:: FUNCTOR3 semantic presentation
theorem Th1: :: FUNCTOR3:1
for b
1 being
categoryfor b
2, b
3, b
4, b
5 being
object of b
1for b
6 being
Morphism of b
2,b
3for b
7 being
Morphism of b
3,b
4for b
8 being
Morphism of b
2,b
5for b
9 being
Morphism of b
5,b
4 holds
( b
7 * b
6 = b
9 * b
8 & b
6 * (b6 " ) = idm b
3 &
(b9 " ) * b
9 = idm b
5 &
<^b2,b3^> <> {} &
<^b3,b2^> <> {} &
<^b3,b4^> <> {} &
<^b4,b5^> <> {} &
<^b5,b4^> <> {} implies b
8 * (b6 " ) = (b9 " ) * b
7 )
theorem Th2: :: FUNCTOR3:2
for b
1 being non
empty transitive AltCatStr for b
2, b
3 being non
empty with_units AltCatStr for b
4 being
feasible Covariant FunctorStr of b
1,b
2for b
5 being
FunctorStr of b
2,b
3for b
6, b
7 being
object of b
1 holds
Morph-Map (b5 * b4),b
6,b
7 = (Morph-Map b5,(b4 . b6),(b4 . b7)) * (Morph-Map b4,b6,b7)
theorem Th3: :: FUNCTOR3:3
for b
1 being non
empty transitive AltCatStr for b
2, b
3 being non
empty with_units AltCatStr for b
4 being
feasible Contravariant FunctorStr of b
1,b
2for b
5 being
FunctorStr of b
2,b
3for b
6, b
7 being
object of b
1 holds
Morph-Map (b5 * b4),b
6,b
7 = (Morph-Map b5,(b4 . b7),(b4 . b6)) * (Morph-Map b4,b6,b7)
Lemma1:
for b1 being set
for b2 being non empty set
for b3 being Function of b1,b2
for b4 being ManySortedSet of b1
for b5 being ManySortedSet of b2 holds
( ( for b6 being set holds
not ( b6 in b1 & b4 . b6 <> {} & not b5 . (b3 . b6) <> {} ) ) implies for b6 being ManySortedFunction of b4,b5 * b3 holds ((id b5) * b3) ** b6 = b6 )
theorem Th4: :: FUNCTOR3:4
theorem Th5: :: FUNCTOR3:5
theorem Th6: :: FUNCTOR3:6
theorem Th7: :: FUNCTOR3:7
theorem Th8: :: FUNCTOR3:8
theorem Th9: :: FUNCTOR3:9
theorem Th10: :: FUNCTOR3:10
definition
let c
1, c
2, c
3 be non
empty transitive with_units AltCatStr ;
let c
4, c
5 be
covariant Functor of c
1,c
2;
let c
6 be
transformation of c
4,c
5;
let c
7 be
covariant Functor of c
2,c
3;
assume E7:
c
4 is_transformable_to c
5
;
func c
7 * c
6 -> transformation of a
7 * a
4,a
7 * a
5 means :
Def1:
:: FUNCTOR3:def 1
for b
1 being
object of a
1 holds a
8 . b
1 = a
7 . (a6 ! b1);
existence
ex b1 being transformation of c7 * c4,c7 * c5 st
for b2 being object of c1 holds b1 . b2 = c7 . (c6 ! b2)
uniqueness
for b1, b2 being transformation of c7 * c4,c7 * c5 holds
( ( for b3 being object of c1 holds b1 . b3 = c7 . (c6 ! b3) ) & ( for b3 being object of c1 holds b2 . b3 = c7 . (c6 ! b3) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines * FUNCTOR3:def 1 :
theorem Th11: :: FUNCTOR3:11
definition
let c
1, c
2, c
3 be non
empty transitive with_units AltCatStr ;
let c
4, c
5 be
covariant Functor of c
2,c
3;
let c
6 be
covariant Functor of c
1,c
2;
let c
7 be
transformation of c
4,c
5;
assume E9:
c
4 is_transformable_to c
5
;
func c
7 * c
6 -> transformation of a
4 * a
6,a
5 * a
6 means :
Def2:
:: FUNCTOR3:def 2
for b
1 being
object of a
1 holds a
8 . b
1 = a
7 ! (a6 . b1);
existence
ex b1 being transformation of c4 * c6,c5 * c6 st
for b2 being object of c1 holds b1 . b2 = c7 ! (c6 . b2)
uniqueness
for b1, b2 being transformation of c4 * c6,c5 * c6 holds
( ( for b3 being object of c1 holds b1 . b3 = c7 ! (c6 . b3) ) & ( for b3 being object of c1 holds b2 . b3 = c7 ! (c6 . b3) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines * FUNCTOR3:def 2 :
theorem Th12: :: FUNCTOR3:12
theorem Th13: :: FUNCTOR3:13
theorem Th14: :: FUNCTOR3:14
theorem Th15: :: FUNCTOR3:15
theorem Th16: :: FUNCTOR3:16
theorem Th17: :: FUNCTOR3:17
theorem Th18: :: FUNCTOR3:18
theorem Th19: :: FUNCTOR3:19
theorem Th20: :: FUNCTOR3:20
theorem Th21: :: FUNCTOR3:21
definition
let c
1, c
2, c
3 be non
empty transitive with_units AltCatStr ;
let c
4, c
5 be
covariant Functor of c
1,c
2;
let c
6, c
7 be
covariant Functor of c
2,c
3;
let c
8 be
transformation of c
4,c
5;
let c
9 be
transformation of c
6,c
7;
func c
9 (#) c
8 -> transformation of a
6 * a
4,a
7 * a
5 equals :: FUNCTOR3:def 3
(a9 * a5) `*` (a6 * a8);
coherence
(c9 * c5) `*` (c6 * c8) is transformation of c6 * c4,c7 * c5
;
end;
:: deftheorem Def3 defines (#) FUNCTOR3:def 3 :
theorem Th22: :: FUNCTOR3:22
theorem Th23: :: FUNCTOR3:23
theorem Th24: :: FUNCTOR3:24
theorem Th25: :: FUNCTOR3:25
theorem Th26: :: FUNCTOR3:26
theorem Th27: :: FUNCTOR3:27
for b
1, b
2, b
3, b
4 being
categoryfor b
5, b
6 being
covariant Functor of b
1,b
2for b
7, b
8 being
covariant Functor of b
2,b
3for b
9, b
10 being
covariant Functor of b
3,b
4for b
11 being
transformation of b
5,b
6for b
12 being
transformation of b
7,b
8for b
13 being
transformation of b
9,b
10 holds
( b
5 is_transformable_to b
6 & b
7 is_transformable_to b
8 & b
9 is_transformable_to b
10 implies
(b13 (#) b12) (#) b
11 = b
13 (#) (b12 (#) b11) )
E21:
now
let c
1, c
2, c
3 be
category;
let c
4, c
5 be
covariant Functor of c
1,c
2;
let c
6, c
7 be
covariant Functor of c
2,c
3;
let c
8 be
natural_transformation of c
6,c
7;
let c
9 be
natural_transformation of c
4,c
5;
assume E22:
c
4 is_naturally_transformable_to c
5
;
then E23:
c
4 is_transformable_to c
5
by FUNCTOR2:def 6;
assume E24:
c
6 is_naturally_transformable_to c
7
;
then E25:
c
6 is_transformable_to c
7
by FUNCTOR2:def 6;
set c
10 = c
8 (#) c
9;
E26:
now
let c
11, c
12 be
object of c
1;
assume E27:
<^c11,c12^> <> {}
;
let c
13 be
Morphism of c
11,c
12;
E28:
(
(c6 * c4) . c
11 = c
6 . (c4 . c11) &
(c7 * c5) . c
11 = c
7 . (c5 . c11) )
by FUNCTOR0:34;
E29:
(
(c6 * c4) . c
12 = c
6 . (c4 . c12) &
(c7 * c5) . c
12 = c
7 . (c5 . c12) )
by FUNCTOR0:34;
E30:
<^(c4 . c12),(c5 . c12)^> <> {}
by E23, FUNCTOR2:def 1;
E31:
<^(c4 . c11),(c4 . c12)^> <> {}
by E27, FUNCTOR0:def 19;
then E32:
<^(c4 . c11),(c5 . c12)^> <> {}
by E30, ALTCAT_1:def 4;
E33:
<^(c5 . c11),(c5 . c12)^> <> {}
by E27, FUNCTOR0:def 19;
E34:
<^(c4 . c11),(c5 . c11)^> <> {}
by E23, FUNCTOR2:def 1;
E35:
<^(c6 . (c4 . c11)),(c7 . (c4 . c11))^> <> {}
by E25, FUNCTOR2:def 1;
E36:
<^(c7 . (c4 . c11)),(c7 . (c5 . c11))^> <> {}
by E34, FUNCTOR0:def 19;
E37:
<^(c7 . (c5 . c11)),(c7 . (c5 . c12))^> <> {}
by E33, FUNCTOR0:def 19;
E38:
<^((c6 * c4) . c11),((c6 * c4) . c12)^> <> {}
by E27, FUNCTOR0:def 19;
<^(c6 . (c4 . c12)),(c6 . (c5 . c12))^> <> {}
by E30, FUNCTOR0:def 19;
then E39:
<^((c6 * c4) . c12),((c6 * c5) . c12)^> <> {}
by E29, FUNCTOR0:34;
<^(c6 . (c5 . c12)),(c7 . (c5 . c12))^> <> {}
by E25, FUNCTOR2:def 1;
then E40:
<^((c6 * c5) . c12),((c7 * c5) . c12)^> <> {}
by E29, FUNCTOR0:34;
E41:
c
6 * c
5 is_transformable_to c
7 * c
5
by E25, Th10;
E42:
c
6 * c
4 is_transformable_to c
6 * c
5
by E23, Th10;
E43:
c
8 ! (c5 . c12) = (c8 * c5) . c
12
by E25, Def2;
E44:
c
8 ! (c5 . c11) = (c8 * c5) . c
11
by E25, Def2;
reconsider c
14 = c
6 . (c9 ! c12) as
Morphism of
((c6 * c4) . c12),
((c6 * c5) . c12) by E29, FUNCTOR0:34;
reconsider c
15 = c
8 ! (c5 . c12) as
Morphism of
((c6 * c5) . c12),
((c7 * c5) . c12) by E29, FUNCTOR0:34;
reconsider c
16 = c
6 . (c4 . c13) as
Morphism of
((c6 * c4) . c11),
((c6 * c4) . c12) by E29, FUNCTOR0:34;
reconsider c
17 = c
6 . ((c9 ! c12) * (c4 . c13)) as
Morphism of
((c6 * c4) . c11),
((c6 * c5) . c12) by E28, FUNCTOR0:34;
reconsider c
18 = c
6 . (c9 ! c11) as
Morphism of
((c6 * c4) . c11),
((c6 * c5) . c11) by E28, FUNCTOR0:34;
reconsider c
19 = c
8 ! (c5 . c11) as
Morphism of
((c6 * c5) . c11),
((c7 * c5) . c11) by E28, FUNCTOR0:34;
reconsider c
20 = c
7 . (c5 . c13) as
Morphism of
((c7 * c5) . c11),
((c7 * c5) . c12) by E28, FUNCTOR0:34;
E45: c
6 . ((c9 ! c12) * (c4 . c13)) =
(c6 . (c9 ! c12)) * (c6 . (c4 . c13))
by E30, E31, FUNCTOR0:def 24
.=
c
14 * c
16
by E28, E29, FUNCTOR0:34
;
thus ((c8 (#) c9) ! c12) * ((c6 * c4) . c13) =
(((c8 * c5) ! c12) * ((c6 * c9) ! c12)) * ((c6 * c4) . c13)
by E41, E42, FUNCTOR2:def 5
.=
(c15 * ((c6 * c9) ! c12)) * ((c6 * c4) . c13)
by E41, E43, FUNCTOR2:def 4
.=
(c15 * c14) * ((c6 * c4) . c13)
by E23, Th11
.=
(c15 * c14) * c
16
by E27, Th6
.=
c
15 * c
17
by E38, E39, E40, E45, ALTCAT_1:25
.=
(c8 ! (c5 . c12)) * (c6 . ((c9 ! c12) * (c4 . c13)))
by E28, E29, FUNCTOR0:34
.=
(c7 . ((c9 ! c12) * (c4 . c13))) * (c8 ! (c4 . c11))
by E24, E32, FUNCTOR2:def 7
.=
(c7 . ((c5 . c13) * (c9 ! c11))) * (c8 ! (c4 . c11))
by E22, E27, FUNCTOR2:def 7
.=
((c7 . (c5 . c13)) * (c7 . (c9 ! c11))) * (c8 ! (c4 . c11))
by E33, E34, FUNCTOR0:def 24
.=
(c7 . (c5 . c13)) * ((c7 . (c9 ! c11)) * (c8 ! (c4 . c11)))
by E35, E36, E37, ALTCAT_1:25
.=
(c7 . (c5 . c13)) * ((c8 ! (c5 . c11)) * (c6 . (c9 ! c11)))
by E24, E34, FUNCTOR2:def 7
.=
c
20 * (c19 * c18)
by E28, E29, FUNCTOR0:34
.=
((c7 * c5) . c13) * (c19 * c18)
by E27, Th6
.=
((c7 * c5) . c13) * (((c8 * c5) ! c11) * c18)
by E41, E44, FUNCTOR2:def 4
.=
((c7 * c5) . c13) * (((c8 * c5) ! c11) * ((c6 * c9) ! c11))
by E23, Th11
.=
((c7 * c5) . c13) * ((c8 (#) c9) ! c11)
by E41, E42, FUNCTOR2:def 5
;
end;
thus E27:
c
6 * c
4 is_naturally_transformable_to c
7 * c
5
thus
c
8 (#) c
9 is
natural_transformation of c
6 * c
4,c
7 * c
5
end;
theorem Th28: :: FUNCTOR3:28
theorem Th29: :: FUNCTOR3:29
theorem Th30: :: FUNCTOR3:30
theorem Th31: :: FUNCTOR3:31
for b
1, b
2, b
3 being
categoryfor b
4, b
5, b
6 being
covariant Functor of b
1,b
2for b
7, b
8, b
9 being
covariant Functor of b
2,b
3for b
10 being
natural_transformation of b
7,b
8for b
11 being
natural_transformation of b
8,b
9for b
12 being
transformation of b
4,b
5for b
13 being
transformation of b
5,b
6 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
(b11 `*` b10) (#) (b13 `*` b12) = (b11 (#) b13) `*` (b10 (#) b12) )
theorem Th32: :: FUNCTOR3:32
definition
let c
1, c
2 be
category;
let c
3, c
4 be
covariant Functor of c
1,c
2;
pred c
3,c
4 are_naturally_equivalent means :
Def4:
:: FUNCTOR3:def 4
( a
3 is_naturally_transformable_to a
4 & a
4 is_transformable_to a
3 & ex b
1 being
natural_transformation of a
3,a
4 st
for b
2 being
object of a
1 holds b
1 ! b
2 is
iso );
reflexivity
for b1 being covariant Functor of c1,c2 holds
( b1 is_naturally_transformable_to b1 & b1 is_transformable_to b1 & ex b2 being natural_transformation of b1,b1 st
for b3 being object of c1 holds b2 ! b3 is iso )
symmetry
for b1, b2 being covariant Functor of c1,c2 holds
( b1 is_naturally_transformable_to b2 & b2 is_transformable_to b1 & ex b3 being natural_transformation of b1,b2 st
for b4 being object of c1 holds b3 ! b4 is iso implies ( b2 is_naturally_transformable_to b1 & b1 is_transformable_to b2 & ex b3 being natural_transformation of b2,b1 st
for b4 being object of c1 holds b3 ! b4 is iso ) )
end;
:: deftheorem Def4 defines are_naturally_equivalent FUNCTOR3:def 4 :
:: deftheorem Def5 defines natural_equivalence FUNCTOR3:def 5 :
theorem Th33: :: FUNCTOR3:33
theorem Th34: :: FUNCTOR3:34
theorem Th35: :: FUNCTOR3:35
theorem Th36: :: FUNCTOR3:36
theorem Th37: :: FUNCTOR3:37
for b
1, b
2, b
3 being
categoryfor b
4, b
5 being
covariant Functor of b
1,b
2for b
6, b
7 being
covariant Functor of b
2,b
3for b
8 being
natural_equivalence of b
4,b
5for b
9 being
natural_equivalence of b
6,b
7 holds
( b
4,b
5 are_naturally_equivalent & b
6,b
7 are_naturally_equivalent implies ( b
6 * b
4,b
7 * b
5 are_naturally_equivalent & b
9 (#) b
8 is
natural_equivalence of b
6 * b
4,b
7 * b
5 ) )
:: deftheorem Def6 defines " FUNCTOR3:def 6 :
theorem Th38: :: FUNCTOR3:38
theorem Th39: :: FUNCTOR3:39
theorem Th40: :: FUNCTOR3:40
theorem Th41: :: FUNCTOR3:41
theorem Th42: :: FUNCTOR3:42
theorem Th43: :: FUNCTOR3:43