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FinGSetsForCAP

The (skeletal) elementary topos of finite G-sets

2026.01-03

18 January 2026

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Julia Mickisch
Email: julia.mickisch@student.uni-siegen.de
Homepage: https://github.com/juliamick/
Address:
Walter-Flex-Str. 3
57068 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 Mathematical background

2 The skeletal category of finite right $G$-sets

2.1-1 AsList
2.1-2 UnderlyingGroup

2.2 Constructors

  2.2-1 FinRightGSet
  2.2-2 MapOfFinGSets
  2.2-3 SkeletalCategoryOfFiniteRightGSets

  2.3-1 Skeletal IsEqualForObjects
  2.3-2 Skeletal PreCompose
  2.3-3 Skeletal IdentityMorphism
  2.3-4 Skeletal Initial and Terminal Objects
  2.3-5 Skeletal LiftAlongMonomorphism
  2.3-6 Skeletal ColiftAlongEpimorphism
  2.3-7 Skeletal Product
  2.3-8 Skeletal Coproduct
  2.3-9 Skeletal Image
  2.3-10 Skeletal Equalizer
  2.3-11 Skeletal Pullback
  2.3-12 Skeletal Coequalizer
  2.3-13 Skeletal Pushout

2.4 GAP categories

  2.4-1 IsSkeletalCategoryOfFiniteRightGSets
  2.4-2 IsObjectInSkeletalCategoryOfFiniteRightGSets
  2.4-3 IsMorphismInSkeletalCategoryOfFiniteRightGSets

4.1 Helper functions

4.1-1 PositionAndConjugatorOfStabilizer

5 Reconstructing G from the skeletal category of finite right $G$-sets

5.1 Reconstruction Tools

  5.1-1 EndAsEqualizer
  5.1-2 EndByLifts
  5.1-3 ReconstructTableOfMarks
  5.1-4 HomSkeletalFinRightGSets
  5.1-5 ForgetfulFunctorSkeletalCategoryOfFiniteRightGSets
  5.1-6 ReconstructGroup

7 Reconstructing G from the skeletal category of finite G-sets

7.1-1 Reconstructing the Table of Marks
7.1-2 Reconstructing the Group

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