:: CAT_2 semantic presentation

definition

let D be non empty set ;
let X be set ;
let E be non empty set ;
let F be FUNCTION_DOMAIN of X,E;
let f be Function of D,F;
let d be Element of D;
:: original: .
redefine func f . d -> Element of F;
coherence

proof end;

end;

theorem Th1: :: CAT_2:1

for C, D, E being non empty set
for f being Function of [:C,D:],E holds curry f is Function of C, Funcs D,E

proof end;

theorem Th2: :: CAT_2:2

for D, C, E being non empty set
for f being Function of [:C,D:],E holds curry' f is Function of D, Funcs C,E

proof end;

definition

let C, D, E be non empty set ;
let f be Function of [:C,D:],E;
:: original: curry
redefine func curry f -> Function of C, Funcs D,E;
coherence by Th1;
:: original: curry'
redefine func curry' f -> Function of D, Funcs C,E;
coherence by Th2;

end;

theorem Th3: :: CAT_2:3

for C, D, E being non empty set
for c being Element of C
for d being Element of D
for f being Function of [:C,D:],E holds f . [c,d] = ((curry f) . c) . d

proof end;

theorem Th4: :: CAT_2:4

for C, D, E being non empty set
for c being Element of C
for d being Element of D
for f being Function of [:C,D:],E holds f . [c,d] = ((curry' f) . d) . c

proof end;

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 Morphisms of B --> (id c);
coherence

proof end;

end;

:: deftheorem defines --> CAT_2:def 1 :

theorem :: CAT_2:5

canceled;

theorem Th6: :: CAT_2:6

for C, B being Category
for c being Object of C
for f being Morphism of B holds (B --> c) . f = id c by FUNCOP_1:13;

theorem :: CAT_2:7

for C, B 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

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines Funct CAT_2:def 2 :

registration

let C, D be Category;
cluster Funct C,D -> non empty ;
coherence

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

proof end;

end;

:: deftheorem Def3 defines FUNCTOR-DOMAIN CAT_2:def 3 :

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 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

proof end;

end;

definition

let C, D be Category;
:: original: Funct
redefine func Funct C,D -> FUNCTOR-DOMAIN of C,D;
coherence

proof end;

end;

definition

let C be Category;
mode Subcategory of C -> Category means :Def4: :: CAT_2:def 4
( the Objects of it c= the Objects of C & ( for a, b being Object of it
for a', b' being Object of C st a = a' & b = b' holds
Hom a,b c= Hom a',b' ) & the Comp of it <= the Comp of C & ( for a being Object of it
for a' being Object of C st a = a' holds
id a = id a' ) );
existence

proof end;

end;

:: deftheorem Def4 defines Subcategory CAT_2:def 4 :

registration

let C be Category;
cluster strict Subcategory of C;
existence

proof end;

end;

theorem :: CAT_2:8

canceled;

theorem :: CAT_2:9

canceled;

theorem :: CAT_2:10

canceled;

theorem :: CAT_2:11

canceled;

theorem Th12: :: CAT_2:12

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 Th13: :: CAT_2:13

for C being Category
for E being Subcategory of C holds the Morphisms of E c= the Morphisms of C

proof end;

theorem Th14: :: CAT_2:14

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 Th15: :: CAT_2:15

for C being Category
for E being Subcategory of C
for f being Morphism of E
for f' being Morphism of C st f = f' holds
( dom f = dom f' & cod f = cod f' )

proof end;

theorem :: CAT_2:16

for C being Category
for E being Subcategory of C
for a, b being Object of E
for a', b' being Object of C
for f being Morphism of a,b st a = a' & b = b' & Hom a,b <> {} holds
f is Morphism of a',b'

proof end;

theorem Th17: :: CAT_2:17

for C being Category
for E being Subcategory of C
for f, g being Morphism of E
for f', g' being Morphism of C st f = f' & g = g' & dom g = cod f holds
g * f = g' * f'

proof end;

theorem :: CAT_2:18

for C being Category holds C is Subcategory of C

proof end;

theorem Th19: :: CAT_2:19

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 by Th19;

end;

:: deftheorem defines incl CAT_2:def 5 :

theorem :: CAT_2:20

canceled;

theorem Th21: :: CAT_2:21

for C being Category
for E being Subcategory of C
for f being Morphism of E holds (incl E) . f = f by CAT_1:115;

theorem Th22: :: CAT_2:22

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 Th23: :: CAT_2:23

for C being Category
for E being Subcategory of C
for a being Object of E holds (incl E) . a = a by Th22;

theorem :: CAT_2:24

for C being Category
for E being Subcategory of C holds incl E is faithful

proof end;

theorem Th25: :: CAT_2:25

for C being Category
for E being Subcategory of C holds
( incl E is full iff for a, b being Object of E
for a', b' being Object of C st a = a' & b = b' holds
Hom a,b = Hom a',b' )

proof end;

definition

let C be CatStr ;
let D be Category;
pred C is_full_subcategory_of D means :Def6: :: CAT_2:def 6
( C is Subcategory of D & ( for a, b being Object of C
for a', b' being Object of D st a = a' & b = b' holds
Hom a,b = Hom a',b' ) );

end;

:: deftheorem Def6 defines is_full_subcategory_of CAT_2:def 6 :

theorem :: CAT_2:26

canceled;

theorem :: CAT_2:27

for C being Category
for E being Subcategory of C holds
( E is_full_subcategory_of C iff incl E is full )

proof end;

theorem Th28: :: CAT_2:28

for C being Category
for O being non empty Subset of the Objects 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 Morphisms of C

proof end;

theorem Th29: :: CAT_2:29

for C being Category
for O being non empty Subset of the Objects 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 Dom of C | M is Function of M,O & the Cod of C | M is Function of M,O & the Comp of C || M is PartFunc of [:M,M:],M & the Id of C | O is Function of O,M )

proof end;

theorem :: CAT_2:30

for C being Category
for O being non empty Subset of the Objects 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 Dom of C | M & c = the Cod of C | M & p = the Comp of C || M & i = the Id of C | O holds
CatStr(# O,M,d,c,p,i #) is_full_subcategory_of C

proof end;

theorem :: CAT_2:31

for C being Category
for O being non empty Subset of the Objects 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 CatStr(# O,M,d,c,p,i #) 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 Dom of C | M & c = the Cod of C | M & p = the Comp of C || M & i = the Id of C | O )

proof end;

definition

let X1, X2, Y1, Y2 be non empty set ;
let f1 be Function of X1,Y1;
let f2 be Function of X2,Y2;
:: original: [:
redefine func [:f1,f2:] -> Function of [:X1,X2:],[:Y1,Y2:];
coherence

proof end;

end;

definition

let A, B be non empty set ;
let f be PartFunc of [:A,A:],A;
let g be PartFunc of [:B,B:],B;
:: original: |:
redefine func |:f,g:| -> PartFunc of [:[:A,B:],[:A,B:]:],[:A,B:];
coherence by FUNCT_4:62;

end;

definition

let C, D be Category;
func [:C,D:] -> Category equals :: CAT_2:def 7
CatStr(# [:the Objects of C,the Objects of D:],[:the Morphisms of C,the Morphisms of D:],[:the Dom of C,the Dom of D:],[:the Cod of C,the Cod of D:],|:the Comp of C,the Comp of D:|,[:the Id of C,the Id of D:] #);
coherence

proof end;

end;

:: deftheorem defines [: CAT_2:def 7 :

registration

let C, D be Category;
cluster [:C,D:] -> strict ;
coherence ;

end;

theorem :: CAT_2:32

canceled;

theorem :: CAT_2:33

for C, D being Category holds
( the Objects of [:C,D:] = [:the Objects of C,the Objects of D:] & the Morphisms of [:C,D:] = [:the Morphisms of C,the Morphisms of D:] & the Dom of [:C,D:] = [:the Dom of C,the Dom of D:] & the Cod of [:C,D:] = [:the Cod of C,the Cod of D:] & the Comp of [:C,D:] = |:the Comp of C,the Comp of D:| & the Id of [:C,D:] = [:the Id of C,the Id of D:] ) ;

theorem Th34: :: CAT_2:34

for C, D being Category
for c being Object of C
for d being Object of D holds [c,d] is Object of [:C,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 by Th34;

end;

theorem Th35: :: CAT_2:35

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_1:9;

theorem Th36: :: CAT_2:36

for C, D being Category
for f being Morphism of C
for g being Morphism of D holds [f,g] is Morphism of [:C,D:] ;

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 by Th36;

end;

theorem Th37: :: CAT_2:37

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_1:9;

theorem Th38: :: CAT_2:38

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)] ) by FUNCT_3:96;

theorem Th39: :: CAT_2:39

for C, D being Category
for f, f' being Morphism of C
for g, g' being Morphism of D st dom f' = cod f & dom g' = cod g holds
[f',g'] * [f,g] = [(f' * f),(g' * g)]

proof end;

theorem Th40: :: CAT_2:40

for C, D being Category
for f, f' being Morphism of C
for g, g' being Morphism of D st dom [f',g'] = cod [f,g] holds
[f',g'] * [f,g] = [(f' * f),(g' * g)]

proof end;

theorem Th41: :: CAT_2:41

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)] by FUNCT_3:96;

theorem :: CAT_2:42

for C, D being Category
for c, c' being Object of C
for d, d' being Object of D holds Hom [c,d],[c',d'] = [:(Hom c,c'),(Hom d,d'):]

proof end;

theorem :: CAT_2:43

for C, D being Category
for c, c' being Object of C
for f being Morphism of c,c'
for d, d' being Object of D
for g being Morphism of d,d' st Hom c,c' <> {} & Hom d,d' <> {} holds
[f,g] is Morphism of [c,d],[c',d']

proof end;

theorem Th44: :: CAT_2:44

for C, C', D being Category
for S being Functor of [:C,C':],D
for c being Object of C holds (curry S) . (id c) is Functor of C',D

proof end;

theorem Th45: :: CAT_2:45

for C, C', D being Category
for S being Functor of [:C,C':],D
for c' being Object of C' holds (curry' S) . (id c') is Functor of C,D

proof end;

definition

let C, C', D be Category;
let S be Functor of [:C,C':],D;
let c be Object of C;
func S ?- c -> Functor of C',D equals :: CAT_2:def 8
(curry S) . (id c);
coherence by Th44;

end;

:: deftheorem defines ?- CAT_2:def 8 :

theorem :: CAT_2:46

canceled;

theorem Th47: :: CAT_2:47

for C, C', D being Category
for S being Functor of [:C,C':],D
for c being Object of C
for f being Morphism of C' holds (S ?- c) . f = S . [(id c),f] by Th3;

theorem :: CAT_2:48

for C, C', D being Category
for S being Functor of [:C,C':],D
for c being Object of C
for c' being Object of C' holds (Obj (S ?- c)) . c' = (Obj S) . [c,c']

proof end;

definition

let C, C', D be Category;
let S be Functor of [:C,C':],D;
let c' be Object of C';
func S -? c' -> Functor of C,D equals :: CAT_2:def 9
(curry' S) . (id c');
coherence by Th45;

end;

:: deftheorem defines -? CAT_2:def 9 :

theorem :: CAT_2:49

canceled;

theorem Th50: :: CAT_2:50

for C, C', D being Category
for S being Functor of [:C,C':],D
for c' being Object of C'
for f being Morphism of C holds (S -? c') . f = S . [f,(id c')] by Th4;

theorem :: CAT_2:51

for C, C', D being Category
for S being Functor of [:C,C':],D
for c being Object of C
for c' being Object of C' holds (Obj (S -? c')) . c = (Obj S) . [c,c']

proof end;

theorem :: CAT_2:52

for C, B, D being Category
for L being Function of the Objects of C, Funct B,D
for M being Function of the Objects 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_2:53

for C, B, D being Category
for L being Function of the Objects of C, Funct B,D
for M being Function of the Objects 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;

theorem Th54: :: CAT_2:54

for C, D being Category holds pr1 the Morphisms of C,the Morphisms of D is Functor of [:C,D:],C

proof end;

theorem Th55: :: CAT_2:55

for C, D being Category holds pr2 the Morphisms of C,the Morphisms of D is Functor of [:C,D:],D

proof end;

definition

let C, D be Category;
func pr1 C,D -> Functor of [:C,D:],C equals :: CAT_2:def 10
pr1 the Morphisms of C,the Morphisms of D;
coherence by Th54;
func pr2 C,D -> Functor of [:C,D:],D equals :: CAT_2:def 11
pr2 the Morphisms of C,the Morphisms of D;
coherence by Th55;

end;

:: deftheorem defines pr1 CAT_2:def 10 :

:: deftheorem defines pr2 CAT_2:def 11 :

theorem :: CAT_2:56

canceled;

theorem :: CAT_2:57

canceled;

theorem Th58: :: CAT_2:58

for C, D being Category
for f being Morphism of C
for g being Morphism of D holds (pr1 C,D) . [f,g] = f by FUNCT_3:def 5;

theorem :: CAT_2:59

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 Th60: :: CAT_2:60

for C, D being Category
for f being Morphism of C
for g being Morphism of D holds (pr2 C,D) . [f,g] = g by FUNCT_3:def 6;

theorem :: CAT_2:61

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;

theorem Th62: :: CAT_2:62

for C, D, D' being Category
for T being Functor of C,D
for T' being Functor of C,D' holds <:T,T':> is Functor of C,[:D,D':]

proof end;

definition

let C, D, D' be Category;
let T be Functor of C,D;
let T' be Functor of C,D';
:: original: <:
redefine func <:T,T':> -> Functor of C,[:D,D':];
coherence by Th62;

end;

theorem :: CAT_2:63

for C, D, D' being Category
for T being Functor of C,D
for T' being Functor of C,D'
for c being Object of C holds (Obj <:T,T':>) . c = [((Obj T) . c),((Obj T') . c)]

proof end;

theorem Th64: :: CAT_2:64

for C, D, C', D' being Category
for T being Functor of C,D
for T' being Functor of C',D' holds [:T,T':] = <:(T * (pr1 C,C')),(T' * (pr2 C,C')):>

proof end;

theorem Th65: :: CAT_2:65

for C, D, C', D' being Category
for T being Functor of C,D
for T' being Functor of C',D' holds [:T,T':] is Functor of [:C,C':],[:D,D':]

proof end;

definition

let C, C', D, D' be Category;
let T be Functor of C,D;
let T' be Functor of C',D';
:: original: [:
redefine func [:T,T':] -> Functor of [:C,C':],[:D,D':];
coherence by Th65;

end;

theorem :: CAT_2:66

for C, D, C', D' being Category
for T being Functor of C,D
for T' being Functor of C',D'
for c being Object of C
for c' being Object of C' holds (Obj [:T,T':]) . [c,c'] = [((Obj T) . c),((Obj T') . c')]

proof end;