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FreydCategoriesForCAP

Freyd categories - Formal (co)kernels for additive categories

2024.04-01

4 April 2024

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
  1.1-6 AddWeakKernelObject
  1.1-7 AddWeakKernelEmbedding
  1.1-8 AddWeakKernelEmbeddingWithGivenWeakKernelObject
  1.1-9 AddWeakKernelLift
  1.1-10 AddWeakKernelLiftWithGivenWeakKernelObject

1.2 Weak cokernel

  1.2-1 WeakCokernelObject
  1.2-2 WeakCokernelProjection
  1.2-3 WeakCokernelProjectionWithGivenWeakCokernelObject
  1.2-4 WeakCokernelColift
  1.2-5 WeakCokernelColiftWithGivenWeakCokernelObject
  1.2-6 AddWeakCokernelObject
  1.2-7 AddWeakCokernelProjection
  1.2-8 AddWeakCokernelProjectionWithGivenWeakCokernelObject
  1.2-9 AddWeakCokernelColift
  1.2-10 AddWeakCokernelColiftWithGivenWeakCokernelObject

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
  1.3-9 AddWeakBiFiberProduct
  1.3-10 AddProjectionInFirstFactorOfWeakBiFiberProduct
  1.3-11 AddProjectionInSecondFactorOfWeakBiFiberProduct
  1.3-12 AddProjectionInFirstFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  1.3-13 AddProjectionInSecondFactorOfWeakBiFiberProductWithGivenWeakBiFiberProduct
  1.3-14 AddUniversalMorphismIntoWeakBiFiberProduct
  1.3-15 AddUniversalMorphismIntoWeakBiFiberProductWithGivenWeakBiFiberProduct
  1.3-16 AddWeakBiFiberProductMorphismToDirectSum

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
  1.4-6 AddBiasedWeakFiberProduct
  1.4-7 AddProjectionOfBiasedWeakFiberProduct
  1.4-8 AddProjectionOfBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct
  1.4-9 AddUniversalMorphismIntoBiasedWeakFiberProduct
  1.4-10 AddUniversalMorphismIntoBiasedWeakFiberProductWithGivenBiasedWeakFiberProduct

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
  1.5-9 AddWeakBiPushout
  1.5-10 AddInjectionOfFirstCofactorOfWeakBiPushout
  1.5-11 AddInjectionOfSecondCofactorOfWeakBiPushout
  1.5-12 AddInjectionOfFirstCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  1.5-13 AddInjectionOfSecondCofactorOfWeakBiPushoutWithGivenWeakBiPushout
  1.5-14 AddUniversalMorphismFromWeakBiPushout
  1.5-15 AddUniversalMorphismFromWeakBiPushoutWithGivenWeakBiPushout
  1.5-16 AddDirectSumMorphismToWeakBiPushout

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
  1.6-6 AddBiasedWeakPushout
  1.6-7 AddInjectionOfBiasedWeakPushout
  1.6-8 AddInjectionOfBiasedWeakPushoutWithGivenBiasedWeakPushout
  1.6-9 AddUniversalMorphismFromBiasedWeakPushout
  1.6-10 AddUniversalMorphismFromBiasedWeakPushoutWithGivenBiasedWeakPushout

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
  1.7-7 AddSomeProjectiveObjectForKernelObject
  1.7-8 AddSomeInjectiveObjectForCokernelObject
  1.7-9 AddEpimorphismFromSomeProjectiveObjectForKernelObject
  1.7-10 AddMonomorphismToSomeInjectiveObjectForCokernelObject
  1.7-11 AddEpimorphismFromSomeProjectiveObjectForKernelObjectWithGivenSomeProjectiveObjectForKernelObject
  1.7-12 AddMonomorphismToSomeInjectiveObjectForCokernelObjectWithGivenSomeInjectiveObjectForCokernelObject

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

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

  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

  9.2-1 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 Adelman category basics for category of rows

12.2 Adelman category basics for category of columns

12.3 Basics based on category of rows

12.4 Basics based on category of columns

12.5 CoFreyd category

12.6 Cokernel image closure in category of rows

12.7 Cokernel image closure in category of columns

12.8 Adelman category basics

12.9 Grade filtration

12.10 Groups as categories

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 Linear closure of categories

12.14 Matrices over ZP K

12.15 Matrices over ZG

12.16 Opposite( CategoryOfRow ) as CategoryOfColumns

12.17 Testing if an object is projective, injective, bijective, and computing projective/injective dimensions

12.20 Category of relations

12.21 Rings as Ab-categories

13 Linear closure of a category

13.1-1 ExtendFunctorToLinearClosureOfSource
13.1-2 ExtendFunctorToLinearClosureOfSource

14 Rings as categories

14.1 Random methods

  14.1-1 RandomObjectByList
  14.1-2 RandomObjectByInteger
  14.1-3 RandomMorphismWithFixedSourceAndRangeByList
  14.1-4 RandomMorphismWithFixedSourceAndRangeByInteger
  14.1-5 RandomMorphismWithFixedSourceByList
  14.1-6 RandomMorphismWithFixedSourceByInteger
  14.1-7 RandomMorphismWithFixedRangeByList
  14.1-8 RandomMorphismWithFixedRangeByInteger
  14.1-9 RandomMorphismByList
  14.1-10 RandomMorphismByInteger

15 Examples on graded rows and columns

15.1 Freyd category of graded rows

15.2 Freyd category of graded columns

15.3 Constructors of objects and reduction of degree lists

15.4 Constructors of morphisms

15.5 The GAP categories

15.6 A few categorical constructions for graded rows

15.7 A few categorical constructions for graded columns

15.8 Additional examples on monoidal structure for graded rows

15.9 Additional examples on monoidal structure for graded columns

15.10 Examples to test Tools methods in graded rows/cols

16 Category of rows and columns over a field

16.1 Abelian operations for rows

17 Example on tensor products in Freyd categories

17.1 Tensor products for categories of rows

17.2 Tensor products for categories of columns

18 The CAP category of graded module presentations for CAP by use of Freyd categories

18.1 CAP categories

18.1-1 FpGradedLeftModules
18.1-2 FpGradedRightModules

18.2 The GAP categories for graded module presentations for CAP

  18.2-1 IsFpGradedLeftOrRightModulesObject
  18.2-2 IsFpGradedLeftModulesObject
  18.2-3 IsFpGradedRightModulesObject

18.3 The GAP categories for graded module presentation morphisms for CAP

  18.3-1 IsFpGradedLeftOrRightModulesMorphism
  18.3-2 IsFpGradedLeftModulesMorphism
  18.3-3 IsFpGradedRightModulesMorphism

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