:: ALTCAT_3 semantic presentation
:: deftheorem Def1 defines is_left_inverse_of ALTCAT_3:def 1 :
:: deftheorem Def2 defines retraction ALTCAT_3:def 2 :
:: deftheorem Def3 defines coretraction ALTCAT_3:def 3 :
theorem Th1: :: ALTCAT_3:1
:: deftheorem Def4 defines " ALTCAT_3:def 4 :
theorem Th2: :: ALTCAT_3:2
theorem Th3: :: ALTCAT_3:3
theorem Th4: :: ALTCAT_3:4
:: deftheorem Def5 defines iso ALTCAT_3:def 5 :
theorem Th5: :: ALTCAT_3:5
theorem Th6: :: ALTCAT_3:6
theorem Th7: :: ALTCAT_3:7
definition
let C be
category;
let o1 be
object of
C,
o2 be
object of
C;
pred c2,
c3 are_iso means :
Def6:
:: ALTCAT_3:def 6
(
<^o1,o2^> <> {} &
<^o2,o1^> <> {} & ex
A being
Morphism of
o1,
o2 st
A is
iso );
reflexivity
for o1 being object of C holds
( <^o1,o1^> <> {} & <^o1,o1^> <> {} & ex A being Morphism of o1,o1 st A is iso )
symmetry
for o1, o2 being object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} & ex A being Morphism of o1,o2 st A is iso holds
( <^o2,o1^> <> {} & <^o1,o2^> <> {} & ex A being Morphism of o2,o1 st A is iso )
end;
:: deftheorem Def6 defines are_iso ALTCAT_3:def 6 :
theorem Th8: :: ALTCAT_3:8
:: deftheorem Def7 defines mono ALTCAT_3:def 7 :
:: deftheorem Def8 defines epi ALTCAT_3:def 8 :
theorem Th9: :: ALTCAT_3:9
theorem Th10: :: ALTCAT_3:10
theorem Th11: :: ALTCAT_3:11
theorem Th12: :: ALTCAT_3:12
theorem Th13: :: ALTCAT_3:13
theorem Th14: :: ALTCAT_3:14
theorem Th15: :: ALTCAT_3:15
theorem Th16: :: ALTCAT_3:16
theorem Th17: :: ALTCAT_3:17
theorem Th18: :: ALTCAT_3:18
theorem Th19: :: ALTCAT_3:19
theorem Th20: :: ALTCAT_3:20
theorem Th21: :: ALTCAT_3:21
theorem Th22: :: ALTCAT_3:22
theorem Th23: :: ALTCAT_3:23
theorem Th24: :: ALTCAT_3:24
:: deftheorem Def9 defines initial ALTCAT_3:def 9 :
theorem Th25: :: ALTCAT_3:25
theorem Th26: :: ALTCAT_3:26
:: deftheorem Def10 defines terminal ALTCAT_3:def 10 :
theorem Th27: :: ALTCAT_3:27
theorem Th28: :: ALTCAT_3:28
:: deftheorem Def11 defines _zero ALTCAT_3:def 11 :
theorem Th29: :: ALTCAT_3:29
:: deftheorem Def12 defines _zero ALTCAT_3:def 12 :
theorem Th30: :: ALTCAT_3:30