:: Basic properties of objects and morphisms. In categories without uniqueness of { \bf cod } and { \bf dom }
:: by Beata Madras
::
:: Received February 14, 1997
:: Copyright (c) 1997-2012 Association of Mizar Users


begin

definition
let C be non empty with_units AltCatStr ;
let o1, o2 be object of C;
let A be Morphism of o1,o2;
let B be Morphism of o2,o1;
predA is_left_inverse_of B means :Def1: :: ALTCAT_3:def 1
A * B = idm o2;
end;

:: deftheorem Def1 defines is_left_inverse_of ALTCAT_3:def_1_:_
 for C being non empty with_units AltCatStr
for o1, o2 being object of C
for A being Morphism of o1,o2
for B being Morphism of o2,o1 holds
( A is_left_inverse_of B iff A * B = idm o2 );

notation
let C be non empty with_units AltCatStr ;
let o1, o2 be object of C;
let A be Morphism of o1,o2;
let B be Morphism of o2,o1;
synonym B is_right_inverse_of A for A is_left_inverse_of B;
end;

definition
let C be non empty with_units AltCatStr ;
let o1, o2 be object of C;
let A be Morphism of o1,o2;
attrA is retraction  means :Def2: :: ALTCAT_3:def 2
 ex B being Morphism of o2,o1 st B is_right_inverse_of A;
end;

:: deftheorem Def2 defines retraction ALTCAT_3:def_2_:_
 for C being non empty with_units AltCatStr
for o1, o2 being object of C
for A being Morphism of o1,o2 holds
( A is retraction iff ex B being Morphism of o2,o1 st B is_right_inverse_of A );

definition
let C be non empty with_units AltCatStr ;
let o1, o2 be object of C;
let A be Morphism of o1,o2;
attrA is coretraction  means :Def3: :: ALTCAT_3:def 3
 ex B being Morphism of o2,o1 st B is_left_inverse_of A;
end;

:: deftheorem Def3 defines coretraction ALTCAT_3:def_3_:_
 for C being non empty with_units AltCatStr
for o1, o2 being object of C
for A being Morphism of o1,o2 holds
( A is coretraction iff ex B being Morphism of o2,o1 st B is_left_inverse_of A );

theorem Th1: :: ALTCAT_3:1
for C being non empty with_units AltCatStr
for o being object of C holds
( idm o is retraction & idm o is coretraction )
proofend;

definition
let C be category;
let o1, o2 be object of C;
assume that
B1: <^o1,o2^> <> {} and
B2: <^o2,o1^> <> {} ;
let A be Morphism of o1,o2;
assume B3: ( A is retraction &amp; A is coretraction ) ;
funcA "  ->  Morphism of o2,o1 means :Def4: :: ALTCAT_3:def 4
( it is_left_inverse_of A &amp; it is_right_inverse_of A );
existence
 ex b1 being Morphism of o2,o1 st
( b1 is_left_inverse_of A &amp; b1 is_right_inverse_of A )
proofend;
uniqueness
 for b1, b2 being Morphism of o2,o1 st b1 is_left_inverse_of A &amp; b1 is_right_inverse_of A &amp; b2 is_left_inverse_of A &amp; b2 is_right_inverse_of A holds
b1 = b2
proofend;
end;

:: deftheorem Def4 defines " ALTCAT_3:def_4_:_
 for C being category
for o1, o2 being object of C st <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is retraction &amp; A is coretraction holds
for b5 being Morphism of o2,o1 holds
( b5 = A " iff ( b5 is_left_inverse_of A &amp; b5 is_right_inverse_of A ) );

theorem Th2: :: ALTCAT_3:2
for C being category
for o1, o2 being object of C st <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is retraction &amp; A is coretraction holds
( (A ") * A = idm o1 &amp; A * (A ") = idm o2 )
proofend;

theorem Th3: :: ALTCAT_3:3
for C being category
for o1, o2 being object of C st <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is retraction &amp; A is coretraction holds
(A ") " = A
proofend;

theorem Th4: :: ALTCAT_3:4
for C being category
for o being object of C holds (idm o) " = idm o
proofend;

definition
let C be category;
let o1, o2 be object of C;
let A be Morphism of o1,o2;
attrA is iso  means :Def5: :: ALTCAT_3:def 5
( A * (A ") = idm o2 &amp; (A ") * A = idm o1 );
end;

:: deftheorem Def5 defines iso ALTCAT_3:def_5_:_
 for C being category
for o1, o2 being object of C
for A being Morphism of o1,o2 holds
( A is iso iff ( A * (A ") = idm o2 &amp; (A ") * A = idm o1 ) );

theorem Th5: :: ALTCAT_3:5
for C being category
for o1, o2 being object of C
for A being Morphism of o1,o2 st A is iso holds
( A is retraction &amp; A is coretraction )
proofend;

theorem Th6: :: ALTCAT_3:6
for C being category
for o1, o2 being object of C st <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 holds
( A is iso iff ( A is retraction &amp; A is coretraction ) )
proofend;

theorem Th7: :: ALTCAT_3:7
for C being category
for o1, o2, o3 being object of C
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st <^o1,o2^> <> {} &amp; <^o2,o3^> <> {} &amp; <^o3,o1^> <> {} &amp; A is iso &amp; B is iso holds
( B * A is iso &amp; (B * A) " = (A ") * (B ") )
proofend;

definition
let C be category;
let o1, o2 be object of C;
predo1,o2 are_iso  means :Def6: :: ALTCAT_3:def 6
( <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} &amp; ex A being Morphism of o1,o2 st A is iso );
reflexivity
 for o1 being object of C holds
( <^o1,o1^> <> {} &amp; <^o1,o1^> <> {} &amp; ex A being Morphism of o1,o1 st A is iso )
proofend;
symmetry
 for o1, o2 being object of C st <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} &amp; ex A being Morphism of o1,o2 st A is iso holds
( <^o2,o1^> <> {} &amp; <^o1,o2^> <> {} &amp; ex A being Morphism of o2,o1 st A is iso )
proofend;
end;

:: deftheorem Def6 defines are_iso ALTCAT_3:def_6_:_
 for C being category
for o1, o2 being object of C holds
( o1,o2 are_iso iff ( <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} &amp; ex A being Morphism of o1,o2 st A is iso ) );

theorem :: ALTCAT_3:8
for C being category
for o1, o2, o3 being object of C st o1,o2 are_iso &amp; o2,o3 are_iso holds
o1,o3 are_iso
proofend;

definition
let C be non empty AltCatStr ;
let o1, o2 be object of C;
let A be Morphism of o1,o2;
attrA is mono  means :Def7: :: ALTCAT_3:def 7
 for o being object of C st <^o,o1^> <> {} holds
for B, C being Morphism of o,o1 st A * B = A * C holds
B = C;
end;

:: deftheorem Def7 defines mono ALTCAT_3:def_7_:_
 for C being non empty AltCatStr
for o1, o2 being object of C
for A being Morphism of o1,o2 holds
( A is mono iff for o being object of C st <^o,o1^> <> {} holds
for B, C being Morphism of o,o1 st A * B = A * C holds
B = C );

definition
let C be non empty AltCatStr ;
let o1, o2 be object of C;
let A be Morphism of o1,o2;
attrA is epi  means :Def8: :: ALTCAT_3:def 8
 for o being object of C st <^o2,o^> <> {} holds
for B, C being Morphism of o2,o st B * A = C * A holds
B = C;
end;

:: deftheorem Def8 defines epi ALTCAT_3:def_8_:_
 for C being non empty AltCatStr
for o1, o2 being object of C
for A being Morphism of o1,o2 holds
( A is epi iff for o being object of C st <^o2,o^> <> {} holds
for B, C being Morphism of o2,o st B * A = C * A holds
B = C );

theorem Th9: :: ALTCAT_3:9
for C being non empty transitive associative AltCatStr
for o1, o2, o3 being object of C st <^o1,o2^> <> {} &amp; <^o2,o3^> <> {} holds
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st A is mono &amp; B is mono holds
B * A is mono
proofend;

theorem Th10: :: ALTCAT_3:10
for C being non empty transitive associative AltCatStr
for o1, o2, o3 being object of C st <^o1,o2^> <> {} &amp; <^o2,o3^> <> {} holds
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st A is epi &amp; B is epi holds
B * A is epi
proofend;

theorem :: ALTCAT_3:11
for C being non empty transitive associative AltCatStr
for o1, o2, o3 being object of C st <^o1,o2^> <> {} &amp; <^o2,o3^> <> {} holds
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st B * A is mono holds
A is mono
proofend;

theorem :: ALTCAT_3:12
for C being non empty transitive associative AltCatStr
for o1, o2, o3 being object of C st <^o1,o2^> <> {} &amp; <^o2,o3^> <> {} holds
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st B * A is epi holds
B is epi
proofend;

Lm1: now__::_thesis:_for_C_being_non_empty_with_units_AltCatStr_
for_a_being_object_of_C_holds_
(_idm_a_is_epi_&amp;_idm_a_is_mono_)
let C be non empty with_units AltCatStr ; ::_thesis: for a being object of C holds
( idm a is epi &amp; idm a is mono )
let a be object of C; ::_thesis: ( idm a is epi &amp; idm a is mono )
thus  idm a is epi ::_thesis: idm a is mono
proof
let o be object of C; :: according toALTCAT_3:def_8 ::_thesis: ( <^a,o^> <> {} implies for B, C being Morphism of a,o st B * (idm a) = C * (idm a) holds
B = C )
assume A1: <^a,o^> <> {} ; ::_thesis: for B, C being Morphism of a,o st B * (idm a) = C * (idm a) holds
B = C
let B, C be Morphism of a,o; ::_thesis: ( B * (idm a) = C * (idm a) implies B = C )
assume A2: B * (idm a) = C * (idm a) ; ::_thesis: B = C
thus B = B * (idm a) by A1, ALTCAT_1:def_17
.= C by A1, A2, ALTCAT_1:def_17 ; ::_thesis: verum
end;
thus  idm a is mono ::_thesis: verum
proof
let o be object of C; :: according toALTCAT_3:def_7 ::_thesis: ( <^o,a^> <> {} implies for B, C being Morphism of o,a st (idm a) * B = (idm a) * C holds
B = C )
assume A3: <^o,a^> <> {} ; ::_thesis: for B, C being Morphism of o,a st (idm a) * B = (idm a) * C holds
B = C
let B, C be Morphism of o,a; ::_thesis: ( (idm a) * B = (idm a) * C implies B = C )
assume A4: (idm a) * B = (idm a) * C ; ::_thesis: B = C
thus B = (idm a) * B by A3, ALTCAT_1:20
.= C by A3, A4, ALTCAT_1:20 ; ::_thesis: verum
end;
end;

theorem :: ALTCAT_3:13
for X being non empty set
for o1, o2 being object of (EnsCat X) st <^o1,o2^> <> {} holds
for A being Morphism of o1,o2
for F being Function of o1,o2 st F = A holds
( A is mono iff F is one-to-one )
proofend;

theorem :: ALTCAT_3:14
for X being non empty with_non-empty_elements set
for o1, o2 being object of (EnsCat X) st <^o1,o2^> <> {} holds
for A being Morphism of o1,o2
for F being Function of o1,o2 st F = A holds
( A is epi iff F is onto )
proofend;

theorem Th15: :: ALTCAT_3:15
for C being category
for o1, o2 being object of C st <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is retraction holds
A is epi
proofend;

theorem Th16: :: ALTCAT_3:16
for C being category
for o1, o2 being object of C st <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is coretraction holds
A is mono
proofend;

theorem :: ALTCAT_3:17
for C being category
for o1, o2 being object of C st <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is iso holds
( A is mono &amp; A is epi )
proofend;

theorem Th18: :: ALTCAT_3:18
for C being category
for o1, o2, o3 being object of C st <^o1,o2^> <> {} &amp; <^o2,o3^> <> {} &amp; <^o3,o1^> <> {} holds
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st A is retraction &amp; B is retraction holds
B * A is retraction
proofend;

theorem Th19: :: ALTCAT_3:19
for C being category
for o1, o2, o3 being object of C st <^o1,o2^> <> {} &amp; <^o2,o3^> <> {} &amp; <^o3,o1^> <> {} holds
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st A is coretraction &amp; B is coretraction holds
B * A is coretraction
proofend;

theorem Th20: :: ALTCAT_3:20
for C being category
for o1, o2 being object of C
for A being Morphism of o1,o2 st A is retraction &amp; A is mono &amp; <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} holds
A is iso
proofend;

theorem :: ALTCAT_3:21
for C being category
for o1, o2 being object of C
for A being Morphism of o1,o2 st A is coretraction &amp; A is epi &amp; <^o1,o2^> <> {} &amp; <^o2,o1^> <> {} holds
A is iso
proofend;

theorem :: ALTCAT_3:22
for C being category
for o1, o2, o3 being object of C
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st <^o1,o2^> <> {} &amp; <^o2,o3^> <> {} &amp; <^o3,o1^> <> {} &amp; B * A is retraction holds
B is retraction
proofend;

theorem :: ALTCAT_3:23
for C being category
for o1, o2, o3 being object of C
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st <^o1,o2^> <> {} &amp; <^o2,o3^> <> {} &amp; <^o3,o1^> <> {} &amp; B * A is coretraction holds
A is coretraction
proofend;

theorem :: ALTCAT_3:24
for C being category st ( for o1, o2 being object of C
for A1 being Morphism of o1,o2 holds A1 is retraction ) holds
for a, b being object of C
for A being Morphism of a,b st <^a,b^> <> {} &amp; <^b,a^> <> {} holds
A is iso
proofend;

registration
let C be non empty with_units AltCatStr ;
let o be object of C;
cluster retraction coretraction mono epi for Element of <^o,o^>;
existence
 ex b1 being Morphism of o,o st
( b1 is mono &amp; b1 is epi &amp; b1 is retraction &amp; b1 is coretraction )
proofend;
end;

registration
let C be category;
let o be object of C;
cluster retraction coretraction iso mono epi for Element of <^o,o^>;
existence
 ex b1 being Morphism of o,o st
( b1 is mono &amp; b1 is epi &amp; b1 is iso &amp; b1 is retraction &amp; b1 is coretraction )
proofend;
end;

registration
let C be category;
let o be object of C;
let A, B be mono Morphism of o,o;
clusterA * B ->  mono ;
coherence
 A * B is mono
proofend;
end;

registration
let C be category;
let o be object of C;
let A, B be epi Morphism of o,o;
clusterA * B ->  epi ;
coherence
 A * B is epi
proofend;
end;

registration
let C be category;
let o be object of C;
let A, B be iso Morphism of o,o;
clusterA * B ->  iso ;
coherence
 A * B is iso
proofend;
end;

registration
let C be category;
let o be object of C;
let A, B be retraction Morphism of o,o;
clusterA * B ->  retraction ;
coherence
 A * B is retraction
proofend;
end;

registration
let C be category;
let o be object of C;
let A, B be coretraction Morphism of o,o;
clusterA * B ->  coretraction ;
coherence
 A * B is coretraction
proofend;
end;

definition
let C be AltGraph ;
let o be object of C;
attro is initial  means :Def9: :: ALTCAT_3:def 9
 for o1 being object of C ex M being Morphism of o,o1 st
( M in <^o,o1^> &amp; <^o,o1^> is trivial );
end;

:: deftheorem Def9 defines initial ALTCAT_3:def_9_:_
 for C being AltGraph
for o being object of C holds
( o is initial iff for o1 being object of C ex M being Morphism of o,o1 st
( M in <^o,o1^> &amp; <^o,o1^> is trivial ) );

theorem :: ALTCAT_3:25
for C being AltGraph
for o being object of C holds
( o is initial iff for o1 being object of C ex M being Morphism of o,o1 st
( M in <^o,o1^> &amp; ( for M1 being Morphism of o,o1 st M1 in <^o,o1^> holds
M = M1 ) ) )
proofend;

theorem Th26: :: ALTCAT_3:26
for C being category
for o1, o2 being object of C st o1 is initial &amp; o2 is initial holds
o1,o2 are_iso
proofend;

definition
let C be AltGraph ;
let o be object of C;
attro is terminal  means :Def10: :: ALTCAT_3:def 10
 for o1 being object of C ex M being Morphism of o1,o st
( M in <^o1,o^> &amp; <^o1,o^> is trivial );
end;

:: deftheorem Def10 defines terminal ALTCAT_3:def_10_:_
 for C being AltGraph
for o being object of C holds
( o is terminal iff for o1 being object of C ex M being Morphism of o1,o st
( M in <^o1,o^> &amp; <^o1,o^> is trivial ) );

theorem :: ALTCAT_3:27
for C being AltGraph
for o being object of C holds
( o is terminal iff for o1 being object of C ex M being Morphism of o1,o st
( M in <^o1,o^> &amp; ( for M1 being Morphism of o1,o st M1 in <^o1,o^> holds
M = M1 ) ) )
proofend;

theorem :: ALTCAT_3:28
for C being category
for o1, o2 being object of C st o1 is terminal &amp; o2 is terminal holds
o1,o2 are_iso
proofend;

definition
let C be AltGraph ;
let o be object of C;
attro is _zero  means :Def11: :: ALTCAT_3:def 11
( o is initial &amp; o is terminal );
end;

:: deftheorem Def11 defines _zero ALTCAT_3:def_11_:_
 for C being AltGraph
for o being object of C holds
( o is _zero iff ( o is initial &amp; o is terminal ) );

theorem :: ALTCAT_3:29
for C being category
for o1, o2 being object of C st o1 is _zero &amp; o2 is _zero holds
o1,o2 are_iso
proofend;

definition
let C be non empty AltCatStr ;
let o1, o2 be object of C;
let M be Morphism of o1,o2;
attrM is _zero  means :Def12: :: ALTCAT_3:def 12
 for o being object of C st o is _zero holds
for A being Morphism of o1,o
for B being Morphism of o,o2 holds M = B * A;
end;

:: deftheorem Def12 defines _zero ALTCAT_3:def_12_:_
 for C being non empty AltCatStr
for o1, o2 being object of C
for M being Morphism of o1,o2 holds
( M is _zero iff for o being object of C st o is _zero holds
for A being Morphism of o1,o
for B being Morphism of o,o2 holds M = B * A );

theorem :: ALTCAT_3:30
for C being category
for o1, o2, o3 being object of C
for M1 being Morphism of o1,o2
for M2 being Morphism of o2,o3 st M1 is _zero &amp; M2 is _zero holds
M2 * M1 is _zero
proofend;