‣ IsCommutative( C ) | ( property ) |
Returns: true or false
Check whether the finitely presented category C is commutative.
‣ IsCounitary( B ) | ( property ) |
Returns: true or false
Check whether B is counitary.
‣ IsCoassociative( B ) | ( property ) |
Returns: true or false
Check whether B is coassociative.
‣ UnderlyingQuiver( C ) | ( attribute ) |
Returns: a QPA quiver
The quiver underlying the finitely presented category C.
‣ UnderlyingQuiverAlgebra( C ) | ( attribute ) |
Returns: a QPA path algebra
The quiver algebra (=path algebra with relations) underlying the finitely presented category C.
‣ Size( C ) | ( attribute ) |
Returns: a nonnegative integer
The number of morphisms in the finitely presented category C.
‣ BasisPathsByVertexIndex( C ) | ( attribute ) |
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.
‣ BasisMorphismsByVertexIndex( A ) | ( attribute ) |
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.
‣ HomStructureOnBasisPaths( C ) | ( attribute ) |
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'.
‣ AssignSetOfObjects( C, label ) | ( operation ) |
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.
‣ SetOfGeneratingMorphisms( C, obj_1, obj_2 ) | ( operation ) |
Returns: a list
The subset of the generating morphisms that start at obj_1 and ends at obj_2.
‣ SetOfGeneratingMorphisms( obj_1, obj_2 ) | ( operation ) |
Returns: a list
The subset of the generating morphisms that start at obj_1 and ends at obj_2.
‣ SetOfGeneratingMorphisms( C, i, j ) | ( operation ) |
Returns: a list
Delegates to SetOfGeneratingMorphisms( C, SetOfObjects(C)[i], SetOfObjects(C)[j] ).
‣ AssignSetOfGeneratingMorphisms( C, label ) | ( operation ) |
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.
‣ RelationsOfFpCategoryDefinedByQuiverAlgebra( C ) | ( attribute ) |
Returns: a QPA path algebra
The relations of the finitely presented category C corresponding to RelationsOfAlgebra( UnderlyingQuiverAlgebra( C ) ).
‣ Antipode( B ) | ( attribute ) |
Returns: a CAP functor
The antipode of the Hopf finitely presented category B.
‣ UnderlyingVertex( obj ) | ( attribute ) |
Returns: a vertex in a QPA quiver
The vertex of the quiver underlying the object obj in a finitely presented category.
‣ UnderlyingQuiverAlgebraElement( mor ) | ( attribute ) |
Returns: an element in a QPA path algebra
The quiver algebra element underlying the morphism mor in a finitely presented category.
‣ UnderlyingAlgebra( C ) | ( attribute ) |
Returns: a ring
The underlying algebra of the finitely presented category C.
‣ Parity( C ) | ( attribute ) |
Returns: a string ("left" or "right")
The parity of the finitely presented category C.
‣ POW( C, n ) | ( operation ) |
Returns: a CAP category
The n-th power of the finitely presented category C. Admissible values for n are \(0,1,2\).
‣ CategoryFromNerveData( C ) | ( attribute ) |
‣ TrivialCategory( str ) | ( operation ) |
2.3-2 \*‣ \*( C, D ) | ( operation ) |
‣ ElementaryTensor( a, b, T ) | ( operation ) |
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.
‣ ElementaryTensor( a, g, T ) | ( operation ) |
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.
‣ ElementaryTensor( f, b, T ) | ( operation ) |
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.
‣ QuiverVertexAsIdentityPath( vertex ) | ( operation ) |
Returns: a path
Simply returns vertex, but with the semantics of being an identity path.
‣ FreeCategory( q ) | ( operation ) |
‣ Category( q, L ) | ( operation ) |
‣ QuotientCategory( C, L ) | ( operation ) |
‣ /( C, L ) | ( operation ) |
Returns: a CAP category
Construct the finitely presented category generated by the quiver q, possibly modulo the relations L.
‣ ObjectInFpCategory( A, V ) | ( operation ) |
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.
‣ MorphismInFpCategory( S, path, T ) | ( operation ) |
‣ MorphismInFpCategory( A, path ) | ( operation ) |
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 \/‣ \/( path, A ) | ( operation ) |
Returns: a morphism in a CAP category
Delegates to MorphismInFpCategory( path ).
‣ CategoryFromDataTables( C ) | ( attribute ) |
Returns: a CAP category
Returns the CategoryFromDataTables of the f.p. category C.
‣ IsFpCategoryDefinedByQuiverAlgebra( arg ) | ( category ) |
Returns: true or false
The GAP category of finitely presented categories.
‣ IsMonoidAsCategory( arg ) | ( category ) |
Returns: true or false
The GAP category of algebras.
‣ IsCellInFpCategory( arg ) | ( category ) |
Returns: true or false
The GAP category of cells in a finitely presented category.
‣ IsObjectInFpCategoryDefinedByQuiverAlgebra( arg ) | ( category ) |
Returns: true or false
The GAP category of objects in a finitely presented category.
‣ IsMorphismInFpCategoryDefinedByQuiverAlgebra( arg ) | ( category ) |
Returns: true or false
The GAP category of morphisms in a finitely presented category.
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