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1 Quivers

1.1 Constructors

1.1-1 FinQuiver

Returns: a CAP quiver

Construct a CAP quiver out of the q_datum which is specified as follows:

• q_datum[1] is a string representing the name of the quiver.

• q_datum[2] is a list consisting of two entries: the number of objects and the list of labels for these objects. This list might contain an optional third entry for the list of LaTeX-strings for the objects.

• q_datum[3] is a list consisting of four entries: the number of morphisms, the list of source-indices, the list of range-indices and the list of labels for these morphisms. This list might contain an optional fifth entry for the list of LaTeX-strings for the morphisms.

gap> LoadPackage( "FpCategories", false );
true
gap> str :=
>   "q(0..5)[x:0->0,s:0->1,a:1->2,c:1->3,e:1->4,b:2->4,d:3->4,t:4->5,y:5->5]";;
gap> q := FinQuiver( str );
FinQuiver( "q(0,1,2,3,4,5)[x:0-≻0,s:0-≻1,a:1-≻2,c:1-≻3,e:1-≻4,b:2-≻4,d:3-≻4,
t:4-≻5,y:5-≻5]" )
gap> QuiverName( q );
"q"
gap> NumberOfObjects( q );
6
gap> LabelsOfObjects( q );
[ "0", "1", "2", "3", "4", "5" ]
gap> SetOfObjects( q );
[ (0), (1), (2), (3), (4), (5) ]
gap> o := q.0;
(0)
gap> ObjectIndex( o );
1
gap> IsIdenticalObj( o, ObjectConstructor( q, 1 ) );
true
gap> ObjectLabel( o );
"0"
gap> LaTeXOutput( o );
"0"
gap> NumberOfMorphisms( q );
9
gap> LabelsOfMorphisms( q );
[ "x", "s", "a", "c", "e", "b", "d", "t", "y" ]
gap> IndicesOfSources( q );
[ 1, 1, 2, 2, 2, 3, 4, 5, 6 ]
gap> IndicesOfTargets( q );
[ 1, 2, 3, 4, 5, 5, 5, 6, 6 ]
gap> SetOfMorphisms( q );
[ x:(0) -≻ (0), s:(0) -≻ (1), a:(1) -≻ (2), c:(1) -≻ (3), e:(1) -≻ (4),
  b:(2) -≻ (4), d:(3) -≻ (4), t:(4) -≻ (5), y:(5) -≻ (5) ]
gap> m := q.y;
y:(5) -≻ (5)
gap> MorphismIndex( m );
9
gap> IsIdenticalObj( m, MorphismConstructor( q, q.("5"), 9, q.("5") ) );
true
gap> MorphismLabel( m );
"y"
gap> LaTeXOutput( m );
"{5}-\\left({y}\\right)\\rightarrow{5}"
gap> MorphismsOfExternalHom( q.("0"), q.("0") );
[ x:(0) -≻ (0) ]
gap> MorphismsOfExternalHom( q.("0"), q.("1") );
[ s:(0) -≻ (1) ]
gap> q_op := OppositeQuiver( q );
FinQuiver( "q_op(0,1,2,3,4,5)[x:0-≻0,s:1-≻0,a:2-≻1,c:3-≻1,e:4-≻1,b:4-≻2,
d:4-≻3,t:5-≻4,y:5-≻5]" )
gap> MorphismsOfExternalHom( q_op.1, q_op.0 );
[ s:(1) -≻ (0) ]
gap> q_op_x_q := TensorProductOfFinQuivers( q_op, q );;
gap> NumberOfObjects( q_op_x_q );
36
gap> NumberOfMorphisms( q_op_x_q );
108

gap> q1 := RandomFinQuiver( 2, 4, false );; # without loops
gap> q2 := RandomFinQuiver( 2, 4, true );;  # with loops
gap> Assert( 0, NumberOfObjects( q1 ) = 2 );
gap> Assert( 0, NumberOfMorphisms( q1 ) = 4 );
gap> Assert( 0, Length( Filtered( SetOfMorphisms( q1 ), IsEndomorphism ) ) = 0 );
gap> Assert( 0, Length( Filtered( SetOfMorphisms( q2 ), IsEndomorphism ) ) > 0 );
gap> q3 := FinQuiver( "q(3)[a:1->2,b:2->3]" );
FinQuiver( "q(1,2,3)[a:1-≻2,b:2-≻3]" )
gap> q4 := FinQuiver( "q(4..6)[a:4->5,b:5->6]" );
FinQuiver( "q(4,5,6)[a:4-≻5,b:5-≻6]" )

1.1-2 ObjectConstructor

Returns: a CAP category object

The input is a CAP quiver \(q\) and a positive integer i. The output is the i'th object in q.

1.1-3 MorphismConstructor

Returns: a CAP category object

The input is a CAP quiver \(q\), an object s, a positive integer i and an object t. The output is the i'th morphism in q (whose source is s and target is t).

1.1-4 RandomFinQuiver

Returns: a CAP quiver

The outout is a random quiver consisting of nr_objs objects and nr_mors morphisms and has loops only if has_loops = true.

1.1-5 TensorProductOfFinQuivers

Returns: a CAP quiver

Returns the tensor product of q1 and q2.

1.2 Attributes

1.2-1 QuiverDatum

Returns: a list

Returns the quiver datum of the quiver q.

1.2-2 QuiverName

Returns: a list

Returns the name of the quiver q.

1.2-3 NumberOfObjects

Returns: a positive integer

Returns the number of objects in the quiver q.

1.2-4 LabelsOfObjects

Returns: a list

Returns the labels of all objects in the quiver q.

1.2-5 LaTeXStringsOfObjects

Returns: a list

Returns the LaTeX-strings of all objects in the quiver q.

1.2-6 NumberOfMorphisms

Returns: a positive integer

Returns the number of morphisms in the quiver q.

1.2-7 LabelsOfMorphisms

Returns: a list

Returns the labels of all morphisms in the quiver q.

1.2-8 LaTeXStringsOfMorphisms

Returns: a list

Returns the LaTeX-strings of all morphisms in the quiver q.

1.2-9 IndicesOfSources

Returns: a list

Returns the indices of sources of all morphisms in the quiver q.

1.2-10 IndicesOfTargets

Returns: a list

Returns the indices of targets of all morphisms in the quiver q.

1.2-11 IndicesPairsOfCompatibleMorphisms

Returns: a list

Returns the indices of all paris of morphisms where the target of the first morphism equals the source of the second morphism.

1.2-12 SetOfObjects

Returns: a list of CAP quiver objects

Returns the set of objects of q.

1.2-13 SetOfMorphisms

Returns: a list of CAP quiver morphisms

Returns the set of morphisms of q.

1.2-14 ExternalHoms

Returns: a list of CAP quiver morphisms

Returns a list of lists where the entry at \([i,j]\) is the set of morphism from the \(i\)'th object to the \(j\)'th object in q.

1.2-15 OppositeQuiver

Returns: CAP quiver

Returns the opposite quiver of q.

1.2-16 ObjectIndex

Returns: a positive integer

Returns the index of the object v.

1.2-17 ObjectLabel

Returns: a string

Returns the label of the object v.

1.2-18 UnderlyingFinQuiver

Returns: a CAP quiver

Returns the quiver containing the object v.

1.2-19 MorphismIndex

Returns: a positive integer

Returns the index of the morphism alpha.

1.2-20 MorphismLabel

Returns: a string

Returns the label of the morphism alpha.

1.2-21 UnderlyingFinQuiver

Returns: a CAP quiver

Returns the quiver containing the morphism alpha.

1.3 GAP categories

1.3-1 IsFinQuiver

Returns: true or false

The GAP category of quivers.

1.3-2 IsFinQuiverObject

Returns: true or false

The GAP category of objects in a quiver.

1.3-3 IsFinQuiverMorphism

Returns: true or false

The GAP category of morphisms in a quiver.

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