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### 5 Algebroids as preadditive categories generated by enhanced quivers

#### 5.1 Properties

##### 5.1-1 IsFinitelyPresentedLinearCategory
 ‣ IsFinitelyPresentedLinearCategory( A ) ( property )

Returns: true or false

Is the linear category A finitely presented. This property is true by construction for algebroids generated by finite quivers.

##### 5.1-2 IsLinearClosureOfACategory
 ‣ IsLinearClosureOfACategory( C ) ( property )

Returns: true or false

The property of C being a linear closure of a category.

##### 5.1-3 IsCommutative
 ‣ IsCommutative( A ) ( property )

Returns: true or false

Check whether the algebroid A is commutative.

##### 5.1-4 IsCounitary
 ‣ IsCounitary( B ) ( property )

Returns: true or false

Check whether B is counitary.

##### 5.1-5 IsCoassociative
 ‣ IsCoassociative( B ) ( property )

Returns: true or false

Check whether B is coassociative.

#### 5.2 Attributes

##### 5.2-1 UnderlyingQuiver
 ‣ UnderlyingQuiver( A ) ( attribute )

Returns: a QPA quiver

The quiver underlying the algebroid A.

##### 5.2-2 UnderlyingQuiverAlgebra
 ‣ UnderlyingQuiverAlgebra( A ) ( attribute )

Returns: a QPA path algebra

The quiver algebra (=path algebra with relations) underlying the algebroid A.

##### 5.2-3 Dimension
 ‣ Dimension( A ) ( attribute )

Returns: a nonnegative integer

The dimension of the underlying quiver algebra (=path algebra with relations) underlying the algebroid A.

##### 5.2-4 BasisPathsByVertexIndex
 ‣ BasisPathsByVertexIndex( A ) ( 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 algebroid A, indexed by the vertex indices of source and target of the path.

##### 5.2-5 BasisMorphismsByVertexIndex
 ‣ 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 algebroid A, indexed by the vertex indices of source and target of the morphism.

##### 5.2-6 HomStructureOnBasisPaths
 ‣ HomStructureOnBasisPaths( A ) ( 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 algebroid A: 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'.

##### 5.2-7 AssignSetOfObjects
 ‣ AssignSetOfObjects( A, label ) ( operation )

Returns: nothing

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

##### 5.2-8 SetOfGeneratingMorphisms
 ‣ SetOfGeneratingMorphisms( A, obj_1, obj_2 ) ( operation )

Returns: a list

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

##### 5.2-9 SetOfGeneratingMorphisms
 ‣ 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.

##### 5.2-10 SetOfGeneratingMorphisms
 ‣ SetOfGeneratingMorphisms( A, i, j ) ( operation )

Returns: a list

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

##### 5.2-11 AssignSetOfGeneratingMorphisms
 ‣ AssignSetOfGeneratingMorphisms( A, label ) ( operation )

Returns: nothing

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

##### 5.2-12 RelationsOfAlgebroid
 ‣ RelationsOfAlgebroid( A ) ( attribute )

Returns: a QPA path algebra

The relations of the algebroid A corresponding to RelationsOfAlgebra( UnderlyingQuiverAlgebra( A ) ).

##### 5.2-13 OppositeAlgebroid
 ‣ OppositeAlgebroid( A ) ( attribute )

Returns: a CAP category

The algebroid defined by the opposite of the underlying quiver algebra.

##### 5.2-14 Antipode
 ‣ Antipode( B ) ( attribute )

Returns: a CAP functor

The antipode of the Hopf algebroid B.

##### 5.2-15 UnderlyingVertex
 ‣ UnderlyingVertex( obj ) ( attribute )

Returns: a vertex in a QPA quiver

The vertex of the quiver underlying the object obj in an algebroid.

##### 5.2-16 UnderlyingQuiverAlgebraElement
 ‣ UnderlyingQuiverAlgebraElement( mor ) ( attribute )

Returns: an element in a QPA path algebra

The quiver algebra element underlying the morphism mor in an algebroid.

##### 5.2-17 UnderlyingAlgebra
 ‣ UnderlyingAlgebra( A ) ( attribute )

Returns: a ring

The underlying algebra of an algebroid.

##### 5.2-18 Parity
 ‣ Parity( A ) ( attribute )

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

The parity of an algebroid.

##### 5.2-19 POW
 ‣ POW( A, n ) ( operation )

Returns: a CAP category

The n-th power of the algebroid A. Admissible values for n are 0,1,2.

#### 5.3 Operations

##### 5.3-1 DecomposeQuiverAlgebraElement
 ‣ DecomposeQuiverAlgebraElement( e ) ( operation )

##### 5.3-2 ApplyToQuiverAlgebraElement
 ‣ ApplyToQuiverAlgebraElement( img_of_objs_func, imgs_of_generating_mors_func, path, covariant ) ( operation )
 ‣ ApplyToQuiverAlgebraElement( F, path ) ( operation )

Returns: a morphism in a CAP category

Apply the functor F (e.g., defined by the functions img_of_objs_func and imgs_of_generating_mors_func) to the quiver algebra element p.

##### 5.3-3 TrivialAlgebroid
 ‣ TrivialAlgebroid( k, str ) ( operation )

##### 5.3-4 \*
 ‣ \*( A, B ) ( operation )

##### 5.3-5 ElementaryTensor
 ‣ ElementaryTensor( a, b, T ) ( operation )

Returns: a morphism in a CAP category

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

##### 5.3-6 ElementaryTensor
 ‣ ElementaryTensor( a, g, T ) ( operation )

Returns: a morphism in a CAP category

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

##### 5.3-7 ElementaryTensor
 ‣ ElementaryTensor( f, b, T ) ( operation )

Returns: a morphism in a CAP category

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

#### 5.4 Constructors

##### 5.4-1 Algebroid
 ‣ Algebroid( Rq ) ( attribute )
 ‣ Algebroid( Rq, L ) ( operation )
 ‣ Algebroid( R, q ) ( operation )
 ‣ Algebroid( C, H ) ( operation )
 ‣ Algebroid( R, C ) ( operation )
 ‣ []( R, C ) ( operation )
 ‣ QuotientCategory( A, L ) ( operation )
 ‣ /( A, L ) ( operation )

Returns: a CAP category

Construct the algebroid associated to the path R-algebra Rq of the quiver q modulo the relations L as an R-linear category.

##### 5.4-2 DescentToZDefinedByBasisPaths
 ‣ DescentToZDefinedByBasisPaths( A ) ( attribute )

Returns: a CAP category

The argument is an algebroid A with left acting domain given by the field of rationals \mathbb{Q}, either realized by Rationals or by HomalgFieldOfRationals (in no external CAS). Futhermore, A has to satisfy another technical condition that is described below. The output is an algebroid A' defined over \mathbb{Z} such that A' \otimes_{\mathbb{Z}} \mathbb{Q} \simeq A. For the construction of A', let T be the multiplication table of the underlying quiver algebra of A expressed with respect to the \mathbb{Q}-basis BasisPaths( CanonicalBasis( A' ) ), denoted by B. Now, we expect the following technical condition to hold: T should only consist of integral linear combinations of paths in B. Then A' is constructed as the algebroid over \mathbb{Z}, realized as HomalgRingOfIntegers, with the same vertices as A, a \mathbb{Z}-basis of paths given by B, and composition is carried out using the rules defined by T.

##### 5.4-3 CapFunctor
 ‣ CapFunctor( A, images_of_objects, images_of_generating_morphisms, B, covariant ) ( operation )
 ‣ CapFunctor( A, images_of_objects, images_of_generating_morphisms, B ) ( operation )
 ‣ CapFunctor( A, rec_images_of_objects, rec_images_of_generating_morphisms, covariant ) ( operation )
 ‣ CapFunctor( A, rec_images_of_objects, rec_images_of_generating_morphisms ) ( operation )
 ‣ CapFunctor( A, obj ) ( operation )
 ‣ CapFunctor( A, images_of_objects, images_of_generating_morphisms, B, covariant ) ( operation )
 ‣ CapFunctor( A, images_of_objects, images_of_generating_morphisms, B ) ( operation )
 ‣ CapFunctor( A, rec_images_of_objects, rec_images_of_generating_morphisms, covariant ) ( operation )
 ‣ CapFunctor( A, rec_images_of_objects, rec_images_of_generating_morphisms ) ( operation )
 ‣ CapFunctor( A, obj ) ( operation )

Returns: a CAP functor

Construct a functor with source the finitely presented algebroid A and target B using the two defining lists of images images_of_objects and images_of_generating_morphisms. The order of their entries must correspond to that of the vertices and arrows of the underlying quiver. If the last boolean argument covariant is not specified it defaults to true. Alternatively one could specify the records of images rec_images_of_objects and rec_images_of_generating_morphisms. The record rec_images_of_objects is supposed to contain the images of the objects of A. The record rec_images_of_generating_morphisms is supposed to contain the images of the set of generating morphisms of A.

In the case of two arguments, where the second argument is an object obj then the output is the constant functor having obj as the value on objects and IdentityMorphism(obj) as the value on morphisms. Construct a functor with source the finitely presented category C and target B using the two defining lists of images images_of_objects and images_of_generating_morphisms. The order of their entries must correspond to that of the vertices and arrows of the underlying quiver. If the last boolean argument covariant is not specified it defaults to true. Alternatively one could specify the records of images rec_images_of_objects and rec_images_of_generating_morphisms. The record rec_images_of_objects is supposed to contain the images of the objects of C. The record rec_images_of_generating_morphisms is supposed to contain the images of the set of generating morphisms of C.

In the case of two arguments, where the second argument is an object obj then the output is the constant functor having obj as the value on objects and IdentityMorphism(obj) as the value on morphisms.

##### 5.4-4 NaturalTransformation
 ‣ NaturalTransformation( eta, F, G ) ( operation )
 ‣ NaturalTransformation( F, eta, G ) ( operation )

Returns: a CAP natural transformation

Construct, using the record (or list) of images eta, a natural transformation from the functor F to the parallel functor G.

##### 5.4-5 ObjectInAlgebroid
 ‣ ObjectInAlgebroid( A, V ) ( operation )

Returns: an object in a CAP category

The constructor of objects in an algebroid A given a vertex V in the underlying quiver.

##### 5.4-6 \/
 ‣ \/( V, A ) ( operation )

Returns: an object in a CAP category

Delegates to ObjectInAlgebroid( A, V ).

##### 5.4-7 MorphismInAlgebroid
 ‣ MorphismInAlgebroid( S, path, T ) ( operation )
 ‣ MorphismInAlgebroid( A, path ) ( operation )

Returns: a morphism in a CAP category

The constructor of morphisms in an algebroid A given the source S, the target T, and the underlying quiver algebra element path. If neither S nor T are provided they are read off from path.

##### 5.4-8 \/
 ‣ \/( path, A ) ( operation )

Returns: a morphism in a CAP category

Delegates to MorphismInAlgebroid( path ).

##### 5.4-9 POW
 ‣ POW( phi, i ) ( operation )

Returns: a CAP morphism

The i-th power of the CAP endomorhpism phi.

#### 5.5 GAP categories

##### 5.5-1 IsCellInAlgebroid
 ‣ IsCellInAlgebroid( arg ) ( filter )

Returns: true or false

The GAP category of cells in an algebroid.

##### 5.5-2 IsObjectInAlgebroid
 ‣ IsObjectInAlgebroid( arg ) ( filter )

Returns: true or false

The GAP category of objects in an algebroid.

##### 5.5-3 IsMorphismInAlgebroid
 ‣ IsMorphismInAlgebroid( arg ) ( filter )

Returns: true or false

The GAP category of morphisms in an algebroid.

##### 5.5-4 IsAlgebroid
 ‣ IsAlgebroid( arg ) ( filter )

Returns: true or false

The GAP category of algebroids.

##### 5.5-5 IsAlgebraAsCategory
 ‣ IsAlgebraAsCategory( arg ) ( filter )

Returns: true or false

The GAP category of algebras.

##### 5.5-6 IsAlgebroidMorphism
 ‣ IsAlgebroidMorphism( arg ) ( filter )

Returns: true or false

The GAP category of morphisms of algebroids.

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