 :
for C, D being Category
for b3 being non empty set holds
( b3 is FUNCTOR-DOMAIN of C,D iff for x being Element of b3 holds x is Functor of C,D );

definition
let C, D be Category;
let F be FUNCTOR-DOMAIN of C,D;
:: original: Element
redefine mode Element of F -> Functor of C,D;
coherence
for b1 being Element of F holds b1 is Functor of C,D
by Def3;
end;

definition
let A be non empty set ;
let C, D be Category;
let F be FUNCTOR-DOMAIN of C,D;
let T be Function of A,F;
let x be Element of A;
:: original: .
redefine func T . x -> Element of F;
coherence
T . x is Element of F
proof end;
end;

definition
let C, D be Category;
:: original: Funct
redefine func Funct (C,D) -> FUNCTOR-DOMAIN of C,D;
coherence
Funct (C,D) is FUNCTOR-DOMAIN of C,D
proof end;
end;

:: Subcategory
definition
let C be Category;
mode Subcategory of C -> Category means :Def4: :: CAT_2:def 4
( the carrier of it c= the carrier of C & ( for a, b being Object of it
for a9, b9 being Object of C st a = a9 & b = b9 holds
Hom (a,b) c= Hom (a9,b9) ) & the Comp of it c= the Comp of C & ( for a being Object of it
for a9 being Object of C st a = a9 holds
id a = id a9 ) );
existence
ex b1 being Category st
( the carrier of b1 c= the carrier of C & ( for a, b being Object of b1
for a9, b9 being Object of C st a = a9 & b = b9 holds
Hom (a,b) c= Hom (a9,b9) ) & the Comp of b1 c= the Comp of C & ( for a being Object of b1
for a9 being Object of C st a = a9 holds
id a = id a9 ) )
proof end;
end;

:: deftheorem Def4 defines Subcategory CAT_2:def 4 :
for C, b2 being Category holds
( b2 is Subcategory of C iff ( the carrier of b2 c= the carrier of C & ( for a, b being Object of b2
for a9, b9 being Object of C st a = a9 & b = b9 holds
Hom (a,b) c= Hom (a9,b9) ) & the Comp of b2 c= the Comp of C & ( for a being Object of b2
for a9 being Object of C st a = a9 holds
id a = id a9 ) ) );

registration
let C be Category;
existence
ex b1 being Subcategory of C st b1 is strict
proof end;
end;

theorem Th2: :: CAT_2:6
for C being Category
for E being Subcategory of C
for e being Object of E holds e is Object of C
proof end;

theorem Th3: :: CAT_2:7
for C being Category
for E being Subcategory of C holds the carrier' of E c= the carrier' of C
proof end;

theorem Th4: :: CAT_2:8
for C being Category
for E being Subcategory of C
for f being Morphism of E holds f is Morphism of C
proof end;

theorem Th5: :: CAT_2:9
for C being Category
for E being Subcategory of C
for f being Morphism of E
for f9 being Morphism of C st f = f9 holds
( dom f = dom f9 & cod f = cod f9 )
proof end;

theorem :: CAT_2:10
for C being Category
for E being Subcategory of C
for a, b being Object of E
for a9, b9 being Object of C
for f being Morphism of a,b st a = a9 & b = b9 & Hom (a,b) <> {} holds
f is Morphism of a9,b9
proof end;

theorem Th7: :: CAT_2:11
for C being Category
for E being Subcategory of C
for f, g being Morphism of E
for f9, g9 being Morphism of C st f = f9 & g = g9 & dom g = cod f holds
g (*) f = g9 (*) f9
proof end;

theorem Th8: :: CAT_2:12
for C being Category holds C is Subcategory of C
proof end;

theorem Th9: :: CAT_2:13
for C being Category
for E being Subcategory of C holds id E is Functor of E,C
proof end;

definition
let C be Category;
let E be Subcategory of C;
func incl E -> Functor of E,C equals :: CAT_2:def 5
id E;
coherence
id E is Functor of E,C
by Th9;
end;

:: deftheorem defines incl CAT_2:def 5 :
for C being Category
for E being Subcategory of C holds incl E = id E;

theorem Th10: :: CAT_2:14
for C being Category
for E being Subcategory of C
for a being Object of E holds (Obj (incl E)) . a = a
proof end;

theorem :: CAT_2:15
for C being Category
for E being Subcategory of C
for a being Object of E holds (incl E) . a = a by Th10;

theorem :: CAT_2:16
for C being Category
for E being Subcategory of C holds incl E is faithful
proof end;

theorem Th13: :: CAT_2:17
for C being Category
for E being Subcategory of C holds
( incl E is full iff for a, b being Object of E
for a9, b9 being Object of C st a = a9 & b = b9 holds
Hom (a,b) = Hom (a9,b9) )
proof end;

definition
let D be Category;
let C be Subcategory of D;
attr C is full means :Def6: :: CAT_2:def 6
for c1, c2 being Object of C
for d1, d2 being Object of D st c1 = d1 & c2 = d2 holds
Hom (c1,c2) = Hom (d1,d2);
end;

:: deftheorem Def6 defines full CAT_2:def 6 :
for D being Category
for C being Subcategory of D holds
( C is full iff for c1, c2 being Object of C
for d1, d2 being Object of D st c1 = d1 & c2 = d2 holds
Hom (c1,c2) = Hom (d1,d2) );

registration
let D be Category;
existence
ex b1 being Subcategory of D st b1 is full
proof end;
end;

theorem :: CAT_2:18
for C being Category
for E being Subcategory of C holds
( E is full iff incl E is full ) by Th13;

theorem Th15: :: CAT_2:19
for C being Category
for O being non empty Subset of the carrier of C holds union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } is non empty Subset of the carrier' of C
proof end;

theorem Th16: :: CAT_2:20
for C being Category
for O being non empty Subset of the carrier of C
for M being non empty set st M = union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } holds
( the Source of C | M is Function of M,O & the Target of C | M is Function of M,O & the Comp of C || M is PartFunc of [:M,M:],M )
proof end;

registration
let O, M be non empty set ;
let d, c be Function of M,O;
let p be PartFunc of [:M,M:],M;
cluster CatStr(# O,M,d,c,p #) -> non empty non void ;
coherence
( not CatStr(# O,M,d,c,p #) is void & not CatStr(# O,M,d,c,p #) is empty )
;
end;

Lm1: for C being Category
for O being non empty Subset of the carrier of C
for M being non empty set
for d, c being Function of M,O
for p being PartFunc of [:M,M:],M
for i being Function of O,M st M = union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } & d = the Source of C | M & c = the Target of C | M & p = the Comp of C || M holds
CatStr(# O,M,d,c,p #) is Category

proof end;

Lm2: for C being Category
for O being non empty Subset of the carrier of C
for M being non empty set
for d, c being Function of M,O
for p being PartFunc of [:M,M:],M
for i being Function of O,M st M = union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } & d = the Source of C | M & c = the Target of C | M & p = the Comp of C || M holds
CatStr(# O,M,d,c,p #) is Subcategory of C

proof end;

theorem :: CAT_2:21
for C being Category
for O being non empty Subset of the carrier of C
for M being non empty set
for d, c being Function of M,O
for p being PartFunc of [:M,M:],M
for i being Function of O,M st M = union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } & d = the Source of C | M & c = the Target of C | M & p = the Comp of C || M holds
CatStr(# O,M,d,c,p #) is full Subcategory of C
proof end;

theorem :: CAT_2:22
for C being Category
for O being non empty Subset of the carrier of C
for M being non empty set
for d, c being Function of M,O
for p being PartFunc of [:M,M:],M st CatStr(# O,M,d,c,p #) is full Subcategory of C holds
( M = union { (Hom (a,b)) where a, b is Object of C : ( a in O & b in O ) } & d = the Source of C | M & c = the Target of C | M & p = the Comp of C || M )
proof end;

:: Product of Categories
definition
let C, D be Category;
func [:C,D:] -> Category equals :: CAT_2:def 7
CatStr(# [: the carrier of C, the carrier of D:],[: the carrier' of C, the carrier' of D:],[: the Source of C, the Source of D:],[: the Target of C, the Target of D:],|: the Comp of C, the Comp of D:| #);
coherence
CatStr(# [: the carrier of C, the carrier of D:],[: the carrier' of C, the carrier' of D:],[: the Source of C, the Source of D:],[: the Target of C, the Target of D:],|: the Comp of C, the Comp of D:| #) is Category
proof end;
end;

:: deftheorem defines [: CAT_2:def 7 :
for C, D being Category holds [:C,D:] = CatStr(# [: the carrier of C, the carrier of D:],[: the carrier' of C, the carrier' of D:],[: the Source of C, the Source of D:],[: the Target of C, the Target of D:],|: the Comp of C, the Comp of D:| #);

theorem :: CAT_2:23
for C, D being Category holds
( the carrier of [:C,D:] = [: the carrier of C, the carrier of D:] & the carrier' of [:C,D:] = [: the carrier' of C, the carrier' of D:] & the Source of [:C,D:] = [: the Source of C, the Source of D:] & the Target of [:C,D:] = [: the Target of C, the Target of D:] & the Comp of [:C,D:] = |: the Comp of C, the Comp of D:| ) ;

definition
let C, D be Category;
let c be Object of C;
let d be Object of D;
:: original: [
redefine func [c,d] -> Object of [:C,D:];
coherence
[c,d] is Object of [:C,D:]
proof end;
end;

theorem :: CAT_2:24
canceled;

::$CT theorem :: CAT_2:25 for C, D being Category for cd being Object of [:C,D:] ex c being Object of C ex d being Object of D st cd = [c,d] by DOMAIN_1:1; definition let C, D be Category; let f be Morphism of C; let g be Morphism of D; :: original: [ redefine func [f,g] -> Morphism of [:C,D:]; coherence [f,g] is Morphism of [:C,D:] proof end; end; theorem :: CAT_2:26 canceled; ::$CT
theorem :: CAT_2:27
for C, D being Category
for fg being Morphism of [:C,D:] ex f being Morphism of C ex g being Morphism of D st fg = [f,g] by DOMAIN_1:1;

theorem Th22: :: CAT_2:28
for C, D being Category
for f being Morphism of C
for g being Morphism of D holds
( dom [f,g] = [(dom f),(dom g)] & cod [f,g] = [(cod f),(cod g)] )
proof end;

theorem Th23: :: CAT_2:29
for C, D being Category
for f, f9 being Morphism of C
for g, g9 being Morphism of D st dom f9 = cod f & dom g9 = cod g holds
[f9,g9] (*) [f,g] = [(f9 (*) f),(g9 (*) g)]
proof end;

theorem Th24: :: CAT_2:30
for C, D being Category
for f, f9 being Morphism of C
for g, g9 being Morphism of D st dom [f9,g9] = cod [f,g] holds
[f9,g9] (*) [f,g] = [(f9 (*) f),(g9 (*) g)]
proof end;

theorem Th25: :: CAT_2:31
for C, D being Category
for c being Object of C
for d being Object of D holds id [c,d] = [(id c),(id d)]
proof end;

theorem :: CAT_2:32
for C, D being Category
for c, c9 being Object of C
for d, d9 being Object of D holds Hom ([c,d],[c9,d9]) = [:(Hom (c,c9)),(Hom (d,d9)):]
proof end;

theorem :: CAT_2:33
for C, D being Category
for c, c9 being Object of C
for f being Morphism of c,c9
for d, d9 being Object of D
for g being Morphism of d,d9 st Hom (c,c9) <> {} & Hom (d,d9) <> {} holds
[f,g] is Morphism of [c,d],[c9,d9]
proof end;

:: Bifunctors
definition
let C, C9, D be Category;
let S be Functor of [:C,C9:],D;
let m be Morphism of C;
let m9 be Morphism of C9;
:: original: .
redefine func S . (m,m9) -> Morphism of D;
coherence
S . (m,m9) is Morphism of D
proof end;
end;

theorem Th28: :: CAT_2:34
for C, D, C9 being Category
for S being Functor of [:C,C9:],D
for c being Object of C holds () . (id c) is Functor of C9,D
proof end;

theorem Th29: :: CAT_2:35
for C, D, C9 being Category
for S being Functor of [:C,C9:],D
for c9 being Object of C9 holds () . (id c9) is Functor of C,D
proof end;

:: Partial Functors
definition
let C, C9, D be Category;
let S be Functor of [:C,C9:],D;
let c be Object of C;
func S ?- c -> Functor of C9,D equals :: CAT_2:def 8
() . (id c);
coherence
() . (id c) is Functor of C9,D
by Th28;
end;

:: deftheorem defines ?- CAT_2:def 8 :
for C, C9, D being Category
for S being Functor of [:C,C9:],D
for c being Object of C holds S ?- c = () . (id c);

theorem :: CAT_2:36
for C, D, C9 being Category
for S being Functor of [:C,C9:],D
for c being Object of C
for f being Morphism of C9 holds (S ?- c) . f = S . ((id c),f) by FUNCT_5:69;

theorem :: CAT_2:37
for C, D, C9 being Category
for S being Functor of [:C,C9:],D
for c being Object of C
for c9 being Object of C9 holds (Obj (S ?- c)) . c9 = (Obj S) . (c,c9)
proof end;

definition
let C, C9, D be Category;
let S be Functor of [:C,C9:],D;
let c9 be Object of C9;
func S -? c9 -> Functor of C,D equals :: CAT_2:def 9
() . (id c9);
coherence
() . (id c9) is Functor of C,D
by Th29;
end;

:: deftheorem defines -? CAT_2:def 9 :
for C, C9, D being Category
for S being Functor of [:C,C9:],D
for c9 being Object of C9 holds S -? c9 = () . (id c9);

theorem :: CAT_2:38
for C, D, C9 being Category
for S being Functor of [:C,C9:],D
for c9 being Object of C9
for f being Morphism of C holds (S -? c9) . f = S . (f,(id c9)) by FUNCT_5:70;

theorem :: CAT_2:39
for C, D, C9 being Category
for S being Functor of [:C,C9:],D
for c being Object of C
for c9 being Object of C9 holds (Obj (S -? c9)) . c = (Obj S) . (c,c9)
proof end;

theorem :: CAT_2:40
for B, C, D being Category
for L being Function of the carrier of C,(Funct (B,D))
for M being Function of the carrier of B,(Funct (C,D)) st ( for c being Object of C
for b being Object of B holds (M . b) . (id c) = (L . c) . (id b) ) & ( for f being Morphism of B
for g being Morphism of C holds ((M . (cod f)) . g) (*) ((L . (dom g)) . f) = ((L . (cod g)) . f) (*) ((M . (dom f)) . g) ) holds
ex S being Functor of [:B,C:],D st
for f being Morphism of B
for g being Morphism of C holds S . (f,g) = ((L . (cod g)) . f) (*) ((M . (dom f)) . g)
proof end;

theorem :: CAT_2:41
for B, C, D being Category
for L being Function of the carrier of C,(Funct (B,D))
for M being Function of the carrier of B,(Funct (C,D)) st ex S being Functor of [:B,C:],D st
for c being Object of C
for b being Object of B holds
( S -? c = L . c & S ?- b = M . b ) holds
for f being Morphism of B
for g being Morphism of C holds ((M . (cod f)) . g) (*) ((L . (dom g)) . f) = ((L . (cod g)) . f) (*) ((M . (dom f)) . g)
proof end;

definition
let C, D be Category;
func pr1 (C,D) -> Functor of [:C,D:],C equals :: CAT_2:def 10
pr1 ( the carrier' of C, the carrier' of D);
coherence
pr1 ( the carrier' of C, the carrier' of D) is Functor of [:C,D:],C
proof end;
func pr2 (C,D) -> Functor of [:C,D:],D equals :: CAT_2:def 11
pr2 ( the carrier' of C, the carrier' of D);
coherence
pr2 ( the carrier' of C, the carrier' of D) is Functor of [:C,D:],D
proof end;
end;

:: deftheorem defines pr1 CAT_2:def 10 :
for C, D being Category holds pr1 (C,D) = pr1 ( the carrier' of C, the carrier' of D);

:: deftheorem defines pr2 CAT_2:def 11 :
for C, D being Category holds pr2 (C,D) = pr2 ( the carrier' of C, the carrier' of D);

theorem :: CAT_2:42
canceled;

theorem :: CAT_2:43
canceled;

::$CT 2 theorem :: CAT_2:44 for C, D being Category for c being Object of C for d being Object of D holds (Obj (pr1 (C,D))) . (c,d) = c proof end; theorem :: CAT_2:45 for C, D being Category for c being Object of C for d being Object of D holds (Obj (pr2 (C,D))) . (c,d) = d proof end; definition let C, D, D9 be Category; let T be Functor of C,D; let T9 be Functor of C,D9; :: original: <: redefine func <:T,T9:> -> Functor of C,[:D,D9:]; coherence <:T,T9:> is Functor of C,[:D,D9:] proof end; end; theorem :: CAT_2:46 canceled; ::$CT
theorem :: CAT_2:47
for C, D, D9 being Category
for T being Functor of C,D
for T9 being Functor of C,D9
for c being Object of C holds (Obj <:T,T9:>) . c = [((Obj T) . c),((Obj T9) . c)]
proof end;

theorem Th39: :: CAT_2:48
for C, D, C9, D9 being Category
for T being Functor of C,D
for T9 being Functor of C9,D9 holds [:T,T9:] = <:(T * (pr1 (C,C9))),(T9 * (pr2 (C,C9))):>
proof end;

definition
let C, C9, D, D9 be Category;
let T be Functor of C,D;
let T9 be Functor of C9,D9;
:: original: [:
redefine func [:T,T9:] -> Functor of [:C,C9:],[:D,D9:];
coherence
[:T,T9:] is Functor of [:C,C9:],[:D,D9:]
proof end;
end;

theorem :: CAT_2:49
canceled;

::\$CT
theorem :: CAT_2:50
for C, D, C9, D9 being Category
for T being Functor of C,D
for T9 being Functor of C9,D9
for c being Object of C
for c9 being Object of C9 holds (Obj [:T,T9:]) . (c,c9) = [((Obj T) . c),((Obj T9) . c9)]
proof end;