:: ALTCAT_2 semantic presentation
theorem Th1: :: ALTCAT_2:1
theorem Th2: :: ALTCAT_2:2
theorem Th3: :: ALTCAT_2:3
theorem Th4: :: ALTCAT_2:4
theorem Th5: :: ALTCAT_2:5
theorem Th6: :: ALTCAT_2:6
:: deftheorem Def1 defines cc= ALTCAT_2:def 1 :
:: deftheorem Def2 defines cc= ALTCAT_2:def 2 :
theorem Th7: :: ALTCAT_2:7
canceled;
theorem Th8: :: ALTCAT_2:8
theorem Th9: :: ALTCAT_2:9
theorem Th10: :: ALTCAT_2:10
theorem Th11: :: ALTCAT_2:11
theorem Th12: :: ALTCAT_2:12
definition
let c
1 be
CatStr ;
func the_hom_sets_of c
1 -> ManySortedSet of
[:the Objects of a1,the Objects of a1:] means :
Def3:
:: ALTCAT_2:def 3
for b
1, b
2 being
Object of a
1 holds a
2 . b
1,b
2 = Hom b
1,b
2;
existence
ex b1 being ManySortedSet of [:the Objects of c1,the Objects of c1:] st
for b2, b3 being Object of c1 holds b1 . b2,b3 = Hom b2,b3
uniqueness
for b1, b2 being ManySortedSet of [:the Objects of c1,the Objects of c1:] holds
( ( for b3, b4 being Object of c1 holds b1 . b3,b4 = Hom b3,b4 ) & ( for b3, b4 being Object of c1 holds b2 . b3,b4 = Hom b3,b4 ) implies b1 = b2 )
end;
:: deftheorem Def3 defines the_hom_sets_of ALTCAT_2:def 3 :
theorem Th13: :: ALTCAT_2:13
definition
let c
1 be
Category;
func the_comps_of c
1 -> BinComp of
(the_hom_sets_of a1) means :
Def4:
:: ALTCAT_2:def 4
for b
1, b
2, b
3 being
Object of a
1 holds a
2 . b
1,b
2,b
3 = the
Comp of a
1 | [:((the_hom_sets_of a1) . b2,b3),((the_hom_sets_of a1) . b1,b2):];
existence
ex b1 being BinComp of (the_hom_sets_of c1) st
for b2, b3, b4 being Object of c1 holds b1 . b2,b3,b4 = the Comp of c1 | [:((the_hom_sets_of c1) . b3,b4),((the_hom_sets_of c1) . b2,b3):]
uniqueness
for b1, b2 being BinComp of (the_hom_sets_of c1) holds
( ( for b3, b4, b5 being Object of c1 holds b1 . b3,b4,b5 = the Comp of c1 | [:((the_hom_sets_of c1) . b4,b5),((the_hom_sets_of c1) . b3,b4):] ) & ( for b3, b4, b5 being Object of c1 holds b2 . b3,b4,b5 = the Comp of c1 | [:((the_hom_sets_of c1) . b4,b5),((the_hom_sets_of c1) . b3,b4):] ) implies b1 = b2 )
end;
:: deftheorem Def4 defines the_comps_of ALTCAT_2:def 4 :
theorem Th14: :: ALTCAT_2:14
theorem Th15: :: ALTCAT_2:15
theorem Th16: :: ALTCAT_2:16
:: deftheorem Def5 defines Alter ALTCAT_2:def 5 :
theorem Th17: :: ALTCAT_2:17
theorem Th18: :: ALTCAT_2:18
theorem Th19: :: ALTCAT_2:19
:: deftheorem Def6 defines reflexive ALTCAT_2:def 6 :
:: deftheorem Def7 defines reflexive ALTCAT_2:def 7 :
definition
let c
1 be non
empty transitive AltCatStr ;
redefine attr a
1 is
associative means :
Def8:
:: ALTCAT_2:def 8
for b
1, b
2, b
3, b
4 being
object of a
1for b
5 being
Morphism of b
1,b
2for b
6 being
Morphism of b
2,b
3for b
7 being
Morphism of b
3,b
4 holds
(
<^b1,b2^> <> {} &
<^b2,b3^> <> {} &
<^b3,b4^> <> {} implies
(b7 * b6) * b
5 = b
7 * (b6 * b5) );
compatibility
( c1 is associative iff for b1, b2, b3, b4 being object of c1
for b5 being Morphism of b1,b2
for b6 being Morphism of b2,b3
for b7 being Morphism of b3,b4 holds
( <^b1,b2^> <> {} & <^b2,b3^> <> {} & <^b3,b4^> <> {} implies (b7 * b6) * b5 = b7 * (b6 * b5) ) )
end;
:: deftheorem Def8 defines associative ALTCAT_2:def 8 :
definition
let c
1 be non
empty AltCatStr ;
redefine attr a
1 is
with_units means :: ALTCAT_2:def 9
for b
1 being
object of a
1 holds
(
<^b1,b1^> <> {} & ex b
2 being
Morphism of b
1,b
1 st
for b
3 being
object of a
1for b
4 being
Morphism of b
3,b
1for b
5 being
Morphism of b
1,b
3 holds
( (
<^b3,b1^> <> {} implies b
2 * b
4 = b
4 ) & (
<^b1,b3^> <> {} implies b
5 * b
2 = b
5 ) ) );
compatibility
( c1 is with_units iff for b1 being object of c1 holds
( <^b1,b1^> <> {} & ex b2 being Morphism of b1,b1 st
for b3 being object of c1
for b4 being Morphism of b3,b1
for b5 being Morphism of b1,b3 holds
( ( <^b3,b1^> <> {} implies b2 * b4 = b4 ) & ( <^b1,b3^> <> {} implies b5 * b2 = b5 ) ) ) )
end;
:: deftheorem Def9 defines with_units ALTCAT_2:def 9 :
Lemma18:
for b1, b2 being strict AltCatStr holds
( the carrier of b1 is empty & the carrier of b2 is empty implies b1 = b2 )
:: deftheorem Def10 defines the_empty_category ALTCAT_2:def 10 :
theorem Th20: :: ALTCAT_2:20
:: deftheorem Def11 defines SubCatStr ALTCAT_2:def 11 :
theorem Th21: :: ALTCAT_2:21
theorem Th22: :: ALTCAT_2:22
theorem Th23: :: ALTCAT_2:23
definition
let c
1 be non
empty AltCatStr ;
let c
2 be
object of c
1;
func ObCat c
2 -> strict SubCatStr of a
1 means :
Def12:
:: ALTCAT_2:def 12
( the
carrier of a
3 = {a2} & the
Arrows of a
3 = a
2,a
2 :-> <^a2,a2^> & the
Comp of a
3 = [a2,a2,a2] .--> (the Comp of a1 . a2,a2,a2) );
existence
ex b1 being strict SubCatStr of c1 st
( the carrier of b1 = {c2} & the Arrows of b1 = c2,c2 :-> <^c2,c2^> & the Comp of b1 = [c2,c2,c2] .--> (the Comp of c1 . c2,c2,c2) )
uniqueness
for b1, b2 being strict SubCatStr of c1 holds
( the carrier of b1 = {c2} & the Arrows of b1 = c2,c2 :-> <^c2,c2^> & the Comp of b1 = [c2,c2,c2] .--> (the Comp of c1 . c2,c2,c2) & the carrier of b2 = {c2} & the Arrows of b2 = c2,c2 :-> <^c2,c2^> & the Comp of b2 = [c2,c2,c2] .--> (the Comp of c1 . c2,c2,c2) implies b1 = b2 )
;
end;
:: deftheorem Def12 defines ObCat ALTCAT_2:def 12 :
theorem Th24: :: ALTCAT_2:24
theorem Th25: :: ALTCAT_2:25
:: deftheorem Def13 defines full ALTCAT_2:def 13 :
:: deftheorem Def14 defines id-inheriting ALTCAT_2:def 14 :
theorem Th26: :: ALTCAT_2:26
theorem Th27: :: ALTCAT_2:27
theorem Th28: :: ALTCAT_2:28
theorem Th29: :: ALTCAT_2:29
theorem Th30: :: ALTCAT_2:30
theorem Th31: :: ALTCAT_2:31
theorem Th32: :: ALTCAT_2:32
theorem Th33: :: ALTCAT_2:33
for b
1 being non
empty transitive AltCatStr for b
2 being non
empty transitive SubCatStr of b
1for b
3, b
4, b
5 being
object of b
2 holds
(
<^b3,b4^> <> {} &
<^b4,b5^> <> {} implies for b
6, b
7, b
8 being
object of b
1 holds
( b
6 = b
3 & b
7 = b
4 & b
8 = b
5 implies for b
9 being
Morphism of b
6,b
7for b
10 being
Morphism of b
7,b
8for b
11 being
Morphism of b
3,b
4for b
12 being
Morphism of b
4,b
5 holds
( b
9 = b
11 & b
10 = b
12 implies b
10 * b
9 = b
12 * b
11 ) ) )
theorem Th34: :: ALTCAT_2:34
theorem Th35: :: ALTCAT_2:35