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AdditiveClosuresForCAP

Additive closures for pre-abelian categories

2025.07-03

3 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
2 Additive closure
3 Example on additive closure
4 Category of rows
5 Example on category of rows
6 Category of columns
7 Example on category of columns
8 Category of rows and columns over a field
9 Category of graded rows and category of graded columns
10 Examples on graded rows and columns
11 Basic operations
12 Examples and Tests
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
Index

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