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### 8 Example on category of columns

#### 8.1 Constructors of objects

gap> S := HomalgRingOfIntegers();
Z
gap> cols := CategoryOfColumns( S );
Columns( Z )
gap> obj1 := CategoryOfColumnsObject( 2, cols );
<A column module over Z of rank 2>
gap> obj2 := CategoryOfColumnsObject( 8, cols );
<A column module over Z of rank 8>


#### 8.2 Constructors of morphisms

gap> obj3 := CategoryOfColumnsObject( 1, cols );
<A column module over Z of rank 1>
gap> IsWellDefined( obj1 );
true
gap> obj4 := CategoryOfColumnsObject( 2, cols );
<A column module over Z of rank 2>
gap> mor := CategoryOfColumnsMorphism( obj3, HomalgMatrix( [[1],[2]], S ), obj4 );
<A morphism in Columns( Z )>
gap> IsWellDefined( mor );
true

gap> Display( Source( mor ) );
A column module over Z of rank 1
gap> Display( Range( mor ) );
A column module over Z of rank 2
gap> Display( UnderlyingMatrix( mor ) );
[ [  1 ],
[  2 ] ]


#### 8.3 A few categorical constructions for category of columns

gap> ZeroObject( cols );
<A column module over Z of rank 0>
gap> obj5 := CategoryOfColumnsObject( 2, cols );
<A column module over Z of rank 2>

gap> Display( ZeroMorphism( ZeroObject( cols ), obj5 ) );
A zero, split monomorphism in Columns( Z )

Source:
A column module over Z of rank 0

Matrix:
(an empty 2 x 0 matrix)

Range:
A column module over Z of rank 2

gap> obj6 := CategoryOfColumnsObject( 1, cols );
<A column module over Z of rank 1>

gap> Display( IdentityMorphism( obj6 ) );
An identity morphism in Columns( Z )

Source:
A column module over Z of rank 1

Matrix:
[ [  1 ] ]

Range:
A column module over Z of rank 1

gap> directSum := DirectSum( [ obj5, obj6 ] );
<A column module over Z of rank 3>

gap> Display( directSum );
A column module over Z of rank 3

gap> i1 := InjectionOfCofactorOfDirectSum( [ obj5, obj6 ], 1 );
<A morphism in Columns( Z )>

gap> Display( i1 );
A morphism in Columns( Z )

Source:
A column module over Z of rank 2

Matrix:
[ [  1,  0 ],
[  0,  1 ],
[  0,  0 ] ]

Range:
A column module over Z of rank 3

gap> i2 := InjectionOfCofactorOfDirectSum( [ obj5, obj6 ], 2 );
<A morphism in Columns( Z )>

gap> Display( i2 );
A morphism in Columns( Z )

Source:
A column module over Z of rank 1

Matrix:
[ [  0 ],
[  0 ],
[  1 ] ]

Range:
A column module over Z of rank 3

gap> proj1 := ProjectionInFactorOfDirectSum( [ obj5, obj6 ], 1 );
<A morphism in Columns( Z )>

gap> Display( proj1 );
A morphism in Columns( Z )

Source:
A column module over Z of rank 3

Matrix:
[ [  1,  0,  0 ],
[  0,  1,  0 ] ]

Range:
A column module over Z of rank 2

gap> proj2 := ProjectionInFactorOfDirectSum( [ obj5, obj6 ], 2 );
<A morphism in Columns( Z )>

gap> Display( proj2 );
A morphism in Columns( Z )

Source:
A column module over Z of rank 3

Matrix:
[ [  0,  0,  1 ] ]

Range:
A column module over Z of rank 1

gap> k := WeakKernelEmbedding( proj1 );
<A morphism in Columns( Z )>

gap> Display( k );
A morphism in Columns( Z )

Source:
A column module over Z of rank 1

Matrix:
[ [  0 ],
[  0 ],
[  1 ] ]

Range:
A column module over Z of rank 3

gap> ck := WeakCokernelProjection( k );
<A morphism in Columns( Z )>

gap> Display( ck );
A morphism in Columns( Z )

Source:
A column module over Z of rank 3

Matrix:
[ [   0,   1,   0 ],
[  -1,   0,   0 ] ]

Range:
A column module over Z of rank 2

gap> IsMonomorphism( k );
true
gap> IsEpimorphism( k );
false
gap> IsMonomorphism( ck );
false
gap> IsEpimorphism( ck );
true
gap> mor1 := CategoryOfColumnsMorphism( obj5, HomalgMatrix( [[ 1, 2 ]], S ), obj6 );
<A morphism in Columns( Z )>

gap> Display( mor1 );
A morphism in Columns( Z )

Source:
A column module over Z of rank 2

Matrix:
[ [  1,  2 ] ]

Range:
A column module over Z of rank 1

gap> mor2 := IdentityMorphism( obj6 );
<An identity morphism in Columns( Z )>

gap> Display( mor2 );
An identity morphism in Columns( Z )

Source:
A column module over Z of rank 1

Matrix:
[ [  1 ] ]

Range:
A column module over Z of rank 1

gap> lift := Lift( mor1, mor2 );
<A morphism in Columns( Z )>

gap> Display( lift );
A morphism in Columns( Z )

Source:
A column module over Z of rank 2

Matrix:
[ [  1,  2 ] ]

Range:
A column module over Z of rank 1

gap> source := CategoryOfColumnsObject( 1, cols );
<A column module over Z of rank 1>
gap> range := CategoryOfColumnsObject( 2, cols );
<A column module over Z of rank 2>
gap> mor := CategoryOfColumnsMorphism( source, HomalgMatrix( [[ 2 ], [ 3 ]], S ), range );
<A morphism in Columns( Z )>
gap> colift := Colift( mor2, mor );
<A morphism in Columns( Z )>

gap> Display( colift );
A morphism in Columns( Z )

Source:
A column module over Z of rank 1

Matrix:
[ [  2 ],
[  3 ] ]

Range:
A column module over Z of rank 2

gap> fp := WeakBiFiberProduct( mor1, mor2 );
<A column module over Z of rank 2>
gap> fp_proj := ProjectionOfBiasedWeakFiberProduct( mor1, mor2 );
<A morphism in Columns( Z )>

gap> Display( fp_proj );
A morphism in Columns( Z )

Source:
A column module over Z of rank 2

Matrix:
[ [  -2,  -1 ],
[   1,   0 ] ]

Range:
A column module over Z of rank 2

gap> po := WeakBiPushout( mor, mor2 );
<A column module over Z of rank 2>
gap> inj_push := InjectionOfBiasedWeakPushout( mor, mor2 );
<A morphism in Columns( Z )>

gap> Display( inj_push );
A morphism in Columns( Z )

Source:
A column module over Z of rank 2

Matrix:
[ [  -3,   2 ],
[   1,  -1 ] ]

Range:
A column module over Z of rank 2

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