:: ISOCAT_1 semantic presentation
theorem Th1: :: ISOCAT_1:1
theorem Th2: :: ISOCAT_1:2
theorem Th3: :: ISOCAT_1:3
theorem Th4: :: ISOCAT_1:4
theorem Th5: :: ISOCAT_1:5
canceled;
theorem Th6: :: ISOCAT_1:6
canceled;
theorem Th7: :: ISOCAT_1:7
theorem Th8: :: ISOCAT_1:8
theorem Th9: :: ISOCAT_1:9
definition
let c
1, c
2 be
Category;
redefine mode Functor of c
1,c
2 -> Relation of the
Morphisms of a
1,the
Morphisms of a
2 means :: ISOCAT_1:def 1
( ( for b
1 being
Object of a
1 holds
ex b
2 being
Object of a
2 st a
3 . (id b1) = id b
2 ) & ( for b
1 being
Morphism of a
1 holds
( a
3 . (id (dom b1)) = id (dom (a3 . b1)) & a
3 . (id (cod b1)) = id (cod (a3 . b1)) ) ) & ( for b
1, b
2 being
Morphism of a
1 holds
(
dom b
2 = cod b
1 implies a
3 . (b2 * b1) = (a3 . b2) * (a3 . b1) ) ) );
compatibility
for b1 being Relation of the Morphisms of c1,the Morphisms of c2 holds
( b1 is Functor of c1,c2 iff ( ( for b2 being Object of c1 holds
ex b3 being Object of c2 st b1 . (id b2) = id b3 ) & ( for b2 being Morphism of c1 holds
( b1 . (id (dom b2)) = id (dom (b1 . b2)) & b1 . (id (cod b2)) = id (cod (b1 . b2)) ) ) & ( for b2, b3 being Morphism of c1 holds
( dom b3 = cod b2 implies b1 . (b3 * b2) = (b1 . b3) * (b1 . b2) ) ) ) )
by CAT_1:96, CAT_1:97, CAT_1:98, CAT_1:99;
end;
:: deftheorem Def1 defines Functor ISOCAT_1:def 1 :
theorem Th10: :: ISOCAT_1:10
theorem Th11: :: ISOCAT_1:11
theorem Th12: :: ISOCAT_1:12
:: deftheorem Def2 defines " ISOCAT_1:def 2 :
:: deftheorem Def3 defines is_an_isomorphism ISOCAT_1:def 3 :
theorem Th13: :: ISOCAT_1:13
theorem Th14: :: ISOCAT_1:14
theorem Th15: :: ISOCAT_1:15
theorem Th16: :: ISOCAT_1:16
theorem Th17: :: ISOCAT_1:17
:: deftheorem Def4 defines are_isomorphic ISOCAT_1:def 4 :
theorem Th18: :: ISOCAT_1:18
canceled;
theorem Th19: :: ISOCAT_1:19
canceled;
theorem Th20: :: ISOCAT_1:20
for b
1, b
2, b
3 being
Category holds
( b
1 ~= b
2 & b
2 ~= b
3 implies b
1 ~= b
3 )
theorem Th21: :: ISOCAT_1:21
theorem Th22: :: ISOCAT_1:22
theorem Th23: :: ISOCAT_1:23
theorem Th24: :: ISOCAT_1:24
:: deftheorem Def5 defines * ISOCAT_1:def 5 :
:: deftheorem Def6 defines * ISOCAT_1:def 6 :
theorem Th25: :: ISOCAT_1:25
theorem Th26: :: ISOCAT_1:26
theorem Th27: :: ISOCAT_1:27
:: deftheorem Def7 defines * ISOCAT_1:def 7 :
theorem Th28: :: ISOCAT_1:28
:: deftheorem Def8 defines * ISOCAT_1:def 8 :
theorem Th29: :: ISOCAT_1:29
theorem Th30: :: ISOCAT_1:30
theorem Th31: :: ISOCAT_1:31
theorem Th32: :: ISOCAT_1:32
theorem Th33: :: ISOCAT_1:33
theorem Th34: :: ISOCAT_1:34
theorem Th35: :: ISOCAT_1:35
theorem Th36: :: ISOCAT_1:36
theorem Th37: :: ISOCAT_1:37
theorem Th38: :: ISOCAT_1:38
theorem Th39: :: ISOCAT_1:39
theorem Th40: :: ISOCAT_1:40
definition
let c
1, c
2, c
3 be
Category;
let c
4, c
5 be
Functor of c
1,c
2;
let c
6, c
7 be
Functor of c
2,c
3;
let c
8 be
natural_transformation of c
4,c
5;
let c
9 be
natural_transformation of c
6,c
7;
func c
9 (#) c
8 -> natural_transformation of a
6 * a
4,a
7 * a
5 equals :: ISOCAT_1:def 9
(a9 * a5) `*` (a6 * a8);
correctness
coherence
(c9 * c5) `*` (c6 * c8) is natural_transformation of c6 * c4,c7 * c5;
;
end;
:: deftheorem Def9 defines (#) ISOCAT_1:def 9 :
theorem Th41: :: ISOCAT_1:41
theorem Th42: :: ISOCAT_1:42
theorem Th43: :: ISOCAT_1:43
theorem Th44: :: ISOCAT_1:44
for b
1, b
2, b
3, b
4 being
Categoryfor b
5, b
6 being
Functor of b
1,b
2for b
7, b
8 being
Functor of b
2,b
3for b
9, b
10 being
Functor of b
3,b
4for b
11 being
natural_transformation of b
5,b
6for b
12 being
natural_transformation of b
7,b
8for b
13 being
natural_transformation of b
9,b
10 holds
( b
5 is_naturally_transformable_to b
6 & b
7 is_naturally_transformable_to b
8 & b
9 is_naturally_transformable_to b
10 implies b
13 (#) (b12 (#) b11) = (b13 (#) b12) (#) b
11 )
theorem Th45: :: ISOCAT_1:45
theorem Th46: :: ISOCAT_1:46
theorem Th47: :: ISOCAT_1:47
for b
1, b
2, b
3 being
Categoryfor b
4, b
5, b
6 being
Functor of b
1,b
2for b
7, b
8, b
9 being
Functor of b
2,b
3for b
10 being
natural_transformation of b
4,b
5for b
11 being
natural_transformation of b
5,b
6for b
12 being
natural_transformation of b
7,b
8for b
13 being
natural_transformation of b
8,b
9 holds
( b
4 is_naturally_transformable_to b
5 & b
5 is_naturally_transformable_to b
6 & b
7 is_naturally_transformable_to b
8 & b
8 is_naturally_transformable_to b
9 implies
(b13 `*` b12) (#) (b11 `*` b10) = (b13 (#) b11) `*` (b12 (#) b10) )
theorem Th48: :: ISOCAT_1:48
for b
1, b
2, b
3, b
4 being
Categoryfor b
5 being
Functor of b
1,b
2for b
6 being
Functor of b
3,b
4for b
7, b
8 being
Functor of b
2,b
3 holds
( b
7 ~= b
8 implies ( b
6 * b
7 ~= b
6 * b
8 & b
7 * b
5 ~= b
8 * b
5 ) )
theorem Th49: :: ISOCAT_1:49
definition
let c
1, c
2 be
Category;
pred c
1 is_equivalent_with c
2 means :
Def10:
:: ISOCAT_1:def 10
ex b
1 being
Functor of a
1,a
2ex b
2 being
Functor of a
2,a
1 st
( b
2 * b
1 ~= id a
1 & b
1 * b
2 ~= id a
2 );
reflexivity
for b1 being Category holds
ex b2, b3 being Functor of b1,b1 st
( b3 * b2 ~= id b1 & b2 * b3 ~= id b1 )
symmetry
for b1, b2 being Category holds
not ( ex b3 being Functor of b1,b2ex b4 being Functor of b2,b1 st
( b4 * b3 ~= id b1 & b3 * b4 ~= id b2 ) & ( for b3 being Functor of b2,b1
for b4 being Functor of b1,b2 holds
not ( b4 * b3 ~= id b2 & b3 * b4 ~= id b1 ) ) )
;
end;
:: deftheorem Def10 defines is_equivalent_with ISOCAT_1:def 10 :
theorem Th50: :: ISOCAT_1:50
theorem Th51: :: ISOCAT_1:51
canceled;
theorem Th52: :: ISOCAT_1:52
canceled;
theorem Th53: :: ISOCAT_1:53
:: deftheorem Def11 defines Equivalence ISOCAT_1:def 11 :
theorem Th54: :: ISOCAT_1:54
theorem Th55: :: ISOCAT_1:55
theorem Th56: :: ISOCAT_1:56
theorem Th57: :: ISOCAT_1:57
theorem Th58: :: ISOCAT_1:58