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Algebroids

Algebroids and bialgebroids as preadditive categories generated by enhanced quivers

2026.06-01

7 June 2026

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 \/
  2.4-5 CategoryFromDataTables
 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 Algebroids as preadditive categories generated by enhanced quivers
 4.1 Properties

  4.1-1 IsFinitelyPresentedLinearCategory
  4.1-2 IsCommutative
  4.1-3 IsCounitary
  4.1-4 IsCoassociative
 4.2 Attributes

  4.2-1 UnderlyingQuiver
  4.2-2 UnderlyingQuiverAlgebra
  4.2-3 Dimension
  4.2-4 BasisPathsByVertexIndex
  4.2-5 BasisMorphismsByVertexIndex
  4.2-6 HomStructureOnBasisPaths
  4.2-7 AssignSetOfObjects
  4.2-8 SetOfGeneratingMorphisms
  4.2-9 SetOfGeneratingMorphisms
  4.2-10 SetOfGeneratingMorphisms
  4.2-11 AssignSetOfGeneratingMorphisms
  4.2-12 RelationsOfAlgebroid
  4.2-13 OppositeAlgebroid
  4.2-14 Antipode
  4.2-15 UnderlyingVertex
  4.2-16 UnderlyingQuiverAlgebraElement
  4.2-17 UnderlyingAlgebra
  4.2-18 Parity
  4.2-19 POW
  4.2-20 DataTablesOfCategory
  4.2-21 IsomorphismOntoAlgebroidFromDataTables
  4.2-22 IsomorphismFromAlgebroidFromDataTables
 4.3 Operations

  4.3-1 DecomposeQuiverAlgebraElement
  4.3-2 ApplyToQuiverAlgebraElement
  4.3-3 TrivialAlgebroid
  4.3-4 \*
  4.3-5 ElementaryTensor
  4.3-6 ElementaryTensor
  4.3-7 ElementaryTensor
 4.4 Constructors

  4.4-1 Algebroid
  4.4-2 DescentToZDefinedByBasisPaths
  4.4-3 CapFunctor
  4.4-4 ObjectInAlgebroid
  4.4-5 \/
  4.4-6 MorphismInAlgebroid
  4.4-7 \/
  4.4-8 POW
 4.5 GAP categories

  4.5-1 IsCellInAlgebroid
  4.5-2 IsObjectInAlgebroid
  4.5-3 IsMorphismInAlgebroid
  4.5-4 IsAlgebroid
  4.5-5 IsAlgebraAsCategory
  4.5-6 IsAlgebroidMorphism
5 Algebroids from data tables
 5.1 GAP categories
6 Category of algebroids
 6.1 Properties

  6.1-1 IsCommutative
  6.1-2 IsCounitary
  6.1-3 IsCoassociative
 6.2 Attributes

  6.2-1 AsCapCategory
  6.2-2 AsCapFunctor
 6.3 Operations

  6.3-1 Twist
 6.4 Constructors

  6.4-1 CategoryOfAlgebroids
  6.4-2 CategoryOfAlgebroidsObject
  6.4-3 CategoryOfAlgebroidsMorphism
  6.4-4 CategoryOfAlgebroidsMorphism
 6.5 GAP categories

  6.5-1 IsCategoryOfAlgebroidsObject
  6.5-2 IsCategoryOfAlgebroidsMorphism
7 Bialgebroids as algebroids with more structure
 7.1 Properties

  7.1-1 IsCocommutative
  7.1-2 IsHopfAlgebroid
 7.2 Attributes

  7.2-1 Counit
  7.2-2 Comultiplication
 7.3 Constructors

  7.3-1 AddBialgebroidStructure
  7.3-2 AddAntipode
 7.4 GAP categories

  7.4-1 IsBialgebroid
8 Linear closures of path categories and their quotients
 8.1 Constructors
9 Category of finitely presented associative unital algebras
 9.1 Attributes

  9.1-1 UnderlyingCategoryOfMatrices
  9.1-2 CoefficientsRing
  9.1-3 CoefficientsRing
  9.1-4 DefiningSeptupleOfFinitelyPresentedAlgebra
  9.1-5 ListOfImages
  9.1-6 NrGenerators
  9.1-7 Generators
  9.1-8 AssociatedLinearClosureOfPathCategory
  9.1-9 DefiningRelations
  9.1-10 GroebnerBasisOfDefiningRelations
  9.1-11 AssociatedQuotientCategoryOfLinearClosureOfPathCategory
  9.1-12 Dimension
  9.1-13 AmbientAlgebra
 9.2 Constructors

  9.2-1 CategoryOfFpAlgebras
  9.2-2 CategoryOfFpAlgebras
 9.3 Operations

  9.3-1 EvaluateFpAlgebraMorphism
  9.3-2 Counit
  9.3-3 Comultiplication
  9.3-4 AssociatedFunctorOfLinearClosuresOfPathCategoriesData
 9.4 GAP categories

  9.4-1 IsCategoryOfFpAlgebras
  9.4-2 IsObjectInCategoryOfFpAlgebras
  9.4-3 IsMorphismInCategoryOfFpAlgebras
10 Category of finitely presented associative unital matrix algebras
 10.1 Attributes

  10.1-1 UnderlyingCategory
  10.1-2 UnderlyingCategoryOfMatrices
  10.1-3 CoefficientsRing
  10.1-4 CoefficientsRing
  10.1-5 DefiningPairOfFinitelyPresentedMatrixAlgebra
  10.1-6 UnderlyingFpAlgebra
  10.1-7 UnderlyingListOfMorphismsInCategoryOfFpAlgebras
  10.1-8 ListOfMatrixImages
  10.1-9 NrGenerators
  10.1-10 Generators
  10.1-11 MatrixGenerators
  10.1-12 AssociatedLinearClosureOfPathCategory
  10.1-13 DefiningRelations
  10.1-14 GroebnerBasisOfDefiningRelations
  10.1-15 AssociatedQuotientCategoryOfLinearClosureOfPathCategory
  10.1-16 Dimension
  10.1-17 AmbientAlgebra
 10.2 Constructors

  10.2-1 CategoryOfFpMatrixAlgebras
  10.2-2 CategoryOfFpMatrixAlgebras
 10.3 Operations

  10.3-1 Counit
  10.3-2 Comultiplication
 10.4 GAP categories

  10.4-1 IsCategoryOfFpMatrixAlgebras
  10.4-2 IsObjectInCategoryOfFpMatrixAlgebras
  10.4-3 IsMorphismInCategoryOfFpMatrixAlgebras
11 Tools
 11.1 Attributes

  11.1-1 Dimension
 11.2 Tools for quivers

  11.2-1 DefiningTripleOfAQuiver
12 Precompilation
13 Examples and Tests
 13.1 Adelman 5 lemma
 13.2 Basics of additive closure
 13.3 Adelman category basics for for additive closure of algebroids
 13.4 AdelmanCategory( AdditiveClosure( Algebroid ) ) vs. AdelmanCategory( QuiverRows )
 13.5 Adelman category basics
 13.6 Adelman snake lemma
 13.7 Exactness conditions
 13.8 Quiver rows bascis
 13.9 Quiver rows over the integers
 13.10 Snake lemma first proof
 13.11 Snake lemma second proof
 13.12 Subobject lattice
 13.13 Adelman category theorem
Index

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