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Locales

Locales, frames, coframes, meet semi-lattices of locally closed subsets, and Boolean algebras of constructible sets

2023.01-02

17 January 2023

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

Markus Lange-Hegermann
Email: markus.lange-hegermann@th-owl.de
Homepage: https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/
Address:
Markus Lange-Hegermann
Hochschule Ostwestfalen-Lippe
Liebigstraße 87
32657 Lemgo
Germany

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

Contents

1 Prosets (preordered sets)
 1.1 Info Class

  1.1-1 InfoLocales
 1.2 Constructors

  1.2-1 UniqueMorphism
 1.3 Operations

  1.3-1 AreIsomorphicForObjectsIfIsHomSetInhabited
 1.4 Add-methods

  1.4-1 AddAreIsomorphicForObjectsIfIsHomSetInhabited
  1.4-2 AddUniqueMorphism
 1.5 GAP Categories

  1.5-1 IsObjectInThinCategory
  1.5-2 IsMorphismInThinCategory
 1.6 GAP Properties

  1.6-1 IsThinCategory
2 Posets (partially ordered sets)
 2.1 Operations

  2.1-1 IsEqualForObjectsIfIsHomSetInhabited
 2.2 Add-methods

  2.2-1 AddIsEqualForObjectsIfIsHomSetInhabited
 2.3 Properties

  2.3-1 IsPosetCategory
3 Lattices
 3.1 Properties

  3.1-1 IsCartesianProset
  3.1-2 IsMeetSemiLattice
  3.1-3 IsCocartesianProset
  3.1-4 IsJoinSemiLattice
  3.1-5 IsBicartesianProset
  3.1-6 IsLattice
  3.1-7 IsDistributiveBicartesianProset
  3.1-8 IsDistributiveLattice
  3.1-9 IsMonoidalLattice
  3.1-10 IsClosedMonoidalLattice
  3.1-11 IsCoclosedMonoidalLattice
 3.2 Attributes

  3.2-1 Factors
  3.2-2 FactorsAttr
 3.3 Operations

  3.3-1 \*
  3.3-2 \+
4 Heyting algebras
 4.1 Operations

  4.1-1 NegationOnObjects
  4.1-2 NegationOnMorphisms
  4.1-3 NegationOnMorphismsWithGivenNegations
  4.1-4 MorphismToDoubleNegation
  4.1-5 MorphismToDoubleNegationWithGivenDoubleNegation
 4.2 Stable internal Hom

  4.2-1 StableInternalHom
 4.3 Add-methods

  4.3-1 AddMorphismToDoubleNegation
  4.3-2 AddMorphismToDoubleNegationWithGivenDoubleNegation
  4.3-3 AddNegationOnMorphisms
  4.3-4 AddNegationOnMorphismsWithGivenNegations
  4.3-5 AddNegationOnObjects
 4.4 Properties

  4.4-1 IsHeytingAlgebroid
  4.4-2 IsHeytingAlgebra
5 Co-Heyting algebras
 5.1 Operations

  5.1-1 ConegationOnObjects
  5.1-2 ConegationOnMorphisms
  5.1-3 ConegationOnMorphismsWithGivenConegations
  5.1-4 MorphismFromDoubleConegation
  5.1-5 MorphismFromDoubleConegationWithGivenDoubleConegation
 5.2 Stable internal coHom

  5.2-1 StableInternalCoHom
 5.3 Add-methods

  5.3-1 AddConegationOnMorphisms
  5.3-2 AddConegationOnMorphismsWithGivenConegations
  5.3-3 AddConegationOnObjects
  5.3-4 AddMorphismFromDoubleConegation
  5.3-5 AddMorphismFromDoubleConegationWithGivenDoubleConegation
 5.4 Properties

  5.4-1 IsCoHeytingAlgebroid
  5.4-2 IsCoHeytingAlgebra
6 The (stable) proset/poset of a category
 6.1 Global variables

  6.1-1 CAP_INTERNAL_METHOD_NAME_LIST_FOR_PREORDERED_SET_OF_CATEGORY
 6.2 Attributes

  6.2-1 AmbientCategory
  6.2-2 UnderlyingCell
 6.3 Constructors

  6.3-1 CreateProsetOrPosetOfCategory
  6.3-2 ProsetOfCategory
  6.3-3 StableProsetOfCategory
  6.3-4 PosetOfCategory
  6.3-5 StablePosetOfCategory
  6.3-6 AsCellOfProset
  6.3-7 AsCellOfStableProset
  6.3-8 AsCellOfPoset
  6.3-9 AsCellOfStablePoset
  6.3-10 /
 6.4 Examples

  6.4-1 Ideal quotient
 6.5 GAP categories

  6.5-1 Poset of category
  6.5-2 Stable poset of category
  6.5-3 IsProsetOrPosetOfCapCategory
  6.5-4 IsCapCategoryCellInProsetOrPosetOfACategory
  6.5-5 IsCapCategoryCellInStableProsetOrPosetOfACategory
  6.5-6 IsCapCategoryObjectInProsetOrPosetOfACategory
  6.5-7 IsProsetOfCapCategory
  6.5-8 IsCapCategoryCellInProsetOfACategory
  6.5-9 IsCapCategoryObjectInProsetOfACategory
  6.5-10 IsCapCategoryMorphismInProsetOfACategory
  6.5-11 IsPosetOfCapCategory
  6.5-12 IsCapCategoryCellInPosetOfACategory
  6.5-13 IsCapCategoryObjectInPosetOfACategory
  6.5-14 IsCapCategoryMorphismInPosetOfACategory
7 Boolean algebras
 7.1 Operations

  7.1-1 MorphismFromDoubleNegation
  7.1-2 MorphismFromDoubleNegationWithGivenDoubleNegation
  7.1-3 MorphismToDoubleConegation
  7.1-4 MorphismToDoubleConegationWithGivenDoubleConegation
 7.2 Add-methods

  7.2-1 AddMorphismFromDoubleNegation
  7.2-2 AddMorphismFromDoubleNegationWithGivenDoubleNegation
  7.2-3 AddMorphismToDoubleConegation
  7.2-4 AddMorphismToDoubleConegationWithGivenDoubleConegation
 7.3 Properties

  7.3-1 IsBiHeytingAlgebroid
  7.3-2 IsBiHeytingAlgebra
  7.3-3 IsBooleanAlgebroid
  7.3-4 IsBooleanAlgebra
8 Meet-semilattice of differences
 8.1 Constructors

  8.1-1 MeetSemilatticeOfSingleDifferences
  8.1-2 \-
 8.2 Operations

  8.2-1 MinuendAndSubtrahendInUnderlyingLattice
  8.2-2 DistinguishedSubtrahend
 8.3 GAP Categories

  8.3-1 IsMeetSemilatticeOfSingleDifferences
  8.3-2 IsObjectInMeetSemilatticeOfSingleDifferences
  8.3-3 IsMorphismInMeetSemilatticeOfSingleDifferences
9 Meet-semilattice of multiple differences
 9.1 Constructors

  9.1-1 MeetSemilatticeOfMultipleDifferences
  9.1-2 AsMultipleDifference
 9.2 Attributes

  9.2-1 AsSingleDifference
 9.3 Operations

  9.3-1 ListOfSingleDifferences
  9.3-2 \[\]
 9.4 GAP Categories

  9.4-1 IsMeetSemilatticeOfMultipleDifferences
  9.4-2 IsObjectInMeetSemilatticeOfMultipleDifferences
  9.4-3 IsMorphismInMeetSemilatticeOfMultipleDifferences
10 Boolean algebra of constructible objects
 10.1 Constructors

  10.1-1 BooleanAlgebraOfConstructibleObjectsAsUnionOfMultipleDifferences
  10.1-2 UnionOfMultipleDifferences
  10.1-3 BooleanAlgebraOfConstructibleObjectsAsUnionOfDifferences
  10.1-4 UnionOfDifferences
 10.2 Properties

  10.2-1 IsOpen
  10.2-2 IsClosed
  10.2-3 IsLocallyClosed
 10.3 Attributes

  10.3-1 UnderlyingCategory
  10.3-2 LocallyClosedPart
  10.3-3 CanonicalObject
  10.3-4 FactorizeObject
  10.3-5 NormalizedObject
  10.3-6 StandardizedObject
  10.3-7 Closure
  10.3-8 ClosureAsConstructibleObject
  10.3-9 Dimension
 10.4 Operations

  10.4-1 NormalizeObject
  10.4-2 StandardizeObject
  10.4-3 Length
  10.4-4 \[\]
  10.4-5 Iterator
  10.4-6 ForAllOp
  10.4-7 ForAnyOp
  10.4-8 ListOfObjectsInMeetSemilatticeOfDifferences
  10.4-9 ListOfObjectsInMeetSemilatticeOfDifferences
 10.5 GAP Categories

  10.5-1 IsBooleanAlgebraOfConstructibleObjects
  10.5-2 IsConstructibleObject
  10.5-3 IsMorphismBetweenConstructibleObjects
  10.5-4 IsBooleanAlgebraOfConstructibleObjectsAsUnionOfMultipleDifferences
  10.5-5 IsConstructibleObjectAsUnionOfMultipleDifferences
  10.5-6 IsMorphismBetweenConstructibleObjectsAsUnionOfMultipleDifferences
  10.5-7 IsBooleanAlgebraOfConstructibleObjectsAsUnionOfSingleDifferences
  10.5-8 IsConstructibleObjectAsUnionOfSingleDifferences
  10.5-9 IsMorphismBetweenConstructibleObjectsAsUnionOfDifferences
11 Tools
 11.1 Info Class

  11.1-1 InfoSquashDatastructureForConstructibleObjects
 11.2 Constructors

  11.2-1 DatastructureForConstructibleObject
  11.2-2 NodeInDatastructureOfConstructibleObject
  11.2-3 Attach
 11.3 Operations

  11.3-1 IsDone
  11.3-2 MinimalLevelOfPreNodes
  11.3-3 Pop
  11.3-4 IsHomSetInhabited
  11.3-5 Remove
  11.3-6 SquashOnce
  11.3-7 RemoveObsoleteSubtrahends
  11.3-8 Squash
  11.3-9 AsUnionOfMultipleDifferences
  11.3-10 DigraphOfParents
  11.3-11 DigraphOfChildren
  11.3-12 Visualize
 11.4 GAP Categories

  11.4-1 IsDatastructureForConstructibleObjects
  11.4-2 IsNodeInDatastructureOfConstructibleObjects
12 Tests
 12.1 GAP Categories

  12.1-1 Precompilation
13 Meet-semilattice of single/multiple differences
 13.1 GAP Categories

  13.1-1 IsMeetSemilatticeOfDifferences
  13.1-2 IsObjectInMeetSemilatticeOfDifferences
  13.1-3 IsMorphismInMeetSemilatticeOfDifferences
 13.2 Properties

  13.2-1 IsClosed
  13.2-2 IsOpen
  13.2-3 IsLocallyClosed
 13.3 Attributes

  13.3-1 UnderlyingCategory
  13.3-2 LocallyClosedPart
  13.3-3 CanonicalObject
  13.3-4 FactorizeObject
  13.3-5 Closure
  13.3-6 Dimension
 13.4 Operations

  13.4-1 NormalizeObject
  13.4-2 StandardizeObject
Index

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