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AdditiveClosuresForCAP

Additive closures for pre-abelian categories

2025.07-05

10 July 2025

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

Martin Bies
Email: martin.bies@alumni.uni-heidelberg.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: fabian.zickgraf@uni-siegen.de
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

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

  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 Example on additive closure

3.1 Using matrix data structures

3.2 The category of rows over a Bézout domain is pre-abelian.

4 Category of rows

4.1 Random Methods in Category of Rows

  4.1-1 RandomObjectByList
  4.1-2 RandomObjectByInteger
  4.1-3 RandomMorphismWithFixedSourceAndRangeByList
  4.1-4 RandomMorphismWithFixedSourceAndRangeByInteger
  4.1-5 RandomMorphismWithFixedSourceByList
  4.1-6 RandomMorphismWithFixedSourceByInteger
  4.1-7 RandomMorphismWithFixedRangeByList
  4.1-8 RandomMorphismWithFixedRangeByInteger
  4.1-9 RandomMorphismByList
  4.1-10 RandomMorphismByInteger

4.2 GAP Categories

4.2-1 IsCategoryOfRowsObject

4.3 Supported CAP operations

  4.3-1 CategoryOfRows of an arbitrary ring
  4.3-2 CategoryOfRows of an exterior algebra over a field
  4.3-3 CategoryOfRows of a commutative ring
  4.3-4 CategoryOfRows of a field

5 Example on category of rows

5.1 Constructors of objects

5.2 Constructors of morphisms

5.3 A few categorical constructions for category of rows

5.4 Simplifications

6 Category of columns

6.1 GAP Categories

6.1-1 IsCategoryOfColumnsObject

7 Example on category of columns

7.1 Constructors of objects

7.2 Constructors of morphisms

7.3 A few categorical constructions for category of columns

8 Category of rows and columns over a field

8.1 Abelian operations

9 Category of graded rows and category of graded columns

9.1 Constructors

  9.1-1 CategoryOfGradedColumns
  9.1-2 CategoryOfGradedRows
  9.1-3 GradedRow
  9.1-4 GradedRow
  9.1-5 GradedColumn
  9.1-6 GradedColumn
  9.1-7 GradedRowOrColumnMorphism
  9.1-8 GradedRowOrColumnMorphism

  9.2-1 UnderlyingHomalgGradedRing
  9.2-2 DegreeList
  9.2-3 RankOfObject
  9.2-4 UnderlyingHomalgGradedRing
  9.2-5 UnderlyingHomalgMatrix

9.3 GAP Categories

  9.3-1 IsGradedRowOrColumn
  9.3-2 IsGradedRow
  9.3-3 IsGradedColumn
  9.3-4 IsGradedRowOrColumnMorphism
  9.3-5 IsGradedRowMorphism
  9.3-6 IsGradedColumnMorphism

9.4 Tools to simplify code

  9.4-1 DeduceMapFromMatrixAndRangeForGradedRows
  9.4-2 DeduceSomeMapFromMatrixAndRangeForGradedRows
  9.4-3 DeduceMapFromMatrixAndSourceForGradedRows
  9.4-4 DeduceSomeMapFromMatrixAndSourceForGradedRows
  9.4-5 DeduceMapFromMatrixAndRangeForGradedCols
  9.4-6 DeduceSomeMapFromMatrixAndRangeForGradedCols
  9.4-7 DeduceMapFromMatrixAndSourceForGradedCols
  9.4-8 DeduceSomeMapFromMatrixAndSourceForGradedCols
  9.4-9 UnzipDegreeList

10 Examples on graded rows and columns

10.1 Constructors of objects and reduction of degree lists

10.2 Constructors of morphisms

10.3 The GAP categories

10.4 A few categorical constructions for graded rows

10.5 A few categorical constructions for graded columns

10.6 Additional examples on monoidal structure for graded rows

10.7 Additional examples on monoidal structure for graded columns

10.8 Examples to test Tools methods in graded rows/cols

11 Basic operations

11.1 Weak kernel

  11.1-1 WeakKernelObject
  11.1-2 WeakKernelEmbedding
  11.1-3 WeakKernelEmbeddingWithGivenWeakKernelObject
  11.1-4 WeakKernelLift
  11.1-5 WeakKernelLiftWithGivenWeakKernelObject

11.2 Weak cokernel

  11.2-1 WeakCokernelObject
  11.2-2 WeakCokernelProjection
  11.2-3 WeakCokernelProjectionWithGivenWeakCokernelObject
  11.2-4 WeakCokernelColift
  11.2-5 WeakCokernelColiftWithGivenWeakCokernelObject

11.3 Weak bi-fiber product

  11.3-1 WeakBiFiberProduct
  11.3-2 ProjectionInFirstFactorOfWeakBiFiberProduct
  11.3-3 ProjectionInFirstFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  11.3-4 ProjectionInSecondFactorOfWeakBiFiberProduct
  11.3-5 ProjectionInSecondFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  11.3-6 UniversalMorphismIntoWeakBiFiberProduct
  11.3-7 UniversalMorphismIntoWeakBiFiberProductWithGivenWeakBiFiberProduct
  11.3-8 WeakBiFiberProductMorphismToDirectSum

11.4 Biased weak fiber product

  11.4-1 BiasedWeakFiberProduct
  11.4-2 ProjectionOfBiasedWeakFiberProduct
  11.4-3 ProjectionOfBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
  11.4-4 UniversalMorphismIntoBiasedWeakFiberProduct
  11.4-5 UniversalMorphismIntoBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct

11.5 Weak bi-pushout

  11.5-1 WeakBiPushout
  11.5-2 InjectionOfFirstCofactorOfWeakBiPushout
  11.5-3 InjectionOfSecondCofactorOfWeakBiPushout
  11.5-4 InjectionOfFirstCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  11.5-5 InjectionOfSecondCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  11.5-6 UniversalMorphismFromWeakBiPushout
  11.5-7 UniversalMorphismFromWeakBiPushoutWithGivenWeakBiPushout
  11.5-8 DirectSumMorphismToWeakBiPushout

11.6 Biased weak pushout

  11.6-1 BiasedWeakPushout
  11.6-2 InjectionOfBiasedWeakPushout
  11.6-3 InjectionOfBiasedWeakPushoutWithGivenBiasedWeakPushout
  11.6-4 UniversalMorphismFromBiasedWeakPushout
  11.6-5 UniversalMorphismFromBiasedWeakPushoutWithGivenBiasedWeakPushout

12 Examples and Tests

12.1 Monoidal structure of AdditiveClosure

12.2 Basics based on category of rows

12.3 Basics based on category of columns

12.4 Homomorphism structure

12.5 Opposite( CategoryOfRow ) as CategoryOfColumns

12.7 Rings as Ab-categories

13 Add Functions

13.1 Available Add functions

  13.1-1 AddBiasedWeakFiberProduct
  13.1-2 AddBiasedWeakPushout
  13.1-3 AddDirectSumMorphismToWeakBiPushout
  13.1-4 AddInjectionOfBiasedWeakPushout
  13.1-5 AddInjectionOfBiasedWeakPushoutWithGivenBiasedWeakPushout
  13.1-6 AddInjectionOfFirstCofactorOfWeakBiPushout
  13.1-7 AddInjectionOfFirstCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  13.1-8 AddInjectionOfSecondCofactorOfWeakBiPushout
  13.1-9 AddInjectionOfSecondCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  13.1-10 AddProjectionInFirstFactorOfWeakBiFiberProduct
  13.1-11 AddProjectionInFirstFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  13.1-12 AddProjectionInSecondFactorOfWeakBiFiberProduct
  13.1-13 AddProjectionInSecondFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  13.1-14 AddProjectionOfBiasedWeakFiberProduct
  13.1-15 AddProjectionOfBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
  13.1-16 AddUniversalMorphismFromBiasedWeakPushout
  13.1-17 AddUniversalMorphismFromBiasedWeakPushoutWithGivenBiasedWeakPushout
  13.1-18 AddUniversalMorphismFromWeakBiPushout
  13.1-19 AddUniversalMorphismFromWeakBiPushoutWithGivenWeakBiPushout
  13.1-20 AddUniversalMorphismIntoBiasedWeakFiberProduct
  13.1-21 AddUniversalMorphismIntoBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
  13.1-22 AddUniversalMorphismIntoWeakBiFiberProduct
  13.1-23 AddUniversalMorphismIntoWeakBiFiberProductWithGivenWeakBiFiberProduct
  13.1-24 AddWeakBiFiberProduct
  13.1-25 AddWeakBiFiberProductMorphismToDirectSum
  13.1-26 AddWeakBiPushout
  13.1-27 AddWeakCokernelColift
  13.1-28 AddWeakCokernelColiftWithGivenWeakCokernelObject
  13.1-29 AddWeakCokernelObject
  13.1-30 AddWeakCokernelProjection
  13.1-31 AddWeakCokernelProjectionWithGivenWeakCokernelObject
  13.1-32 AddWeakKernelEmbedding
  13.1-33 AddWeakKernelEmbeddingWithGivenWeakKernelObject
  13.1-34 AddWeakKernelLift
  13.1-35 AddWeakKernelLiftWithGivenWeakKernelObject
  13.1-36 AddWeakKernelObject

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