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FreydCategoriesForCAP

Freyd categories - Formal (co)kernels for additive categories

2025.07-02

6 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

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 Abelian constructions

  1.1-1 SomeProjectiveObjectForKernelObject
  1.1-2 EpimorphismFromSomeProjectiveObjectForKernelObject
  1.1-3 EpimorphismFromSomeProjectiveObjectForKernelObjectWithGivenSomeProjectiveObjectForKernelObject
  1.1-4 SomeInjectiveObjectForCokernelObject
  1.1-5 MonomorphismToSomeInjectiveObjectForCokernelObject
  1.1-6 MonomorphismToSomeInjectiveObjectForCokernelObjectWithGivenSomeInjectiveObjectForCokernelObject
2 Additive closure
3 Example on additive closure
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
6 Example on category of rows
 6.1 Simplifications
7 Category of columns
8 Example on category of columns
9 Category of graded rows and category of graded columns
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 Homomorphisms between f.p. functors based on category of rows
 12.11 Homomorphisms between f.p. functors based on category of columns
 12.12 Matrices over ZP K
 12.13 Testing if an object is projective, injective, bijective, and computing projective/injective dimensions
 12.14 Prosets
 12.15 Category of relations
13 Add Functions
 13.1 Available Add functions

  13.1-1 AddEpimorphismFromSomeProjectiveObjectForKernelObject
  13.1-2 AddEpimorphismFromSomeProjectiveObjectForKernelObjectWithGivenSomeProjectiveObjectForKernelObject
  13.1-3 AddMonomorphismToSomeInjectiveObjectForCokernelObject
  13.1-4 AddMonomorphismToSomeInjectiveObjectForCokernelObjectWithGivenSomeInjectiveObjectForCokernelObject
  13.1-5 AddSomeInjectiveObjectForCokernelObject
  13.1-6 AddSomeProjectiveObjectForKernelObject
14 Examples on graded rows and columns
 14.1 Freyd category of graded rows
 14.2 Freyd category of graded columns
15 Example on tensor products in Freyd categories
 15.1 Tensor products for categories of rows
 15.2 Tensor products for categories of columns
16 The CAP category of graded module presentations for CAP by use of Freyd categories
 16.1 CAP categories

  16.1-1 FpGradedLeftModules
  16.1-2 FpGradedRightModules
 16.2 The GAP categories for graded module presentations for CAP

  16.2-1 IsFpGradedLeftOrRightModulesObject
  16.2-2 IsFpGradedLeftModulesObject
  16.2-3 IsFpGradedRightModulesObject
 16.3 The GAP categories for graded module presentation morphisms for CAP

  16.3-1 IsFpGradedLeftOrRightModulesMorphism
  16.3-2 IsFpGradedLeftModulesMorphism
  16.3-3 IsFpGradedRightModulesMorphism
Index

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