:: INDEX_1 semantic presentation
theorem Th1: :: INDEX_1:1
theorem Th2: :: INDEX_1:2
:: deftheorem Def1 defines Category-yielding INDEX_1:def 1 :
:: deftheorem Def2 defines Objs INDEX_1:def 2 :
:: deftheorem Def3 defines Mphs INDEX_1:def 3 :
theorem Th3: :: INDEX_1:3
:: deftheorem Def4 defines ManySortedSet INDEX_1:def 4 :
:: deftheorem Def5 defines Category-yielding_on_first INDEX_1:def 5 :
:: deftheorem Def6 defines Function-yielding_on_second INDEX_1:def 6 :
:: deftheorem Def7 defines ManySortedFunctor INDEX_1:def 7 :
:: deftheorem Def8 defines Indexing INDEX_1:def 8 :
theorem Th4: :: INDEX_1:4
theorem Th5: :: INDEX_1:5
E75:
now
let A be non
empty set ,
B be non
empty set ;
let F be
Function of
B,
A,
G be
Function of
B,
A;
let I be
Indexing of
F,
G;
E23:
dom (I `1 ) = A
by PBOOLE:def 3;
consider C being
strict Categorial full Category such that E26:
the
Objects of
C = rng (I `1 )
by CAT_5:20;
take C =
C;
thus
for
a being
Element of
A holds
(I `1 ) . a is
Object of
C
by , , FUNCT_1:def 5;
let b be
Element of
B;
(
(I `1 ) . (F . b) is
Object of
C &
(I `1 ) . (G . b) is
Object of
C &
(I `2 ) . b is
Functor of
(I `1 ) . (F . b),
(I `1 ) . (G . b) )
by , , , FUNCT_1:def 5;
hence
[[((I `1 ) . (F . b)),((I `1 ) . (G . b))],((I `2 ) . b)] is
Morphism of
C
by CAT_5:def 8;
end;
:: deftheorem Def9 defines TargetCat INDEX_1:def 9 :
Lemma78:
for C being Category holds id C = (id C) * (id C)
by FUNCT_2:23;
definition
let A be non
empty set ,
B be non
empty set ;
let F be
Function of
B,
A,
G be
Function of
B,
A;
let c be
PartFunc of
[:B,B:],
B;
let i be
Function of
A,
B;
given C being
Category such that E23:
C = CatStr(#
A,
B,
F,
G,
c,
i #)
;
mode Indexing of
c3,
c4,
c5,
c6 -> Indexing of
a3,
a4 means :
Def10:
:: INDEX_1:def 10
( ( for
a being
Element of
A holds
(it `2 ) . (i . a) = id ((it `1 ) . a) ) & ( for
m1,
m2 being
Element of
B st
F . m2 = G . m1 holds
(it `2 ) . (c . [m2,m1]) = ((it `2 ) . m2) * ((it `2 ) . m1) ) );
existence
ex b1 being Indexing of F,G st
( ( for a being Element of A holds (b1 `2 ) . (i . a) = id ((b1 `1 ) . a) ) & ( for m1, m2 being Element of B st F . m2 = G . m1 holds
(b1 `2 ) . (c . [m2,m1]) = ((b1 `2 ) . m2) * ((b1 `2 ) . m1) ) )
end;
:: deftheorem Def10 defines Indexing INDEX_1:def 10 :
for
A,
B being non
empty set for
F,
G being
Function of
B,
A for
c being
PartFunc of
[:B,B:],
B for
i being
Function of
A,
B st ex
C being
Category st
C = CatStr(#
A,
B,
F,
G,
c,
i #) holds
for
b7 being
Indexing of
F,
G holds
(
b7 is
Indexing of
F,
G,
c,
i iff ( ( for
a being
Element of
A holds
(b7 `2 ) . (i . a) = id ((b7 `1 ) . a) ) & ( for
m1,
m2 being
Element of
B st
F . m2 = G . m1 holds
(b7 `2 ) . (c . [m2,m1]) = ((b7 `2 ) . m2) * ((b7 `2 ) . m1) ) ) );
theorem Th6: :: INDEX_1:6
theorem Th7: :: INDEX_1:7
theorem Th8: :: INDEX_1:8
theorem Th9: :: INDEX_1:9
theorem Th10: :: INDEX_1:10
definition
let C be
Category;
let I be
Indexing of
C;
let T be
TargetCat of
I;
func c2 -functor c1,
c3 -> Functor of
a1,
a3 means :
Def11:
:: INDEX_1:def 11
for
f being
Morphism of
C holds
it . f = [[((I `1 ) . (dom f)),((I `1 ) . (cod f))],((I `2 ) . f)];
existence
ex b1 being Functor of C,T st
for f being Morphism of C holds b1 . f = [[((I `1 ) . (dom f)),((I `1 ) . (cod f))],((I `2 ) . f)]
uniqueness
for b1, b2 being Functor of C,T st ( for f being Morphism of C holds b1 . f = [[((I `1 ) . (dom f)),((I `1 ) . (cod f))],((I `2 ) . f)] ) & ( for f being Morphism of C holds b2 . f = [[((I `1 ) . (dom f)),((I `1 ) . (cod f))],((I `2 ) . f)] ) holds
b1 = b2
end;
:: deftheorem Def11 defines -functor INDEX_1:def 11 :
Lemma97:
for C being Category
for I being Indexing of C
for T being TargetCat of I holds Obj (I -functor C,T) = I `1
theorem Th11: :: INDEX_1:11
theorem Th12: :: INDEX_1:12
theorem Th13: :: INDEX_1:13
:: deftheorem Def12 defines rng INDEX_1:def 12 :
theorem Th14: :: INDEX_1:14
:: deftheorem Def13 defines . INDEX_1:def 13 :
:: deftheorem Def14 defines . INDEX_1:def 14 :
E105:
now
let C be
Category,
D be
Category;
set F = the
Objects of
C --> D;
set G = the
Morphisms of
C --> (id D);
set H =
[(the Objects of C --> D),(the Morphisms of C --> (id D))];
let m be
Morphism of
C;
dom (([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Dom of C) = the
Morphisms of
C
by PBOOLE:def 3;
then E23:
(([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Dom of C) . m =
([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) . (the Dom of C . m)
by FUNCT_1:22
.=
(the Objects of C --> D) . (the Dom of C . m)
by MCART_1:7
.=
D
by FUNCOP_1:13
;
dom (([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Cod of C) = the
Morphisms of
C
by PBOOLE:def 3;
then E26:
(([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Cod of C) . m =
([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) . (the Cod of C . m)
by FUNCT_1:22
.=
(the Objects of C --> D) . (the Cod of C . m)
by MCART_1:7
.=
D
by FUNCOP_1:13
;
([(the Objects of C --> D),(the Morphisms of C --> (id D))] `2 ) . m =
(the Morphisms of C --> (id D)) . m
by MCART_1:7
.=
id D
by FUNCOP_1:13
;
hence
([(the Objects of C --> D),(the Morphisms of C --> (id D))] `2 ) . m is
Functor of
(([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Dom of C) . m,
(([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Cod of C) . m
by , ;
end;
E106:
now
let C be
Category,
D be
Category;
set F = the
Objects of
C --> D;
set G = the
Morphisms of
C --> (id D);
set H =
[(the Objects of C --> D),(the Morphisms of C --> (id D))];
let m be
Morphism of
C;
dom (([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Dom of C) = the
Morphisms of
C
by PBOOLE:def 3;
then E23:
(([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Dom of C) . m =
([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) . (the Dom of C . m)
by FUNCT_1:22
.=
(the Objects of C --> D) . (the Dom of C . m)
by MCART_1:7
.=
D
by FUNCOP_1:13
;
dom (([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Cod of C) = the
Morphisms of
C
by PBOOLE:def 3;
then E26:
(([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Cod of C) . m =
([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) . (the Cod of C . m)
by FUNCT_1:22
.=
(the Objects of C --> D) . (the Cod of C . m)
by MCART_1:7
.=
D
by FUNCOP_1:13
;
([(the Objects of C --> D),(the Morphisms of C --> (id D))] `2 ) . m =
(the Morphisms of C --> (id D)) . m
by MCART_1:7
.=
id D
by FUNCOP_1:13
;
hence
([(the Objects of C --> D),(the Morphisms of C --> (id D))] `2 ) . m is
Functor of
(([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Cod of C) . m,
(([(the Objects of C --> D),(the Morphisms of C --> (id D))] `1 ) * the Dom of C) . m
by , E26;
end;
theorem Th15: :: INDEX_1:15
theorem Th16: :: INDEX_1:16
:: deftheorem Def15 defines -indexing_of INDEX_1:def 15 :
theorem Th17: :: INDEX_1:17
theorem Th18: :: INDEX_1:18
theorem Th19: :: INDEX_1:19
theorem Th20: :: INDEX_1:20
theorem Th21: :: INDEX_1:21
definition
let C be
Category,
D be
Category,
E be
Category;
let F be
Functor of
C,
D;
let I be
Indexing of
E;
assume E23:
Image F is
Subcategory of
E
;
func c5 * c4 -> Indexing of
a1 means :
Def16:
:: INDEX_1:def 16
for
F' being
Functor of
C,
E st
F' = F holds
it = ((I -functor E,(rng I)) * F') -indexing_of C;
existence
ex b1 being Indexing of C st
for F' being Functor of C,E st F' = F holds
b1 = ((I -functor E,(rng I)) * F') -indexing_of C
uniqueness
for b1, b2 being Indexing of C st ( for F' being Functor of C,E st F' = F holds
b1 = ((I -functor E,(rng I)) * F') -indexing_of C ) & ( for F' being Functor of C,E st F' = F holds
b2 = ((I -functor E,(rng I)) * F') -indexing_of C ) holds
b1 = b2
end;
:: deftheorem Def16 defines * INDEX_1:def 16 :
theorem Th22: :: INDEX_1:22
theorem Th23: :: INDEX_1:23
theorem Th24: :: INDEX_1:24
theorem Th25: :: INDEX_1:25
theorem Th26: :: INDEX_1:26
definition
let C be
Category;
let I be
Indexing of
C;
let D be
Categorial Category;
assume E23:
D is
TargetCat of
I
;
let E be
Categorial Category;
let F be
Functor of
D,
E;
func c5 * c2 -> Indexing of
a1 means :
Def17:
:: INDEX_1:def 17
for
T being
TargetCat of
I for
G being
Functor of
T,
E st
T = D &
G = F holds
it = (G * (I -functor C,T)) -indexing_of C;
existence
ex b1 being Indexing of C st
for T being TargetCat of I
for G being Functor of T,E st T = D & G = F holds
b1 = (G * (I -functor C,T)) -indexing_of C
uniqueness
for b1, b2 being Indexing of C st ( for T being TargetCat of I
for G being Functor of T,E st T = D & G = F holds
b1 = (G * (I -functor C,T)) -indexing_of C ) & ( for T being TargetCat of I
for G being Functor of T,E st T = D & G = F holds
b2 = (G * (I -functor C,T)) -indexing_of C ) holds
b1 = b2
end;
:: deftheorem Def17 defines * INDEX_1:def 17 :
theorem Th27: :: INDEX_1:27
theorem Th28: :: INDEX_1:28
theorem Th29: :: INDEX_1:29
theorem Th30: :: INDEX_1:30
theorem Th31: :: INDEX_1:31
:: deftheorem Def18 defines * INDEX_1:def 18 :
theorem Th32: :: INDEX_1:32
theorem Th33: :: INDEX_1:33
theorem Th34: :: INDEX_1:34
theorem Th35: :: INDEX_1:35
theorem Th36: :: INDEX_1:36
theorem Th37: :: INDEX_1:37
theorem Th38: :: INDEX_1:38