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Algebroids

Algebroids and bialgebroids as preadditive categories generated by enhanced quivers

2025.07-14

16 July 2025

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Florian Heiderich
Email: heiderich@mathematik.uni-siegen.de
Homepage: https://www.heiderich.org/math/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Sebastian Posur
Email: sebastian.posur@uni-siegen.de
Homepage: https://sebastianpos.github.io/
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57068 Siegen
Germany

Kamal Saleh
Email: kamal.saleh@uni-siegen.de
Homepage: https://github.com/kamalsaleh
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57072 Siegen
Germany

Fabian Zickgraf
Email: fabian.zickgraf@uni-siegen.de
Homepage: https://algebra.mathematik.uni-siegen.de/zickgraf
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

1 Examples
 1.1 The natural numbers as a f.g. monoid
 1.2 The integers as f.g. monoid
 1.3 A f.p. category
 1.4 Algebroids
 1.5 Bialgebroids
 1.6 Hopf algebra generated by group-like elements
 1.7 Hopf algebra generated by a primitive element
 1.8 A non-counitary coalgebra
 1.9 Sweedlers 4-dimensional Hopf algebra
 1.10 Homomorphism structure
 1.11 Algebroids over the integers
 1.12 Unitors
 1.13 Associators
2 Finitely presented categories generated by enhanced quivers
 2.1 Properties

  2.1-1 IsCommutative
  2.1-2 IsCounitary
  2.1-3 IsCoassociative
 2.2 Attributes

  2.2-1 UnderlyingQuiver
  2.2-2 UnderlyingQuiverAlgebra
  2.2-3 Size
  2.2-4 BasisPathsByVertexIndex
  2.2-5 BasisMorphismsByVertexIndex
  2.2-6 HomStructureOnBasisPaths
  2.2-7 AssignSetOfObjects
  2.2-8 SetOfGeneratingMorphisms
  2.2-9 SetOfGeneratingMorphisms
  2.2-10 SetOfGeneratingMorphisms
  2.2-11 AssignSetOfGeneratingMorphisms
  2.2-12 RelationsOfFpCategory
  2.2-13 OppositeFpCategory
  2.2-14 Antipode
  2.2-15 UnderlyingVertex
  2.2-16 UnderlyingQuiverAlgebraElement
  2.2-17 UnderlyingAlgebra
  2.2-18 Parity
  2.2-19 POW
  2.2-20 CategoryFromNerveData
  2.2-21 TruthMorphismOfTrueToSieveFunctor
  2.2-22 SieveFunctor
 2.3 Operations

  2.3-1 TrivialCategory
  2.3-2 \*
  2.3-3 ElementaryTensor
  2.3-4 ElementaryTensor
  2.3-5 ElementaryTensor
  2.3-6 QuiverVertexAsIdentityPath
 2.4 Constructors

  2.4-1 FreeCategory
  2.4-2 ObjectInFpCategory
  2.4-3 MorphismInFpCategory
  2.4-4 \/
 2.5 GAP categories

  2.5-1 IsFpCategory
  2.5-2 IsMonoidAsCategory
  2.5-3 IsCellInFpCategory
  2.5-4 IsObjectInFpCategory
  2.5-5 IsMorphismInFpCategory
3 Finite categories from nerve data
 3.1 GAP categories
4 Finite categories from data tables
 4.1 Attributes

  4.1-1 DataTables
  4.1-2 Size
  4.1-3 OppositeCategoryFromDataTables
 4.2 Constructors

  4.2-1 CategoryFromDataTables
  4.2-2 CategoryFromDataTables
  4.2-3 CategoryFromDataTables
  4.2-4 CategoryFromDataTables
  4.2-5 CategoryFromDataTables
  4.2-6 CreateObject
  4.2-7 CreateMorphism
 4.3 GAP categories

  4.3-1 IsCategoryFromDataTables
  4.3-2 IsCellInCategoryFromDataTables
  4.3-3 IsObjectInCategoryFromDataTables
  4.3-4 IsMorphismInCategoryFromDataTables
5 Algebroids as preadditive categories generated by enhanced quivers
 5.1 Properties

  5.1-1 IsFinitelyPresentedLinearCategory
  5.1-2 IsLinearClosureOfACategory
  5.1-3 IsCommutative
  5.1-4 IsCounitary
  5.1-5 IsCoassociative
 5.2 Attributes

  5.2-1 UnderlyingQuiver
  5.2-2 UnderlyingQuiverAlgebra
  5.2-3 Dimension
  5.2-4 BasisPathsByVertexIndex
  5.2-5 BasisMorphismsByVertexIndex
  5.2-6 HomStructureOnBasisPaths
  5.2-7 AssignSetOfObjects
  5.2-8 SetOfGeneratingMorphisms
  5.2-9 SetOfGeneratingMorphisms
  5.2-10 SetOfGeneratingMorphisms
  5.2-11 AssignSetOfGeneratingMorphisms
  5.2-12 RelationsOfAlgebroid
  5.2-13 OppositeAlgebroid
  5.2-14 Antipode
  5.2-15 UnderlyingVertex
  5.2-16 UnderlyingQuiverAlgebraElement
  5.2-17 UnderlyingAlgebra
  5.2-18 Parity
  5.2-19 POW
 5.3 Operations

  5.3-1 DecomposeQuiverAlgebraElement
  5.3-2 ApplyToQuiverAlgebraElement
  5.3-3 TrivialAlgebroid
  5.3-4 \*
  5.3-5 ElementaryTensor
  5.3-6 ElementaryTensor
  5.3-7 ElementaryTensor
 5.4 Constructors

  5.4-1 Algebroid
  5.4-2 DescentToZDefinedByBasisPaths
  5.4-3 CapFunctor
  5.4-4 ObjectInAlgebroid
  5.4-5 \/
  5.4-6 MorphismInAlgebroid
  5.4-7 \/
  5.4-8 POW
 5.5 GAP categories

  5.5-1 IsCellInAlgebroid
  5.5-2 IsObjectInAlgebroid
  5.5-3 IsMorphismInAlgebroid
  5.5-4 IsAlgebroid
  5.5-5 IsAlgebraAsCategory
  5.5-6 IsAlgebroidMorphism
6 Algebroids from data tables
 6.1 Properties

  6.1-1 IsAdmissibleAlgebroid
 6.2 Attributes

  6.2-1 DataTablesOfCategory
  6.2-2 IsomorphismOntoAlgebroidFromDataTables
  6.2-3 IsomorphismFromAlgebroidFromDataTables
  6.2-4 QuotientCategory
  6.2-5 CategoryDatum
  6.2-6 BasesElements
  6.2-7 DecompositionIndicesOfBasesElements
  6.2-8 LabelsOfBasesElements
  6.2-9 LaTeXStringsOfBasesElements
  6.2-10 PowerOfArrowIdeal
 6.3 Operations

  6.3-1 AssignSetOfObjects
  6.3-2 AssignSetOfGeneratingMorphisms
  6.3-3 ObjectIndex
  6.3-4 CoefficientsList
  6.3-5 IndicesOfSupportMorphisms
  6.3-6 CoefficientsOfSupportMorphisms
  6.3-7 SupportMorphisms
  6.3-8 DecompositionOfMorphismInAlgebroid
  6.3-9 DecompositionIndicesOfMorphismInAlgebroid
  6.3-10 OppositeAlgebroid
  6.3-11 TensorProductOfAlgebroids
  6.3-12 \*
  6.3-13 ElementaryTensor
  6.3-14 ElementaryTensor
 6.4 Constructors

  6.4-1 AlgebroidFromDataTables
 6.5 GAP categories

  6.5-1 IsAlgebroidFromDataTables
  6.5-2 IsCellInAlgebroidFromDataTables
  6.5-3 IsObjectInAlgebroidFromDataTables
  6.5-4 IsMorphismInAlgebroidFromDataTables
7 Category of algebroids
 7.1 Properties

  7.1-1 IsCommutative
  7.1-2 IsCounitary
  7.1-3 IsCoassociative
 7.2 Attributes

  7.2-1 AsCapCategory
  7.2-2 AsCapFunctor
 7.3 Operations

  7.3-1 Twist
 7.4 Constructors

  7.4-1 CategoryOfAlgebroids
  7.4-2 CategoryOfAlgebroidsObject
  7.4-3 CategoryOfAlgebroidsMorphism
  7.4-4 CategoryOfAlgebroidsMorphism
 7.5 GAP categories

  7.5-1 IsCategoryOfAlgebroidsObject
  7.5-2 IsCategoryOfAlgebroidsMorphism
8 Bialgebroids as algebroids with more structure
 8.1 Properties

  8.1-1 IsCocommutative
  8.1-2 IsHopfAlgebroid
 8.2 Attributes

  8.2-1 Counit
  8.2-2 Comultiplication
 8.3 Constructors

  8.3-1 AddBialgebroidStructure
  8.3-2 AddAntipode
 8.4 GAP categories

  8.4-1 IsBialgebroid
9 Linear closures of path categories and their quotients
 9.1 Constructors

  9.1-1 LinearClosure
  9.1-2 GroebnerBasis
  9.1-3 ReducedGroebnerBasis
  9.1-4 QuotientCategory
10 Category of finitely presented associative unital algebras
 10.1 Attributes

  10.1-1 CoefficientsRing
  10.1-2 CoefficientsRing
  10.1-3 DefiningSeptupleOfFinitelyPresentedAlgebra
  10.1-4 ListOfImages
  10.1-5 NrGenerators
  10.1-6 Generators
  10.1-7 AssociatedLinearClosureOfPathCategory
  10.1-8 DefiningRelations
  10.1-9 GroebnerBasisOfDefiningRelations
  10.1-10 AssociatedQuotientCategoryOfLinearClosureOfPathCategory
  10.1-11 Dimension
  10.1-12 AmbientAlgebra
  10.1-13 AssociatedFunctorOfLinearClosuresOfPathCategoriesData
 10.2 Constructors

  10.2-1 CategoryOfFpAlgebras
 10.3 Operations

  10.3-1 Counit
  10.3-2 Comultiplication
 10.4 GAP categories

  10.4-1 IsCategoryOfFinitelyPresentedAlgebras
  10.4-2 IsObjectInCategoryOfFpAlgebras
  10.4-3 IsMorphismInCategoryOfFpAlgebras
11 Category of finitely presented associative unital matrix algebras
 11.1 Attributes

  11.1-1 UnderlyingCategoryOfFpAlgebras
  11.1-2 UnderlyingCategoryOfMatrices
  11.1-3 CoefficientsRing
  11.1-4 CoefficientsRing
  11.1-5 DefiningPairOfFinitelyPresentedMatrixAlgebra
  11.1-6 UnderlyingMorphismInCategoryOfFpAlgebras
  11.1-7 ListOfMatrixImages
  11.1-8 NrGenerators
  11.1-9 Generators
  11.1-10 MatrixGenerators
  11.1-11 AssociatedLinearClosureOfPathCategory
  11.1-12 DefiningRelations
  11.1-13 GroebnerBasisOfDefiningRelations
  11.1-14 AssociatedQuotientCategoryOfLinearClosureOfPathCategory
  11.1-15 Dimension
  11.1-16 AmbientAlgebra
 11.2 Constructors

  11.2-1 CategoryOfFpMatrixAlgebras
  11.2-2 CategoryOfFpMatrixAlgebras
 11.3 Operations

  11.3-1 Counit
  11.3-2 Comultiplication
 11.4 GAP categories

  11.4-1 IsCategoryOfFinitelyPresentedMatrixAlgebras
  11.4-2 IsObjectInCategoryOfFpMatrixAlgebras
  11.4-3 IsMorphismInCategoryOfFpMatrixAlgebras
12 Tools
 12.1 Attributes

  12.1-1 Dimension
 12.2 Tools for quivers

  12.2-1 DefiningTripleOfAQuiver
13 Precompilation
14 Examples and Tests
 14.1 Adelman 5 lemma
 14.2 Basics of additive closure
 14.3 Adelman category basics for for additive closure of algebroids
 14.4 AdelmanCategory( AdditiveClosure( Algebroid ) ) vs. AdelmanCategory( QuiverRows )
 14.5 Adelman category basics
 14.6 Adelman snake lemma
 14.7 Exactness conditions
 14.8 Quiver rows bascis
 14.9 Quiver rows over the integers
 14.10 Snake lemma first proof
 14.11 Snake lemma second proof
 14.12 Subobject lattice
 14.13 Adelman category theorem
Index

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