:: 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
E12:
now
let c
1, c
2 be non
empty set ;
let c
3, c
4 be
Function of c
2,c
1;
let c
5 be
Indexing of c
3,c
4;
E13:
dom (c5 `1 ) = c
1
by PBOOLE:def 3;
consider c
6 being
strict Categorial full Category such that E14:
the
Objects of c
6 = rng (c5 `1 )
by CAT_5:20;
take c
7 = c
6;
thus
for b
1 being
Element of c
1 holds
(c5 `1 ) . b
1 is
Object of c
7
by E13, E14, FUNCT_1:def 5;
let c
8 be
Element of c
2;
(
(c5 `1 ) . (c3 . c8) is
Object of c
7 &
(c5 `1 ) . (c4 . c8) is
Object of c
7 &
(c5 `2 ) . c
8 is
Functor of
(c5 `1 ) . (c3 . c8),
(c5 `1 ) . (c4 . c8) )
by E13, E14, Th4, FUNCT_1:def 5;
hence
[[((c5 `1 ) . (c3 . c8)),((c5 `1 ) . (c4 . c8))],((c5 `2 ) . c8)] is
Morphism of c
7
by CAT_5:def 8;
end;
:: deftheorem Def9 defines TargetCat INDEX_1:def 9 :
Lemma14:
for b1 being Category holds id b1 = (id b1) * (id b1)
by FUNCT_2:23;
definition
let c
1, c
2 be non
empty set ;
let c
3, c
4 be
Function of c
2,c
1;
let c
5 be
PartFunc of
[:c2,c2:],c
2;
let c
6 be
Function of c
1,c
2;
given c
7 being
Category such that E15:
c
7 = CatStr(# c
1,c
2,c
3,c
4,c
5,c
6 #)
;
mode Indexing of c
3,c
4,c
5,c
6 -> Indexing of a
3,a
4 means :
Def10:
:: INDEX_1:def 10
( ( for b
1 being
Element of a
1 holds
(a7 `2 ) . (a6 . b1) = id ((a7 `1 ) . b1) ) & ( for b
1, b
2 being
Element of a
2 holds
( a
3 . b
2 = a
4 . b
1 implies
(a7 `2 ) . (a5 . [b2,b1]) = ((a7 `2 ) . b2) * ((a7 `2 ) . b1) ) ) );
existence
ex b1 being Indexing of c3,c4 st
( ( for b2 being Element of c1 holds (b1 `2 ) . (c6 . b2) = id ((b1 `1 ) . b2) ) & ( for b2, b3 being Element of c2 holds
( c3 . b3 = c4 . b2 implies (b1 `2 ) . (c5 . [b3,b2]) = ((b1 `2 ) . b3) * ((b1 `2 ) . b2) ) ) )
end;
:: deftheorem Def10 defines Indexing INDEX_1:def 10 :
for b
1, b
2 being non
empty set for b
3, b
4 being
Function of b
2,b
1for b
5 being
PartFunc of
[:b2,b2:],b
2for b
6 being
Function of b
1,b
2 holds
( ex b
7 being
Category st b
7 = CatStr(# b
1,b
2,b
3,b
4,b
5,b
6 #) implies for b
7 being
Indexing of b
3,b
4 holds
( b
7 is
Indexing of b
3,b
4,b
5,b
6 iff ( ( for b
8 being
Element of b
1 holds
(b7 `2 ) . (b6 . b8) = id ((b7 `1 ) . b8) ) & ( for b
8, b
9 being
Element of b
2 holds
( b
3 . b
9 = b
4 . b
8 implies
(b7 `2 ) . (b5 . [b9,b8]) = ((b7 `2 ) . b9) * ((b7 `2 ) . b8) ) ) ) ) );
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
1 be
Category;
let c
2 be
Indexing of c
1;
let c
3 be
TargetCat of c
2;
func c
2 -functor c
1,c
3 -> Functor of a
1,a
3 means :
Def11:
:: INDEX_1:def 11
for b
1 being
Morphism of a
1 holds a
4 . b
1 = [[((a2 `1 ) . (dom b1)),((a2 `1 ) . (cod b1))],((a2 `2 ) . b1)];
existence
ex b1 being Functor of c1,c3 st
for b2 being Morphism of c1 holds b1 . b2 = [[((c2 `1 ) . (dom b2)),((c2 `1 ) . (cod b2))],((c2 `2 ) . b2)]
uniqueness
for b1, b2 being Functor of c1,c3 holds
( ( for b3 being Morphism of c1 holds b1 . b3 = [[((c2 `1 ) . (dom b3)),((c2 `1 ) . (cod b3))],((c2 `2 ) . b3)] ) & ( for b3 being Morphism of c1 holds b2 . b3 = [[((c2 `1 ) . (dom b3)),((c2 `1 ) . (cod b3))],((c2 `2 ) . b3)] ) implies b1 = b2 )
end;
:: deftheorem Def11 defines -functor INDEX_1:def 11 :
Lemma19:
for b1 being Category
for b2 being Indexing of b1
for b3 being TargetCat of b2 holds Obj (b2 -functor b1,b3) = b2 `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 :
E23:
now
let c
1, c
2 be
Category;
set c
3 = the
Objects of c
1 --> c
2;
set c
4 = the
Morphisms of c
1 --> (id c2);
set c
5 =
[(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))];
let c
6 be
Morphism of c
1;
dom (([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Dom of c1) = the
Morphisms of c
1
by PBOOLE:def 3;
then E24:
(([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Dom of c1) . c
6 =
([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) . (the Dom of c1 . c6)
by FUNCT_1:22
.=
(the Objects of c1 --> c2) . (the Dom of c1 . c6)
by MCART_1:7
.=
c
2
by FUNCOP_1:13
;
dom (([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Cod of c1) = the
Morphisms of c
1
by PBOOLE:def 3;
then E25:
(([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Cod of c1) . c
6 =
([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) . (the Cod of c1 . c6)
by FUNCT_1:22
.=
(the Objects of c1 --> c2) . (the Cod of c1 . c6)
by MCART_1:7
.=
c
2
by FUNCOP_1:13
;
([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `2 ) . c
6 =
(the Morphisms of c1 --> (id c2)) . c
6
by MCART_1:7
.=
id c
2
by FUNCOP_1:13
;
hence
([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `2 ) . c
6 is
Functor of
(([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Dom of c1) . c
6,
(([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Cod of c1) . c
6
by E24, E25;
end;
E24:
now
let c
1, c
2 be
Category;
set c
3 = the
Objects of c
1 --> c
2;
set c
4 = the
Morphisms of c
1 --> (id c2);
set c
5 =
[(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))];
let c
6 be
Morphism of c
1;
dom (([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Dom of c1) = the
Morphisms of c
1
by PBOOLE:def 3;
then E25:
(([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Dom of c1) . c
6 =
([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) . (the Dom of c1 . c6)
by FUNCT_1:22
.=
(the Objects of c1 --> c2) . (the Dom of c1 . c6)
by MCART_1:7
.=
c
2
by FUNCOP_1:13
;
dom (([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Cod of c1) = the
Morphisms of c
1
by PBOOLE:def 3;
then E26:
(([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Cod of c1) . c
6 =
([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) . (the Cod of c1 . c6)
by FUNCT_1:22
.=
(the Objects of c1 --> c2) . (the Cod of c1 . c6)
by MCART_1:7
.=
c
2
by FUNCOP_1:13
;
([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `2 ) . c
6 =
(the Morphisms of c1 --> (id c2)) . c
6
by MCART_1:7
.=
id c
2
by FUNCOP_1:13
;
hence
([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `2 ) . c
6 is
Functor of
(([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Cod of c1) . c
6,
(([(the Objects of c1 --> c2),(the Morphisms of c1 --> (id c2))] `1 ) * the Dom of c1) . c
6
by E25, 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
1, c
2, c
3 be
Category;
let c
4 be
Functor of c
1,c
2;
let c
5 be
Indexing of c
3;
assume E30:
Image c
4 is
Subcategory of c
3
;
func c
5 * c
4 -> Indexing of a
1 means :
Def16:
:: INDEX_1:def 16
for b
1 being
Functor of a
1,a
3 holds
( b
1 = a
4 implies a
6 = ((a5 -functor a3,(rng a5)) * b1) -indexing_of a
1 );
existence
ex b1 being Indexing of c1 st
for b2 being Functor of c1,c3 holds
( b2 = c4 implies b1 = ((c5 -functor c3,(rng c5)) * b2) -indexing_of c1 )
uniqueness
for b1, b2 being Indexing of c1 holds
( ( for b3 being Functor of c1,c3 holds
( b3 = c4 implies b1 = ((c5 -functor c3,(rng c5)) * b3) -indexing_of c1 ) ) & ( for b3 being Functor of c1,c3 holds
( b3 = c4 implies b2 = ((c5 -functor c3,(rng c5)) * b3) -indexing_of c1 ) ) implies 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
1 be
Category;
let c
2 be
Indexing of c
1;
let c
3 be
Categorial Category;
assume E35:
c
3 is
TargetCat of c
2
;
let c
4 be
Categorial Category;
let c
5 be
Functor of c
3,c
4;
func c
5 * c
2 -> Indexing of a
1 means :
Def17:
:: INDEX_1:def 17
for b
1 being
TargetCat of a
2for b
2 being
Functor of b
1,a
4 holds
( b
1 = a
3 & b
2 = a
5 implies a
6 = (b2 * (a2 -functor a1,b1)) -indexing_of a
1 );
existence
ex b1 being Indexing of c1 st
for b2 being TargetCat of c2
for b3 being Functor of b2,c4 holds
( b2 = c3 & b3 = c5 implies b1 = (b3 * (c2 -functor c1,b2)) -indexing_of c1 )
uniqueness
for b1, b2 being Indexing of c1 holds
( ( for b3 being TargetCat of c2
for b4 being Functor of b3,c4 holds
( b3 = c3 & b4 = c5 implies b1 = (b4 * (c2 -functor c1,b3)) -indexing_of c1 ) ) & ( for b3 being TargetCat of c2
for b4 being Functor of b3,c4 holds
( b3 = c3 & b4 = c5 implies b2 = (b4 * (c2 -functor c1,b3)) -indexing_of c1 ) ) implies 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