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Algebroids

Algebroids and bialgebroids as preadditive categories generated by enhanced quivers

2025.06-03

13 June 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

1.1 The natural numbers as a f.g. monoid

1.2 The integers as f.g. monoid

1.3 A f.p. category

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

1.13 Associators

2 Finitely presented categories generated by enhanced quivers

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

  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

  4.1-1 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

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

  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

6.1-1 IsAdmissibleAlgebroid

  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

  6.3-1 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

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

7.2-1 AsCapCategory
7.2-2 AsCapFunctor

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

8.1-1 IsCocommutative
8.1-2 IsHopfAlgebroid

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

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

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