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Toposes

Elementary toposes

2025.08-06

25 August 2025

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io
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

Mario Román
Email: mromang08@gmail.com
Homepage: https://mroman42.github.io/
Address:
Mathematical Institute
University of Oxford
Oxford, United Kingdom

Tom Kuhmichel
Email: tom.kuhmichel@student.uni-siegen.de
Homepage: https://github.com/TKuh
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

1.1 Exact cover with global elements

  1.1-1 ExactCoverWithGlobalElements
  1.1-2 IndexOfNonliftableMorphismFromDistinguishedObject
  1.1-3 NonliftableMorphismFromDistinguishedObject

1.2 Subobject classifier

  1.2-1 IsElementaryTopos
  1.2-2 SubobjectClassifier
  1.2-3 CartesianSquareOfSubobjectClassifier
  1.2-4 ClassifyingMorphismOfSubobject
  1.2-5 ClassifyingMorphismOfSubobjectWithGivenSubobjectClassifier
  1.2-6 SubobjectOfClassifyingMorphism
  1.2-7 ListOfSubobjects

1.3 Truth morphisms

  1.3-1 TruthMorphismOfTrue
  1.3-2 TruthMorphismOfTrueWithGivenObjects
  1.3-3 TruthMorphismOfFalse
  1.3-4 TruthMorphismOfFalseWithGivenObjects
  1.3-5 TruthMorphismOfNot
  1.3-6 TruthMorphismOfNotWithGivenObjects
  1.3-7 TruthMorphismOfAnd
  1.3-8 TruthMorphismOfAndWithGivenObjects
  1.3-9 TruthMorphismOfOr
  1.3-10 TruthMorphismOfOrWithGivenObjects
  1.3-11 TruthMorphismOfImplies
  1.3-12 TruthMorphismOfImpliesWithGivenObjects

1.4 Power object

  1.4-1 PowerObject
  1.4-2 PowerObjectFunctorial
  1.4-3 PowerObjectFunctorialWithGivenPowerObjects
  1.4-4 PowerObjectRightEvaluationMorphism
  1.4-5 PowerObjectRightEvaluationMorphismWithGivenObjects
  1.4-6 PowerObjectLeftEvaluationMorphism
  1.4-7 PowerObjectLeftEvaluationMorphismWithGivenObjects
  1.4-8 PRightTransposeMorphism
  1.4-9 PRightTransposeMorphismWithGivenRange
  1.4-10 PLeftTransposeMorphism
  1.4-11 PLeftTransposeMorphismWithGivenRange
  1.4-12 UpperSegmentOfRelation
  1.4-13 UpperSegmentOfRelationWithGivenRange
  1.4-14 LowerSegmentOfRelation
  1.4-15 LowerSegmentOfRelationWithGivenRange
  1.4-16 SingletonMorphism
  1.4-17 SingletonMorphismWithGivenPowerObject
  1.4-18 IsomorphismOntoCartesianSquareOfPowerObject
  1.4-19 IsomorphismOntoCartesianSquareOfPowerObjectWithGivenObjects
  1.4-20 IntersectWithPreimagesWithGivenObjects
  1.4-21 EmbeddingOfRelativePowerObject
  1.4-22 RelativePowerObjectFibrationMorphism
  1.4-23 RelativePowerObjectLeftEvaluationMorphism
  1.4-24 RightFiberMorphism
  1.4-25 RightFiberMorphismWithGivenObjects
  1.4-26 LeftFiberMorphism
  1.4-27 LeftFiberMorphismWithGivenObjects
  1.4-28 SingletonSupportOfRelationsWithGivenObjects

1.5 Relative truth morphisms

  1.5-1 RelativeTruthMorphismOfTrue
  1.5-2 RelativeTruthMorphismOfTrueWithGivenObjects
  1.5-3 RelativeTruthMorphismOfFalse
  1.5-4 RelativeTruthMorphismOfFalseWithGivenObjects
  1.5-5 RelativeTruthMorphismOfNot
  1.5-6 RelativeTruthMorphismOfNotWithGivenObjects
  1.5-7 RelativeTruthMorphismOfAnd
  1.5-8 RelativeTruthMorphismOfAndWithGivenObjects
  1.5-9 RelativeTruthMorphismOfOr
  1.5-10 RelativeTruthMorphismOfOrWithGivenObjects
  1.5-11 RelativeTruthMorphismOfImplies
  1.5-12 RelativeTruthMorphismOfImpliesWithGivenObjects

1.6 Heyting algebra of subobjects

  1.6-1 PseudoComplementSubobject
  1.6-2 EmbeddingOfPseudoComplementSubobject
  1.6-3 EmbeddingOfPseudoComplementSubobjectWithGivenPseudoComplement
  1.6-4 IntersectionSubobject
  1.6-5 EmbeddingOfIntersectionSubobject
  1.6-6 EmbeddingOfIntersectionSubobjectWithGivenIntersection
  1.6-7 UnionSubobject
  1.6-8 EmbeddingOfUnionSubobject
  1.6-9 EmbeddingOfUnionSubobjectWithGivenUnion
  1.6-10 RelativePseudoComplementSubobject
  1.6-11 EmbeddingOfRelativePseudoComplementSubobject
  1.6-12 EmbeddingOfRelativePseudoComplementSubobjectWithGivenImplication

1.7 Pushout complements

  1.7-1 Example in FinSets
  1.7-2 Example in SkeletalFinSets
  1.7-3 HasPushoutComplement
  1.7-4 PushoutComplement

1.8 Double-pushout rewriting

1.9 Lawvere-Tierney topologies

  1.9-1 LawvereTierneyLocalModalityOperators
  1.9-2 LawvereTierneySubobjects
  1.9-3 LawvereTierneyEmbeddingsOfSubobjectClassifiers

1.10 Add-methods

  1.10-1 AddCartesianSquareOfSubobjectClassifier
  1.10-2 AddClassifyingMorphismOfSubobject
  1.10-3 AddClassifyingMorphismOfSubobjectWithGivenSubobjectClassifier
  1.10-4 AddCoproductComplement
  1.10-5 AddDirectProductComplement
  1.10-6 AddEmbeddingOfIntersectionSubobject
  1.10-7 AddEmbeddingOfIntersectionSubobjectWithGivenIntersection
  1.10-8 AddEmbeddingOfPseudoComplementSubobject
  1.10-9 AddEmbeddingOfPseudoComplementSubobjectWithGivenPseudoComplement
  1.10-10 AddEmbeddingOfRelativePseudoComplementSubobject
  1.10-11 AddEmbeddingOfRelativePseudoComplementSubobjectWithGivenImplication
  1.10-12 AddEmbeddingOfUnionSubobject
  1.10-13 AddEmbeddingOfUnionSubobjectWithGivenUnion
  1.10-14 AddExactCoverWithGlobalElements
  1.10-15 AddHasPushoutComplement
  1.10-16 AddIndexOfNonliftableMorphismFromDistinguishedObject
  1.10-17 AddInjectionOfCoproductComplement
  1.10-18 AddInjectionOfCoproductComplementWithGivenCoproductComplement
  1.10-19 AddIntersectionSubobject
  1.10-20 AddIsomorphismOntoCartesianSquareOfPowerObject
  1.10-21 AddIsomorphismOntoCartesianSquareOfPowerObjectWithGivenObjects
  1.10-22 AddLawvereTierneyEmbeddingsOfSubobjectClassifiers
  1.10-23 AddLawvereTierneyLocalModalityOperators
  1.10-24 AddLawvereTierneySubobjects
  1.10-25 AddLeftFiberMorphism
  1.10-26 AddLeftFiberMorphismWithGivenObjects
  1.10-27 AddListOfSubobjects
  1.10-28 AddLowerSegmentOfRelation
  1.10-29 AddLowerSegmentOfRelationWithGivenRange
  1.10-30 AddNonliftableMorphismFromDistinguishedObject
  1.10-31 AddPLeftTransposeMorphism
  1.10-32 AddPLeftTransposeMorphismWithGivenRange
  1.10-33 AddPRightTransposeMorphism
  1.10-34 AddPRightTransposeMorphismWithGivenRange
  1.10-35 AddPowerObject
  1.10-36 AddPowerObjectFunctorial
  1.10-37 AddPowerObjectFunctorialWithGivenPowerObjects
  1.10-38 AddPowerObjectLeftEvaluationMorphism
  1.10-39 AddPowerObjectLeftEvaluationMorphismWithGivenObjects
  1.10-40 AddPowerObjectRightEvaluationMorphism
  1.10-41 AddPowerObjectRightEvaluationMorphismWithGivenObjects
  1.10-42 AddProjectionInDirectProductComplement
  1.10-43 AddProjectionInDirectProductComplementWithGivenDirectProductComplement
  1.10-44 AddPseudoComplementSubobject
  1.10-45 AddPushoutComplement
  1.10-46 AddRelativePseudoComplementSubobject
  1.10-47 AddRelativeTruthMorphismOfAnd
  1.10-48 AddRelativeTruthMorphismOfAndWithGivenObjects
  1.10-49 AddRelativeTruthMorphismOfFalse
  1.10-50 AddRelativeTruthMorphismOfFalseWithGivenObjects
  1.10-51 AddRelativeTruthMorphismOfImplies
  1.10-52 AddRelativeTruthMorphismOfImpliesWithGivenObjects
  1.10-53 AddRelativeTruthMorphismOfNot
  1.10-54 AddRelativeTruthMorphismOfNotWithGivenObjects
  1.10-55 AddRelativeTruthMorphismOfOr
  1.10-56 AddRelativeTruthMorphismOfOrWithGivenObjects
  1.10-57 AddRelativeTruthMorphismOfTrue
  1.10-58 AddRelativeTruthMorphismOfTrueWithGivenObjects
  1.10-59 AddRightFiberMorphism
  1.10-60 AddRightFiberMorphismWithGivenObjects
  1.10-61 AddSingletonMorphism
  1.10-62 AddSingletonMorphismWithGivenPowerObject
  1.10-63 AddSubobjectClassifier
  1.10-64 AddSubobjectOfClassifyingMorphism
  1.10-65 AddTruthMorphismOfAnd
  1.10-66 AddTruthMorphismOfAndWithGivenObjects
  1.10-67 AddTruthMorphismOfFalse
  1.10-68 AddTruthMorphismOfFalseWithGivenObjects
  1.10-69 AddTruthMorphismOfImplies
  1.10-70 AddTruthMorphismOfImpliesWithGivenObjects
  1.10-71 AddTruthMorphismOfNot
  1.10-72 AddTruthMorphismOfNotWithGivenObjects
  1.10-73 AddTruthMorphismOfOr
  1.10-74 AddTruthMorphismOfOrWithGivenObjects
  1.10-75 AddTruthMorphismOfTrue
  1.10-76 AddTruthMorphismOfTrueWithGivenObjects
  1.10-77 AddUnionSubobject
  1.10-78 AddUpperSegmentOfRelation
  1.10-79 AddUpperSegmentOfRelationWithGivenRange

1.11 Coproduct and direct product complements

  1.11-1 CoproductComplement
  1.11-2 InjectionOfCoproductComplement
  1.11-3 InjectionOfCoproductComplementWithGivenCoproductComplement
  1.11-4 DirectProductComplement
  1.11-5 ProjectionInDirectProductComplement
  1.11-6 ProjectionInDirectProductComplementWithGivenDirectProductComplement

3.1 Visualizing posets of subobjects

3.1-1 DigraphOfSubobjects
3.1-2 SvgString

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