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FreydCategoriesForCAP

Freyd categories - Formal (co)kernels for additive categories

2025.06-06

27 June 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

Fabian Zickgraf
Email: fabian.zickgraf@uni-siegen.de
Homepage: https://github.com/zickgraf/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

1 Basic operations
 1.1 Weak kernel

  1.1-1 WeakKernelObject
  1.1-2 WeakKernelEmbedding
  1.1-3 WeakKernelEmbeddingWithGivenWeakKernelObject
  1.1-4 WeakKernelLift
  1.1-5 WeakKernelLiftWithGivenWeakKernelObject
 1.2 Weak cokernel

  1.2-1 WeakCokernelObject
  1.2-2 WeakCokernelProjection
  1.2-3 WeakCokernelProjectionWithGivenWeakCokernelObject
  1.2-4 WeakCokernelColift
  1.2-5 WeakCokernelColiftWithGivenWeakCokernelObject
 1.3 Weak bi-fiber product

  1.3-1 WeakBiFiberProduct
  1.3-2 ProjectionInFirstFactorOfWeakBiFiberProduct
  1.3-3 ProjectionInFirstFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  1.3-4 ProjectionInSecondFactorOfWeakBiFiberProduct
  1.3-5 ProjectionInSecondFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  1.3-6 UniversalMorphismIntoWeakBiFiberProduct
  1.3-7 UniversalMorphismIntoWeakBiFiberProductWithGivenWeakBiFiberProduct
  1.3-8 WeakBiFiberProductMorphismToDirectSum
 1.4 Biased weak fiber product

  1.4-1 BiasedWeakFiberProduct
  1.4-2 ProjectionOfBiasedWeakFiberProduct
  1.4-3 ProjectionOfBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
  1.4-4 UniversalMorphismIntoBiasedWeakFiberProduct
  1.4-5 UniversalMorphismIntoBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
 1.5 Weak bi-pushout

  1.5-1 WeakBiPushout
  1.5-2 InjectionOfFirstCofactorOfWeakBiPushout
  1.5-3 InjectionOfSecondCofactorOfWeakBiPushout
  1.5-4 InjectionOfFirstCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  1.5-5 InjectionOfSecondCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  1.5-6 UniversalMorphismFromWeakBiPushout
  1.5-7 UniversalMorphismFromWeakBiPushoutWithGivenWeakBiPushout
  1.5-8 DirectSumMorphismToWeakBiPushout
 1.6 Biased weak pushout

  1.6-1 BiasedWeakPushout
  1.6-2 InjectionOfBiasedWeakPushout
  1.6-3 InjectionOfBiasedWeakPushoutWithGivenBiasedWeakPushout
  1.6-4 UniversalMorphismFromBiasedWeakPushout
  1.6-5 UniversalMorphismFromBiasedWeakPushoutWithGivenBiasedWeakPushout
 1.7 Abelian constructions

  1.7-1 SomeProjectiveObjectForKernelObject
  1.7-2 EpimorphismFromSomeProjectiveObjectForKernelObject
  1.7-3 EpimorphismFromSomeProjectiveObjectForKernelObjectWithGivenSomeProjectiveObjectForKernelObject
  1.7-4 SomeInjectiveObjectForCokernelObject
  1.7-5 MonomorphismToSomeInjectiveObjectForCokernelObject
  1.7-6 MonomorphismToSomeInjectiveObjectForCokernelObjectWithGivenSomeInjectiveObjectForCokernelObject
2 Additive closure
 2.1 GAP Categories

  2.1-1 IsAdditiveClosureCategory
  2.1-2 IsAdditiveClosureObject
  2.1-3 IsAdditiveClosureMorphism
 2.2 Constructors

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

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

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

  2.6-1 NullMatImmutable
  2.6-2 UnionOfRowsListList
  2.6-3 UnionOfColumnsListList
3 Example on additive closure
 3.1 Using matrix data structures
4 Adelman category
 4.1 GAP Categories

  4.1-1 IsAdelmanCategoryObject
  4.1-2 IsAdelmanCategoryMorphism
  4.1-3 IsAdelmanCategory
 4.2 Constructors

  4.2-1 AdelmanCategory
  4.2-2 AdelmanCategoryObject
  4.2-3 AdelmanCategoryMorphism
  4.2-4 AsAdelmanCategoryObject
  4.2-5 AsAdelmanCategoryMorphism
  4.2-6 /
  4.2-7 /
 4.3 Attributes and Properties

  4.3-1 UnderlyingCategory
  4.3-2 RelationMorphism
  4.3-3 CorelationMorphism
  4.3-4 UnderlyingMorphism
  4.3-5 RelationWitness
  4.3-6 CorelationWitness
  4.3-7 WitnessPairForBeingCongruentToZero
  4.3-8 MereExistenceOfWitnessPairForBeingCongruentToZero
  4.3-9 IsSequenceAsAdelmanCategoryObject
 4.4 Exactness properties: objects as functors

  4.4-1 IsExact
  4.4-2 IsLeftExact
  4.4-3 IsRightExact
  4.4-4 IsMonoPreserving
  4.4-5 IsEpiPreserving
  4.4-6 IsImagePreserving
5 Category of rows
 5.1 Supported CAP operations

  5.1-1 CategoryOfRows of an arbitrary ring
  5.1-2 CategoryOfRows of an exterior algebra over a field
  5.1-3 CategoryOfRows of a commutative ring
  5.1-4 CategoryOfRows of a field
 5.2 GAP Categories

  5.2-1 IsCategoryOfRowsObject
 5.3 Random Methods in Category of Rows

  5.3-1 RandomObjectByList
  5.3-2 RandomObjectByInteger
  5.3-3 RandomMorphismWithFixedSourceAndRangeByList
  5.3-4 RandomMorphismWithFixedSourceAndRangeByInteger
  5.3-5 RandomMorphismWithFixedSourceByList
  5.3-6 RandomMorphismWithFixedSourceByInteger
  5.3-7 RandomMorphismWithFixedRangeByList
  5.3-8 RandomMorphismWithFixedRangeByInteger
  5.3-9 RandomMorphismByList
  5.3-10 RandomMorphismByInteger
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 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 Attributes

  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 Cokernel image closure
11 Freyd category
 11.1 Random methods in Freyd categories

  11.1-1 RandomObjectByList
  11.1-2 RandomObjectByInteger
  11.1-3 RandomMorphismWithFixedSourceAndRangeByList
  11.1-4 RandomMorphismWithFixedSourceAndRangeByInteger
  11.1-5 RandomMorphismWithFixedSourceByList
  11.1-6 RandomMorphismWithFixedSourceByInteger
  11.1-7 RandomMorphismWithFixedRangeByList
  11.1-8 RandomMorphismWithFixedRangeByInteger
  11.1-9 RandomMorphismByList
  11.1-10 RandomMorphismByInteger
 11.2 Internal Hom-Embedding

  11.2-1 INTERNAL_HOM_EMBEDDING
 11.3 Convenient methods for tensor products of freyd objects and morphisms

  11.3-1 *
  11.3-2 ^
  11.3-3 *
  11.3-4 ^
12 Examples and Tests
 12.1 Monoidal structure of AdditiveClosure
 12.2 Adelman category basics for category of rows
 12.3 Adelman category basics for category of columns
 12.4 Basics based on category of rows
 12.5 Basics based on category of columns
 12.6 CoFreyd category
 12.7 Cokernel image closure in category of rows
 12.8 Cokernel image closure in category of columns
 12.9 Adelman category basics
 12.10 Grade filtration
 12.11 Homomorphisms between f.p. functors based on category of rows
 12.12 Homomorphisms between f.p. functors based on category of columns
 12.13 Homomorphism structure
 12.14 Linear closure of categories
 12.15 Matrices over ZP K
 12.16 Matrices over ZG
 12.17 Opposite( CategoryOfRow ) as CategoryOfColumns
 12.18 Testing if an object is projective, injective, bijective, and computing projective/injective dimensions
 12.19 Prosets
 12.20 Tests
 12.21 Category of relations
 12.22 Rings as Ab-categories
13 Linear closure of a category
 13.1 Functors

  13.1-1 ExtendFunctorToLinearClosureOfSource
  13.1-2 ExtendFunctorToLinearClosureOfSource
14 Add Functions
 14.1 Available Add functions

  14.1-1 AddBiasedWeakFiberProduct
  14.1-2 AddBiasedWeakPushout
  14.1-3 AddDirectSumMorphismToWeakBiPushout
  14.1-4 AddEpimorphismFromSomeProjectiveObjectForKernelObject
  14.1-5 AddEpimorphismFromSomeProjectiveObjectForKernelObjectWithGivenSomeProjectiveObjectForKernelObject
  14.1-6 AddInjectionOfBiasedWeakPushout
  14.1-7 AddInjectionOfBiasedWeakPushoutWithGivenBiasedWeakPushout
  14.1-8 AddInjectionOfFirstCofactorOfWeakBiPushout
  14.1-9 AddInjectionOfFirstCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  14.1-10 AddInjectionOfSecondCofactorOfWeakBiPushout
  14.1-11 AddInjectionOfSecondCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  14.1-12 AddMonomorphismToSomeInjectiveObjectForCokernelObject
  14.1-13 AddMonomorphismToSomeInjectiveObjectForCokernelObjectWithGivenSomeInjectiveObjectForCokernelObject
  14.1-14 AddProjectionInFirstFactorOfWeakBiFiberProduct
  14.1-15 AddProjectionInFirstFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  14.1-16 AddProjectionInSecondFactorOfWeakBiFiberProduct
  14.1-17 AddProjectionInSecondFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  14.1-18 AddProjectionOfBiasedWeakFiberProduct
  14.1-19 AddProjectionOfBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
  14.1-20 AddSomeInjectiveObjectForCokernelObject
  14.1-21 AddSomeProjectiveObjectForKernelObject
  14.1-22 AddUniversalMorphismFromBiasedWeakPushout
  14.1-23 AddUniversalMorphismFromBiasedWeakPushoutWithGivenBiasedWeakPushout
  14.1-24 AddUniversalMorphismFromWeakBiPushout
  14.1-25 AddUniversalMorphismFromWeakBiPushoutWithGivenWeakBiPushout
  14.1-26 AddUniversalMorphismIntoBiasedWeakFiberProduct
  14.1-27 AddUniversalMorphismIntoBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
  14.1-28 AddUniversalMorphismIntoWeakBiFiberProduct
  14.1-29 AddUniversalMorphismIntoWeakBiFiberProductWithGivenWeakBiFiberProduct
  14.1-30 AddWeakBiFiberProduct
  14.1-31 AddWeakBiFiberProductMorphismToDirectSum
  14.1-32 AddWeakBiPushout
  14.1-33 AddWeakCokernelColift
  14.1-34 AddWeakCokernelColiftWithGivenWeakCokernelObject
  14.1-35 AddWeakCokernelObject
  14.1-36 AddWeakCokernelProjection
  14.1-37 AddWeakCokernelProjectionWithGivenWeakCokernelObject
  14.1-38 AddWeakKernelEmbedding
  14.1-39 AddWeakKernelEmbeddingWithGivenWeakKernelObject
  14.1-40 AddWeakKernelLift
  14.1-41 AddWeakKernelLiftWithGivenWeakKernelObject
  14.1-42 AddWeakKernelObject
15 Rings as categories
 15.1 Random methods

  15.1-1 RandomObjectByList
  15.1-2 RandomObjectByInteger
  15.1-3 RandomMorphismWithFixedSourceAndRangeByList
  15.1-4 RandomMorphismWithFixedSourceAndRangeByInteger
  15.1-5 RandomMorphismWithFixedSourceByList
  15.1-6 RandomMorphismWithFixedSourceByInteger
  15.1-7 RandomMorphismWithFixedRangeByList
  15.1-8 RandomMorphismWithFixedRangeByInteger
  15.1-9 RandomMorphismByList
  15.1-10 RandomMorphismByInteger
16 Examples on graded rows and columns
 16.1 Freyd category of graded rows
 16.2 Freyd category of graded columns
 16.3 Constructors of objects and reduction of degree lists
 16.4 Constructors of morphisms
 16.5 The GAP categories
 16.6 A few categorical constructions for graded rows
 16.7 A few categorical constructions for graded columns
 16.8 Additional examples on monoidal structure for graded rows
 16.9 Additional examples on monoidal structure for graded columns
 16.10 Examples to test Tools methods in graded rows/cols
17 Category of rows and columns over a field
 17.1 Abelian operations for rows
18 Example on tensor products in Freyd categories
 18.1 Tensor products for categories of rows
 18.2 Tensor products for categories of columns
19 The CAP category of graded module presentations for CAP by use of Freyd categories
 19.1 CAP categories

  19.1-1 FpGradedLeftModules
  19.1-2 FpGradedRightModules
 19.2 The GAP categories for graded module presentations for CAP

  19.2-1 IsFpGradedLeftOrRightModulesObject
  19.2-2 IsFpGradedLeftModulesObject
  19.2-3 IsFpGradedRightModulesObject
 19.3 The GAP categories for graded module presentation morphisms for CAP

  19.3-1 IsFpGradedLeftOrRightModulesMorphism
  19.3-2 IsFpGradedLeftModulesMorphism
  19.3-3 IsFpGradedRightModulesMorphism
Index

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