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2 Finitely presented categories generated by enhanced quivers

2.1 Properties

2.1-1 IsCommutative

Returns: true or false

Check whether the finitely presented category C is commutative.

2.1-2 IsCounitary

Returns: true or false

Check whether B is counitary.

2.1-3 IsCoassociative

Returns: true or false

Check whether B is coassociative.

2.2 Attributes

2.2-1 UnderlyingQuiver

Returns: a QPA quiver

The quiver underlying the finitely presented category C.

2.2-2 UnderlyingQuiverAlgebra

Returns: a QPA path algebra

The quiver algebra (=path algebra with relations) underlying the finitely presented category C.

2.2-3 Size

Returns: a nonnegative integer

The number of morphisms in the finitely presented category C.

2.2-4 BasisPathsByVertexIndex

Returns: a matrix of basis paths of a QPA path algebra

The matrix of basis paths of the canonical basis of the quiver algebra (=path algebra with relations) underlying the f.p. category C, indexed by the vertex indices of source and target of the path.

2.2-5 BasisMorphismsByVertexIndex

Returns: a matrix of basis morphisms

The matrix of basis morphisms of the canonical basis of the quiver algebra (=path algebra with relations) underlying the f.p. category C, indexed by the vertex indices of source and target of the morphism.

2.2-6 HomStructureOnBasisPaths

Returns: a six-dimensional matrix of matrices

The hom structure on basis paths of the canonical basis of the quiver algebra (=path algebra with relations) underlying the f.p. category C: HomStructureOnBasisPaths( A )[ v_index ][ w_index ][ v'_index ][ w'_index ][ basis_path_1_index ][ basis_path_2_index ] = [ Hom(v,w) -> Hom(v',w'): x -> basis_path_1 * x * basis_path_2 ] for basis_path_1: v' -> v and basis_path_2: w -> w'.

2.2-7 AssignSetOfObjects

Returns: nothing

Assigns the objects of the finitely presented category C to global variables. Names of the variables are the concatenation of label with the names of the defining vertices.

2.2-8 SetOfGeneratingMorphisms

Returns: a list

The subset of the generating morphisms that start at obj_1 and ends at obj_2.

2.2-9 SetOfGeneratingMorphisms

Returns: a list

The subset of the generating morphisms that start at obj_1 and ends at obj_2.

2.2-10 SetOfGeneratingMorphisms

Returns: a list

Delegates to SetOfGeneratingMorphisms( C, SetOfObjects(C)[i], SetOfObjects(C)[j] ).

2.2-11 AssignSetOfGeneratingMorphisms

Returns: nothing

Assigns the generating morphisms of the finitely presented category C to global variables. Names of the variables are the concatenation of label with the names of the defining arrows.

2.2-12 RelationsOfFpCategory

Returns: a QPA path algebra

The relations of the finitely presented category C corresponding to RelationsOfAlgebra( UnderlyingQuiverAlgebra( C ) ).

2.2-13 OppositeFpCategory

Returns: a CAP category

The finitely presented category defined by the opposite of the underlying quiver with relations.

2.2-14 Antipode

Returns: a CAP functor

The antipode of the Hopf finitely presented category B.

2.2-15 UnderlyingVertex

Returns: a vertex in a QPA quiver

The vertex of the quiver underlying the object obj in a finitely presented category.

2.2-16 UnderlyingQuiverAlgebraElement

Returns: an element in a QPA path algebra

The quiver algebra element underlying the morphism mor in a finitely presented category.

2.2-17 UnderlyingAlgebra

Returns: a ring

The underlying algebra of the finitely presented category C.

2.2-18 Parity

Returns: a string ("left" or "right")

The parity of the finitely presented category C.

2.2-19 POW

Returns: a CAP category

The n-th power of the finitely presented category C. Admissible values for n are \(0,1,2\).

2.2-20 CategoryFromNerveData

2.2-21 TruthMorphismOfTrueToSieveFunctor

Returns: a CAP functor

Return the truth morphism of true from terminal functor to the functor of sieves from OppositeFpCategory( B ) to RangeCategoryOfHomomorphismStructure( B ).

gap> q := RightQuiver( "q(1)[a:1->1]" );
q(1)[a:1->1]
gap> Fq := FreeCategory( q );
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) )
gap> M0 := Fq / [ [ Fq.a^3, Fq.1 ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
gap> S0 := SieveFunctor( M0 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
-> SkeletalFinSets
gap> M0op := AsCapCategory( Source( S0 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
gap> S0( M0op.1 );
|2|
gap> Display( S0( M0op.1 ) );
{ 0, 1 }
gap> S0( M0op.a );
|2| → |2|
gap> Display( S0( M0op.a ) );
{ 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1 }
gap> M1 := Fq / [ [ Fq.a^3, Fq.a ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a ]
gap> S1 := SieveFunctor( M1 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ]
-> SkeletalFinSets
gap> M1op := AsCapCategory( Source( S1 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ]
gap> S1( M1op.1 );
|3|
gap> Display( S1( M1op.1 ) );
{ 0, 1, 2 }
gap> S1( M1op.a );
|3| → |3|
gap> Display( S1( M1op.a ) );
{ 0, 1, 2 } ⱶ[ 0, 2, 2 ]→ { 0, 1, 2 }
gap> M2 := Fq / [ [ Fq.a^3, Fq.a^2 ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
gap> S2 := SieveFunctor( M2 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> M2op := AsCapCategory( Source( S2 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
gap> S2( M2op.1 );
|4|
gap> Display( S2( M2op.1 ) );
{ 0 ,..., 3 }
gap> S2( M2op.a );
|4| → |4|
gap> Display( S2( M2op.a ) );
{ 0,..., 3 } ⱶ[ 0, 2, 3, 3 ]→ { 0,..., 3 }
gap> eta2 := TruthMorphismOfTrueToSieveFunctor( M2 );
Natural transformation from
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
->
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> eta2( M2op.1 );
|1| → |4|
gap> Display( eta2( M2op.1 ) );
{ 0 } ⱶ[ 3 ]→ { 0,..., 3 }
gap> iota2 := EmbeddingOfSieveFunctor( M2 );
Natural transformation from
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
->
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> iota2( M2op.1 );
|4| → |8|
gap> Display( iota2( M2op.1 ) );
{ 0,..., 3 } ⱶ[ 0, 4, 6, 7 ]→ { 0,..., 7 }

2.2-22 SieveFunctor

Returns: a CAP functor

Return the functor of sieves from OppositeFpCategory( B ) to RangeCategoryOfHomomorphismStructure( B ).

gap> q := RightQuiver( "q(1)[a:1->1]" );
q(1)[a:1->1]
gap> Fq := FreeCategory( q );
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) )
gap> M0 := Fq / [ [ Fq.a^3, Fq.1 ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
gap> S0 := SieveFunctor( M0 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
-> SkeletalFinSets
gap> M0op := AsCapCategory( Source( S0 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = 1 ]
gap> S0( M0op.1 );
|2|
gap> Display( S0( M0op.1 ) );
{ 0, 1 }
gap> S0( M0op.a );
|2| → |2|
gap> Display( S0( M0op.a ) );
{ 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1 }
gap> M1 := Fq / [ [ Fq.a^3, Fq.a ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a ]
gap> S1 := SieveFunctor( M1 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ]
-> SkeletalFinSets
gap> M1op := AsCapCategory( Source( S1 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a ]
gap> S1( M1op.1 );
|3|
gap> Display( S1( M1op.1 ) );
{ 0, 1, 2 }
gap> S1( M1op.a );
|3| → |3|
gap> Display( S1( M1op.a ) );
{ 0, 1, 2 } ⱶ[ 0, 2, 2 ]→ { 0, 1, 2 }
gap> M2 := Fq / [ [ Fq.a^3, Fq.a^2 ] ];
FreeCategory( RightQuiver( "q(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
gap> S2 := SieveFunctor( M2 );
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> M2op := AsCapCategory( Source( S2 ) );
FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
gap> S2( M2op.1 );
|4|
gap> Display( S2( M2op.1 ) );
{ 0 ,..., 3 }
gap> S2( M2op.a );
|4| → |4|
gap> Display( S2( M2op.a ) );
{ 0,..., 3 } ⱶ[ 0, 2, 3, 3 ]→ { 0,..., 3 }
gap> eta2 := TruthMorphismOfTrueToSieveFunctor( M2 );
Natural transformation from
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
->
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> eta2( M2op.1 );
|1| → |4|
gap> Display( eta2( M2op.1 ) );
{ 0 } ⱶ[ 3 ]→ { 0,..., 3 }
gap> iota2 := EmbeddingOfSieveFunctor( M2 );
Natural transformation from
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
->
Functor from FreeCategory( RightQuiver( "q_op(1)[a:1->1]" ) ) / [ a*a*a = a*a ]
-> SkeletalFinSets
gap> iota2( M2op.1 );
|4| → |8|
gap> Display( iota2( M2op.1 ) );
{ 0,..., 3 } ⱶ[ 0, 4, 6, 7 ]→ { 0,..., 7 }

2.3 Operations

2.3-1 TrivialCategory

2.3-2 \*

2.3-3 ElementaryTensor

Returns: a morphism in a CAP category

Given an object a in a finitely presented category A and an object b in a finitely presented category B and the tensor product T of A and B, return the tensor product of a and b in T.

2.3-4 ElementaryTensor

Returns: a morphism in a CAP category

Given an object a in a finitely presented category A and a morphism g in a finitely presented category B and the tensor product T of A and B, return the tensor product of a and g in T.

2.3-5 ElementaryTensor

Returns: a morphism in a CAP category

Given a morphism f in a finitely presented category A and an object b in a finitely presented category B and the tensor product T of A and B, return the tensor product of f and b in T.

2.3-6 QuiverVertexAsIdentityPath

Returns: a path

Simply returns vertex, but with the semantics of being an identity path.

2.4 Constructors

2.4-1 FreeCategory

Returns: a CAP category

Construct the finitely presented category generated by the quiver q, possibly modulo the relations L.

2.4-2 ObjectInFpCategory

Returns: an object in a CAP category

The constructor of objects in a finitely presented category C given a vertex V in the underlying quiver.

2.4-3 MorphismInFpCategory

Returns: an object in a CAP category a morphism in a CAP category

Delegates to ObjectInFpCategory( C, V ). The constructor of morphisms in a finitely presented category C given the source S, the target T, and the underlying path path. If neither S nor T are provided they are read off from path.

2.4-4 \/

Returns: a morphism in a CAP category

Delegates to MorphismInFpCategory( path ).

2.5 GAP categories

2.5-1 IsFpCategory

Returns: true or false

The GAP category of finitely presented categories.

2.5-2 IsMonoidAsCategory

Returns: true or false

The GAP category of algebras.

2.5-3 IsCellInFpCategory

Returns: true or false

The GAP category of cells in a finitely presented category.

2.5-4 IsObjectInFpCategory

Returns: true or false

The GAP category of objects in a finitely presented category.

2.5-5 IsMorphismInFpCategory

Returns: true or false

The GAP category of morphisms in a finitely presented category.

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