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Algebroids

Algebroids and bialgebroids as preadditive categories generated by enhanced quivers

2024.04-01

22 April 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 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 The full subcategory of the simplicial category truncated in degree 2
 1.5 Algebroids
 1.6 Bialgebroids
 1.7 Hopf algebra generated by group-like elements
 1.8 Hopf algebra generated by a primitive element
 1.9 A non-counitary coalgebra
 1.10 Sweedlers 4-dimensional Hopf algebra
 1.11 Homomorphism structure
 1.12 Algebroids over the integers
 1.13 Unitors
 1.14 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 NerveTruncatedInDegree2AsFunctor
  2.2-21 YonedaFibrationAsNaturalTransformation
  2.2-22 YonedaProjectionAsNaturalEpimorphism
  2.2-23 YonedaCompositionAsNaturalEpimorphism
  2.2-24 TruthMorphismOfTrueToSieveFunctor
  2.2-25 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 Global functions

  2.5-1 SimplicialCategoryTruncatedInDegree
 2.6 GAP categories

  2.6-1 IsFpCategory
  2.6-2 IsMonoidAsCategory
  2.6-3 IsCellInFpCategory
  2.6-4 IsObjectInFpCategory
  2.6-5 IsMorphismInFpCategory
3 Finite categories from nerve data
 3.1 Attributes

  3.1-1 NerveData
  3.1-2 Size
  3.1-3 OppositeCategoryFromNerveData
 3.2 Constructors

  3.2-1 CategoryFromNerveData
  3.2-2 CategoryFromNerveData
  3.2-3 CategoryFromNerveData
  3.2-4 CreateObject
 3.3 GAP categories

  3.3-1 IsCategoryFromNerveData
  3.3-2 IsCellInCategoryFromNerveData
  3.3-3 IsObjectInCategoryFromNerveData
  3.3-4 IsMorphismInCategoryFromNerveData
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 CreateObject
  4.2-5 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 NaturalTransformation
  5.4-5 ObjectInAlgebroid
  5.4-6 \/
  5.4-7 MorphismInAlgebroid
  5.4-8 \/
  5.4-9 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 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 Tools
 11.1 Tools for quivers

  11.1-1 DefiningTripleOfAQuiver
 11.2 Tools for categories

  11.2-1 NerveTruncatedInDegree2Data
  11.2-2 IndicesOfGeneratingMorphismsFromHomStructure
  11.2-3 OppositeFiniteCategory
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
Index

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