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FinSetsForCAP

The (skeletal) elementary topos of finite sets

2025.05-01

2 May 2025

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: f.zickgraf@dashdos.com
Homepage: https://github.com/zickgraf/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

1 The category of finite sets
 1.1 Attributes

  1.1-1 AsList
  1.1-2 Length
  1.1-3 AsList
 1.2 Constructors

  1.2-1 CategoryOfFiniteSets
  1.2-2 FinSets
  1.2-3 FinSet
  1.2-4 FinSet
  1.2-5 FinSetNC
  1.2-6 FinSetNC
  1.2-7 MapOfFinSets
  1.2-8 MapOfFinSetsNC
 1.3 Tools

  1.3-1 IsEqualForElementsOfFinSets
  1.3-2 \in
  1.3-3 []
  1.3-4 Iterator
  1.3-5 UnionOfFinSets
  1.3-6 ListOp
  1.3-7 FilteredOp
  1.3-8 First
  1.3-9 EmbeddingOfFinSets
  1.3-10 ProjectionOfFinSets
  1.3-11 Preimage
  1.3-12 ImageObject
  1.3-13 CallFuncList
  1.3-14 ListOp
 1.4 GAP Categories

  1.4-1 IsCategoryOfFiniteSets
  1.4-2 IsObjectInCategoryOfFiniteSets
  1.4-3 IsMorphismInCategoryOfFiniteSets
 1.5 Examples

  1.5-1 IsHomSetInhabited
  1.5-2 Morphisms of external Hom
  1.5-3 Exact cover with global elements
  1.5-4 PreCompose
  1.5-5 IsEpimorphism and IsMonomorhism
  1.5-6 Initial and Terminal Objects
  1.5-7 Projective and Injective Objects
  1.5-8 Product
  1.5-9 Coproduct
  1.5-10 Image
  1.5-11 Coimage
  1.5-12 Equalizer
  1.5-13 Pullback
  1.5-14 Coequalizer
  1.5-15 Pushout
  1.5-16 Cartesian Lambda Introduction
  1.5-17 Lift
  1.5-18 Singleton morphism
  1.5-19 Topos properties
  1.5-20 Subobject Classifier
  1.5-21 Pushout Complement
2 The skeletal category of finite sets
 2.1 Attributes

  2.1-1 Length
  2.1-2 AsList
  2.1-3 AsList
 2.2 Constructors

  2.2-1 SkeletalCategoryOfFiniteSets
  2.2-2 SkeletalFinSets
  2.2-3 FinSet
  2.2-4 FinSet
  2.2-5 MapOfFinSets
 2.3 Tools

  2.3-1 ListOp
  2.3-2 EmbeddingOfFinSets
  2.3-3 Preimage
  2.3-4 ImageObject
  2.3-5 CallFuncList
  2.3-6 GeometricSum
  2.3-7 GeometricSumDiff1
  2.3-8 RemIntWithDomain
  2.3-9 QuoIntWithDomain
  2.3-10 DivIntWithGivenQuotient
  2.3-11 DigitInPositionalNotation
 2.4 GAP Categories

  2.4-1 IsSkeletalCategoryOfFiniteSets
  2.4-2 IsObjectInSkeletalCategoryOfFiniteSets
  2.4-3 IsMorphismInSkeletalCategoryOfFiniteSets
 2.5 Examples

  2.5-1 SkeletalIsHomSetInhabited
  2.5-2 Skeletal WellDefined
  2.5-3 Skeletal PreCompose
  2.5-4 Skeletal Monomophisms and Epimophisms
  2.5-5 Skeletal Initial and Terminal Objects
  2.5-6 Projective and Injective Objects
  2.5-7 Skeletal Product
  2.5-8 Skeletal Coproduct
  2.5-9 Skeletal Image
  2.5-10 Skeletal Coimage
  2.5-11 Skeletal Preimage
  2.5-12 Skeletal Equalizer
  2.5-13 Skeletal Pullback
  2.5-14 Skeletal Coequalizer
  2.5-15 Skeletal Pushout
  2.5-16 Skeletal Cartesian Lambda Introduction
  2.5-17 Skeletal Lift
  2.5-18 Skeletal Colift
  2.5-19 Skeletal singleton morphism
  2.5-20 Skeletal topos properties
  2.5-21 Skeletal Subobject Classifier
  2.5-22 Skeletal Pushout Complement
3 Precompilation
 3.1 Precompiling the skeletal category of finite sets
 3.2 Precompiling ProjectionInFactorOfBinaryDirectProduct
Index

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