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Algebroids

Algebroids and bialgebroids as preadditive categories generated by enhanced quivers

2024.11-01

9 November 2024

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 Precompilation

10 Linear closures of path categories and their quotients

10.1 Constructors

  10.1-1 LinearClosure
  10.1-2 GroebnerBasis
  10.1-3 ReducedGroebnerBasis
  10.1-4 QuotientCategory

11.1 Attributes

11.1-1 Dimension

11.2 Tools for quivers

11.2-1 DefiningTripleOfAQuiver

12 Examples and Tests

12.1 Adelman 5 lemma

12.2 Basics of additive closure

12.3 Adelman category basics for for additive closure of algebroids

12.4 AdelmanCategory( AdditiveClosure( Algebroid ) ) vs. AdelmanCategory( QuiverRows )

12.5 Adelman category basics

12.6 Adelman snake lemma

12.7 Exactness conditions

12.8 Quiver rows bascis

12.9 Quiver rows over the integers

12.10 Snake lemma first proof

12.11 Snake lemma second proof

12.12 Subobject lattice

12.13 Adelman category theorem

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