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AdditiveClosuresForCAP

Additive closures for pre-additive categories

2026.07-01

7 July 2026

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

Martin Bies
Email: martin.bies@rptu.de
Homepage: https://www.ulb.ac.be/sciences/ptm/pmif/people.html
Address:
Physique Théorique et Mathématique
Université Libre de Bruxelles
Campus Plaine - CP 231
Building NO - Level 6 - Office O.6.111
1050 Brussels
Belgium

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

Contents

1 Rings as categories
 1.1 Random methods

  1.1-1 RandomObjectByList
  1.1-2 RandomObjectByInteger
  1.1-3 RandomMorphismWithFixedSourceAndRangeByList
  1.1-4 RandomMorphismWithFixedSourceAndRangeByInteger
  1.1-5 RandomMorphismWithFixedSourceByList
  1.1-6 RandomMorphismWithFixedSourceByInteger
  1.1-7 RandomMorphismWithFixedRangeByList
  1.1-8 RandomMorphismWithFixedRangeByInteger
  1.1-9 RandomMorphismByList
  1.1-10 RandomMorphismByInteger
2 Additive closure
 2.1 Constructors

  2.1-1 AdditiveClosure
  2.1-2 ADDITIVE_CLOSURE
  2.1-3 AdditiveClosureObject
  2.1-4 AsAdditiveClosureObject
  2.1-5 AdditiveClosureMorphism
  2.1-6 AsAdditiveClosureMorphism
  2.1-7 InclusionFunctorInAdditiveClosure
  2.1-8 ExtendFunctorToAdditiveClosures
  2.1-9 ExtendFunctorWithAdditiveRangeToFunctorFromAdditiveClosureOfSource
  2.1-10 ExtendFunctorToAdditiveClosureOfSource
  2.1-11 ExtendNaturalTransformationToAdditiveClosureOfSource
 2.2 Attributes

  2.2-1 UnderlyingCategory
  2.2-2 ObjectList
  2.2-3 MorphismMatrix
  2.2-4 NumberRows
  2.2-5 NumberColumns
 2.3 Operators

  2.3-1 []
  2.3-2 [
  2.3-3 /
  2.3-4 /
 2.4 Global functions

  2.4-1 NullMatImmutable
  2.4-2 UnionOfRowsListList
  2.4-3 UnionOfColumnsListList
 2.5 Random methods for additive closure category

  2.5-1 RandomObjectByList
  2.5-2 RandomObjectByInteger
  2.5-3 RandomMorphismWithFixedSourceAndRangeByList
  2.5-4 RandomMorphismWithFixedSourceAndRangeByInteger
  2.5-5 RandomMorphismWithFixedSourceByList
  2.5-6 RandomMorphismWithFixedSourceByInteger
  2.5-7 RandomMorphismWithFixedRangeByList
  2.5-8 RandomMorphismWithFixedRangeByInteger
  2.5-9 RandomMorphismByList
  2.5-10 RandomMorphismByInteger
 2.6 GAP Categories

  2.6-1 IsAdditiveClosureCategory
  2.6-2 IsAdditiveClosureObject
  2.6-3 IsAdditiveClosureMorphism
3 Reinterpret a linear category over a commutative semiring with free f.g. external Homs
 3.1 Constructors

  3.1-1 LinearCategoryWithMorphismsByCoefficients
 3.2 Attributes

  3.2-1 UnderlyingDatum
  3.2-2 PairOfIntAndListOfCoefficients
 3.3 GAP Categories
 3.4 GAP categories

  3.4-1 IsLinearCategoryWithMorphismsByCoefficients
  3.4-2 IsObjectInLinearCategoryWithMorphismsByCoefficients
  3.4-3 IsMorphismInLinearCategoryWithMorphismsByCoefficients
4 Example on additive closure
 4.1 Using matrix data structures
 4.2 The category of rows over a Bézout domain is pre-abelian.
5 Category of rows
 5.1 Random Methods in Category of Rows

  5.1-1 RandomObjectByList
  5.1-2 RandomObjectByInteger
  5.1-3 RandomMorphismWithFixedSourceAndRangeByList
  5.1-4 RandomMorphismWithFixedSourceAndRangeByInteger
  5.1-5 RandomMorphismWithFixedSourceByList
  5.1-6 RandomMorphismWithFixedSourceByInteger
  5.1-7 RandomMorphismWithFixedRangeByList
  5.1-8 RandomMorphismWithFixedRangeByInteger
  5.1-9 RandomMorphismByList
  5.1-10 RandomMorphismByInteger
 5.2 GAP Categories

  5.2-1 IsCategoryOfRowsObject
 5.3 Supported CAP operations

  5.3-1 CategoryOfRows of an arbitrary ring
  5.3-2 CategoryOfRows of an exterior algebra over a field
  5.3-3 CategoryOfRows of a commutative ring
  5.3-4 CategoryOfRows of a field
6 Example on category of rows
 6.1 Constructors of objects
 6.2 Constructors of morphisms
 6.3 A few categorical constructions for category of rows
 6.4 Simplifications
7 Category of columns
 7.1 GAP Categories

  7.1-1 IsCategoryOfColumnsObject
8 Example on category of columns
 8.1 Constructors of objects
 8.2 Constructors of morphisms
 8.3 A few categorical constructions for category of columns
9 Category of rows and columns over a field
 9.1 Abelian operations
10 Category of graded rows and category of graded columns
 10.1 Constructors

  10.1-1 CategoryOfGradedColumns
  10.1-2 CategoryOfGradedRows
  10.1-3 GradedRow
  10.1-4 GradedRow
  10.1-5 GradedColumn
  10.1-6 GradedColumn
  10.1-7 GradedRowOrColumnMorphism
  10.1-8 GradedRowOrColumnMorphism
 10.2 Attributes

  10.2-1 UnderlyingHomalgGradedRing
  10.2-2 DegreeList
  10.2-3 RankOfObject
  10.2-4 UnderlyingHomalgGradedRing
  10.2-5 UnderlyingHomalgMatrix
 10.3 GAP Categories

  10.3-1 IsGradedRowOrColumn
  10.3-2 IsGradedRow
  10.3-3 IsGradedColumn
  10.3-4 IsGradedRowOrColumnMorphism
  10.3-5 IsGradedRowMorphism
  10.3-6 IsGradedColumnMorphism
 10.4 Tools to simplify code

  10.4-1 DeduceMapFromMatrixAndRangeForGradedRows
  10.4-2 DeduceSomeMapFromMatrixAndRangeForGradedRows
  10.4-3 DeduceMapFromMatrixAndSourceForGradedRows
  10.4-4 DeduceSomeMapFromMatrixAndSourceForGradedRows
  10.4-5 DeduceMapFromMatrixAndRangeForGradedCols
  10.4-6 DeduceSomeMapFromMatrixAndRangeForGradedCols
  10.4-7 DeduceMapFromMatrixAndSourceForGradedCols
  10.4-8 DeduceSomeMapFromMatrixAndSourceForGradedCols
  10.4-9 UnzipDegreeList
11 Examples on graded rows and columns
 11.1 Constructors of objects and reduction of degree lists
 11.2 Constructors of morphisms
 11.3 The GAP categories
 11.4 A few categorical constructions for graded rows
 11.5 A few categorical constructions for graded columns
 11.6 Additional examples on monoidal structure for graded rows
 11.7 Additional examples on monoidal structure for graded columns
 11.8 Examples to test Tools methods in graded rows/cols
12 Basic operations
 12.1 Weak kernel

  12.1-1 WeakKernelObject
  12.1-2 WeakKernelEmbedding
  12.1-3 WeakKernelEmbeddingWithGivenWeakKernelObject
  12.1-4 WeakKernelLift
  12.1-5 WeakKernelLiftWithGivenWeakKernelObject
 12.2 Weak cokernel

  12.2-1 WeakCokernelObject
  12.2-2 WeakCokernelProjection
  12.2-3 WeakCokernelProjectionWithGivenWeakCokernelObject
  12.2-4 WeakCokernelColift
  12.2-5 WeakCokernelColiftWithGivenWeakCokernelObject
 12.3 Weak bi-fiber product

  12.3-1 WeakBiFiberProduct
  12.3-2 ProjectionInFirstFactorOfWeakBiFiberProduct
  12.3-3 ProjectionInFirstFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  12.3-4 ProjectionInSecondFactorOfWeakBiFiberProduct
  12.3-5 ProjectionInSecondFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  12.3-6 UniversalMorphismIntoWeakBiFiberProduct
  12.3-7 UniversalMorphismIntoWeakBiFiberProductWithGivenWeakBiFiberProduct
  12.3-8 WeakBiFiberProductMorphismToDirectSum
 12.4 Biased weak fiber product

  12.4-1 BiasedWeakFiberProduct
  12.4-2 ProjectionOfBiasedWeakFiberProduct
  12.4-3 ProjectionOfBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
  12.4-4 UniversalMorphismIntoBiasedWeakFiberProduct
  12.4-5 UniversalMorphismIntoBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
 12.5 Weak bi-pushout

  12.5-1 WeakBiPushout
  12.5-2 InjectionOfFirstCofactorOfWeakBiPushout
  12.5-3 InjectionOfSecondCofactorOfWeakBiPushout
  12.5-4 InjectionOfFirstCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  12.5-5 InjectionOfSecondCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  12.5-6 UniversalMorphismFromWeakBiPushout
  12.5-7 UniversalMorphismFromWeakBiPushoutWithGivenWeakBiPushout
  12.5-8 DirectSumMorphismToWeakBiPushout
 12.6 Biased weak pushout

  12.6-1 BiasedWeakPushout
  12.6-2 InjectionOfBiasedWeakPushout
  12.6-3 InjectionOfBiasedWeakPushoutWithGivenBiasedWeakPushout
  12.6-4 UniversalMorphismFromBiasedWeakPushout
  12.6-5 UniversalMorphismFromBiasedWeakPushoutWithGivenBiasedWeakPushout
13 Examples and Tests
 13.1 Monoidal structure of AdditiveClosure
 13.2 Basics based on category of rows
 13.3 Basics based on category of columns
 13.4 Homomorphism structure
 13.5 Opposite( CategoryOfRow ) as CategoryOfColumns
 13.6 Tests
 13.7 Rings as Ab-categories
14 Add Functions
 14.1 Available Add functions

  14.1-1 AddBiasedWeakFiberProduct
  14.1-2 AddBiasedWeakPushout
  14.1-3 AddDirectSumMorphismToWeakBiPushout
  14.1-4 AddInjectionOfBiasedWeakPushout
  14.1-5 AddInjectionOfBiasedWeakPushoutWithGivenBiasedWeakPushout
  14.1-6 AddInjectionOfFirstCofactorOfWeakBiPushout
  14.1-7 AddInjectionOfFirstCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  14.1-8 AddInjectionOfSecondCofactorOfWeakBiPushout
  14.1-9 AddInjectionOfSecondCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  14.1-10 AddProjectionInFirstFactorOfWeakBiFiberProduct
  14.1-11 AddProjectionInFirstFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  14.1-12 AddProjectionInSecondFactorOfWeakBiFiberProduct
  14.1-13 AddProjectionInSecondFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  14.1-14 AddProjectionOfBiasedWeakFiberProduct
  14.1-15 AddProjectionOfBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
  14.1-16 AddUniversalMorphismFromBiasedWeakPushout
  14.1-17 AddUniversalMorphismFromBiasedWeakPushoutWithGivenBiasedWeakPushout
  14.1-18 AddUniversalMorphismFromWeakBiPushout
  14.1-19 AddUniversalMorphismFromWeakBiPushoutWithGivenWeakBiPushout
  14.1-20 AddUniversalMorphismIntoBiasedWeakFiberProduct
  14.1-21 AddUniversalMorphismIntoBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
  14.1-22 AddUniversalMorphismIntoWeakBiFiberProduct
  14.1-23 AddUniversalMorphismIntoWeakBiFiberProductWithGivenWeakBiFiberProduct
  14.1-24 AddWeakBiFiberProduct
  14.1-25 AddWeakBiFiberProductMorphismToDirectSum
  14.1-26 AddWeakBiPushout
  14.1-27 AddWeakCokernelColift
  14.1-28 AddWeakCokernelColiftWithGivenWeakCokernelObject
  14.1-29 AddWeakCokernelObject
  14.1-30 AddWeakCokernelProjection
  14.1-31 AddWeakCokernelProjectionWithGivenWeakCokernelObject
  14.1-32 AddWeakKernelEmbedding
  14.1-33 AddWeakKernelEmbeddingWithGivenWeakKernelObject
  14.1-34 AddWeakKernelLift
  14.1-35 AddWeakKernelLiftWithGivenWeakKernelObject
  14.1-36 AddWeakKernelObject
Index

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