:: 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) 1997-2021 Association of Mizar Users

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;

pred A is_left_inverse_of B means :: ALTCAT_3:def 1
A * B = idm o2;

end;

:: deftheorem 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;

attr A is retraction means :: ALTCAT_3:def 2
ex B being Morphism of o2,o1 st B is_right_inverse_of A;

end;

:: deftheorem 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;

attr A is coretraction means :: ALTCAT_3:def 3
ex B being Morphism of o2,o1 st B is_left_inverse_of A;

end;

:: deftheorem 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 )

proof end;

definition

let C be category;
let o1, o2 be Object of C;
assume that
A1: <^o1,o2^> <> {} and
A2: <^o2,o1^> <> {} ;
let A be Morphism of o1,o2;
assume A3: ( A is retraction & A is coretraction ) ;

func A " -> Morphism of o2,o1 means :Def4: :: ALTCAT_3:def 4
( it is_left_inverse_of A & it is_right_inverse_of A );

existence
ex b₁ being Morphism of o2,o1 st
( b₁ is_left_inverse_of A & b₁ is_right_inverse_of A )

proof end;

uniqueness
for b₁, b₂ being Morphism of o2,o1 st b₁ is_left_inverse_of A & b₁ is_right_inverse_of A & b₂ is_left_inverse_of A & b₂ is_right_inverse_of A holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines " ALTCAT_3:def 4 :
for C being category
for o1, o2 being Object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is retraction & A is coretraction holds
for b₅ being Morphism of o2,o1 holds
( b₅ = A " iff ( b₅ is_left_inverse_of A & 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^> <> {} & <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is retraction & A is coretraction holds
( (A ") * A = idm o1 & A * (A ") = idm o2 )

proof end;

theorem Th3: :: ALTCAT_3:3

for C being category
for o1, o2 being Object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is retraction & A is coretraction holds
(A ") " = A

proof end;

theorem Th4: :: ALTCAT_3:4

for C being category
for o being Object of C holds (idm o) " = idm o

proof end;

definition

let C be category;
let o1, o2 be Object of C;
let A be Morphism of o1,o2;

attr A is iso means :: ALTCAT_3:def 5
( A * (A ") = idm o2 & (A ") * A = idm o1 );

end;

:: deftheorem 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 & (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 & A is coretraction )

proof end;

theorem Th6: :: ALTCAT_3:6

for C being category
for o1, o2 being Object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 holds
( A is iso iff ( A is retraction & A is coretraction ) )

proof end;

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^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} & A is iso & B is iso holds
( B * A is iso & (B * A) " = (A ") * (B ") )

proof end;

definition

let C be category;
let o1, o2 be Object of C;

pred o1,o2 are_iso means :: 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 )

proof end;

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 )

proof end;

end;

:: deftheorem 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^> <> {} & <^o2,o1^> <> {} & 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 & o2,o3 are_iso holds
o1,o3 are_iso

proof end;

definition

let C be non empty AltCatStr ;
let o1, o2 be Object of C;
let A be Morphism of o1,o2;

attr A is mono means :: 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 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;

attr A is epi means :: 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 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^> <> {} & <^o2,o3^> <> {} holds
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st A is mono & B is mono holds
B * A is mono

proof end;

theorem Th10: :: ALTCAT_3:10

for C being non empty transitive associative AltCatStr
for o1, o2, o3 being Object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st A is epi & B is epi holds
B * A is epi

proof end;

theorem :: ALTCAT_3:11

for C being non empty transitive associative AltCatStr
for o1, o2, o3 being Object of C st <^o1,o2^> <> {} & <^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

proof end;

theorem :: ALTCAT_3:12

for C being non empty transitive associative AltCatStr
for o1, o2, o3 being Object of C st <^o1,o2^> <> {} & <^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

proof end;

Lm1: now :: thesis: for C being non empty with_units AltCatStr
for a being Object of C holds
( idm a is epi & idm a is mono )

let C be non empty with_units AltCatStr ; :: thesis: for a being Object of C holds
( idm a is epi & idm a is mono )
let a be Object of C; :: thesis: ( idm a is epi & idm a is mono )
thus idm a is epi :: thesis: idm a is mono

proof

let o be Object of C; :: according to ALTCAT_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 to ALTCAT_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 )

proof end;

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 )

proof end;

theorem Th15: :: ALTCAT_3:15

for C being category
for o1, o2 being Object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is retraction holds
A is epi

proof end;

theorem Th16: :: ALTCAT_3:16

for C being category
for o1, o2 being Object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is coretraction holds
A is mono

proof end;

theorem :: ALTCAT_3:17

for C being category
for o1, o2 being Object of C st <^o1,o2^> <> {} & <^o2,o1^> <> {} holds
for A being Morphism of o1,o2 st A is iso holds
( A is mono & A is epi )

proof end;

theorem Th18: :: ALTCAT_3:18

for C being category
for o1, o2, o3 being Object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} holds
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st A is retraction & B is retraction holds
B * A is retraction

proof end;

theorem Th19: :: ALTCAT_3:19

for C being category
for o1, o2, o3 being Object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} holds
for A being Morphism of o1,o2
for B being Morphism of o2,o3 st A is coretraction & B is coretraction holds
B * A is coretraction

proof end;

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 & A is mono & <^o1,o2^> <> {} & <^o2,o1^> <> {} holds
A is iso

proof end;

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 & A is epi & <^o1,o2^> <> {} & <^o2,o1^> <> {} holds
A is iso

proof end;

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^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} & B * A is retraction holds
B is retraction

proof end;

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^> <> {} & <^o2,o3^> <> {} & <^o3,o1^> <> {} & B * A is coretraction holds
A is coretraction

proof end;

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^> <> {} & <^b,a^> <> {} holds
A is iso

proof end;

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 & b₁ is epi & b₁ is retraction & b₁ is coretraction )

proof end;

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 & b₁ is epi & b₁ is iso & b₁ is retraction & b₁ is coretraction )

proof end;

end;

registration

let C be category;
let o be Object of C;
let A, B be mono Morphism of o,o;

cluster A * B -> mono ;

coherence
A * B is mono

proof end;

end;

registration

let C be category;
let o be Object of C;
let A, B be epi Morphism of o,o;

cluster A * B -> epi ;

coherence
A * B is epi

proof end;

end;

registration

let C be category;
let o be Object of C;
let A, B be iso Morphism of o,o;

cluster A * B -> iso ;

coherence
A * B is iso

proof end;

end;

registration

let C be category;
let o be Object of C;
let A, B be retraction Morphism of o,o;

cluster A * B -> retraction ;

coherence
A * B is retraction

proof end;

end;

registration

let C be category;
let o be Object of C;
let A, B be coretraction Morphism of o,o;

cluster A * B -> coretraction ;

coherence
A * B is coretraction

proof end;

end;

definition

let C be AltGraph ;
let o be Object of C;

attr o is initial means :: ALTCAT_3:def 9
for o1 being Object of C ex M being Morphism of o,o1 st
( M in <^o,o1^> & <^o,o1^> is trivial );

end;

:: deftheorem 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^> & <^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^> & ( for M1 being Morphism of o,o1 st M1 in <^o,o1^> holds
M = M1 ) ) )

proof end;

theorem Th26: :: ALTCAT_3:26

for C being category
for o1, o2 being Object of C st o1 is initial & o2 is initial holds
o1,o2 are_iso

proof end;

definition

let C be AltGraph ;
let o be Object of C;

attr o is terminal means :: ALTCAT_3:def 10
for o1 being Object of C ex M being Morphism of o1,o st
( M in <^o1,o^> & <^o1,o^> is trivial );

end;

:: deftheorem 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^> & <^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^> & ( for M1 being Morphism of o1,o st M1 in <^o1,o^> holds
M = M1 ) ) )

proof end;

theorem :: ALTCAT_3:28

for C being category
for o1, o2 being Object of C st o1 is terminal & o2 is terminal holds
o1,o2 are_iso

proof end;

definition

let C be AltGraph ;
let o be Object of C;

attr o is _zero means :: ALTCAT_3:def 11
( o is initial & o is terminal );

end;

:: deftheorem defines _zero ALTCAT_3:def 11 :
for C being AltGraph
for o being Object of C holds
( o is _zero iff ( o is initial & o is terminal ) );

theorem :: ALTCAT_3:29

for C being category
for o1, o2 being Object of C st o1 is _zero & o2 is _zero holds
o1,o2 are_iso by Th26;

definition

let C be non empty AltCatStr ;
let o1, o2 be Object of C;
let M be Morphism of o1,o2;

attr M is _zero means :: 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 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 & M2 is _zero holds
M2 * M1 is _zero

proof end;