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GeneralizedMorphismsForCAP

Implementations of generalized morphisms for the CAP project

2024.09-03

4 October 2024

Sebastian Gutsche
Email: gutsche@mathematik.uni-siegen.de
Homepage: https://sebasguts.github.io/
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57068 Siegen
Germany

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

Contents

1 Generalized Morphism Category
 1.1 GAP Categories

  1.1-1 IsGeneralizedMorphismCategory
  1.1-2 IsGeneralizedMorphismCategoryObject
  1.1-3 IsGeneralizedMorphism
 1.2 Attributes

  1.2-1 UnderlyingHonestObject
  1.2-2 DomainEmbedding
  1.2-3 GeneralizedImageEmbedding
  1.2-4 DefectEmbedding
  1.2-5 GeneralizedKernelEmbedding
  1.2-6 CodomainProjection
  1.2-7 GeneralizedCokernelProjection
  1.2-8 CodefectProjection
  1.2-9 GeneralizedCoimageProjection
  1.2-10 AssociatedMorphism
  1.2-11 DomainAssociatedMorphismCodomainTriple
  1.2-12 HonestRepresentative
  1.2-13 GeneralizedInverse
  1.2-14 IdempotentDefinedBySubobject
  1.2-15 IdempotentDefinedByFactorobject
  1.2-16 UnderlyingHonestCategory
 1.3 Operations

  1.3-1 GeneralizedMorphismFromFactorToSubobject
  1.3-2 CommonRestriction
  1.3-3 ConcatenationProduct
 1.4 Properties

  1.4-1 IsHonest
  1.4-2 HasFullDomain
  1.4-3 HasFullCodomain
  1.4-4 IsSingleValued
  1.4-5 IsTotal
 1.5 Convenience methods

  1.5-1 GeneralizedMorphismCategory
  1.5-2 GeneralizedMorphismObject
  1.5-3 AsGeneralizedMorphism
  1.5-4 GeneralizedMorphism
  1.5-5 GeneralizedMorphism
  1.5-6 GeneralizedMorphismWithRangeAid
  1.5-7 GeneralizedMorphismWithSourceAid
2 Generalized Morphism Category by Cospans
 2.1 GAP Categories

  2.1-1 IsGeneralizedMorphismCategoryByCospans
  2.1-2 IsGeneralizedMorphismCategoryByCospansObject
  2.1-3 IsGeneralizedMorphismByCospan
 2.2 Properties

  2.2-1 HasIdentityAsReversedArrow
 2.3 Attributes

  2.3-1 UnderlyingHonestObject
  2.3-2 Arrow
  2.3-3 ReversedArrow
  2.3-4 NormalizedCospanTuple
  2.3-5 PseudoInverse
  2.3-6 GeneralizedInverseByCospan
  2.3-7 IdempotentDefinedBySubobjectByCospan
  2.3-8 IdempotentDefinedByFactorobjectByCospan
  2.3-9 NormalizedCospan
 2.4 Operations

  2.4-1 GeneralizedMorphismFromFactorToSubobjectByCospan
 2.5 Constructors

  2.5-1 GeneralizedMorphismByCospan
  2.5-2 GeneralizedMorphismByCospan
  2.5-3 GeneralizedMorphismByCospanWithSourceAid
  2.5-4 AsGeneralizedMorphismByCospan
  2.5-5 GeneralizedMorphismCategoryByCospans
  2.5-6 GeneralizedMorphismByCospansObject
 2.6 Constructors of lifts of exact functors and natrual (iso)morphisms

  2.6-1 AsGeneralizedMorphismByCospan
3 Generalized Morphism Category by Spans
 3.1 GAP Categories

  3.1-1 IsGeneralizedMorphismCategoryBySpans
  3.1-2 IsGeneralizedMorphismCategoryBySpansObject
  3.1-3 IsGeneralizedMorphismBySpan
 3.2 Properties

  3.2-1 HasIdentityAsReversedArrow
 3.3 Attributes

  3.3-1 UnderlyingHonestObject
  3.3-2 Arrow
  3.3-3 ReversedArrow
  3.3-4 NormalizedSpanTuple
  3.3-5 PseudoInverse
  3.3-6 GeneralizedInverseBySpan
  3.3-7 IdempotentDefinedBySubobjectBySpan
  3.3-8 IdempotentDefinedByFactorobjectBySpan
  3.3-9 NormalizedSpan
 3.4 Operations

  3.4-1 GeneralizedMorphismFromFactorToSubobjectBySpan
 3.5 Constructors

  3.5-1 GeneralizedMorphismBySpan
  3.5-2 GeneralizedMorphismBySpan
  3.5-3 GeneralizedMorphismBySpanWithRangeAid
  3.5-4 AsGeneralizedMorphismBySpan
  3.5-5 GeneralizedMorphismCategoryBySpans
  3.5-6 GeneralizedMorphismBySpansObject
4 Generalized Morphism Category by Three Arrows
 4.1 GAP Categories

  4.1-1 IsGeneralizedMorphismCategoryByThreeArrows
  4.1-2 IsGeneralizedMorphismCategoryByThreeArrowsObject
  4.1-3 IsGeneralizedMorphismByThreeArrows
 4.2 Properties

  4.2-1 HasIdentitiesAsReversedArrows
  4.2-2 HasIdentityAsSourceAid
  4.2-3 HasIdentityAsRangeAid
 4.3 Attributes

  4.3-1 UnderlyingHonestObject
  4.3-2 SourceAid
  4.3-3 RangeAid
  4.3-4 Arrow
  4.3-5 PseudoInverse
  4.3-6 GeneralizedInverseByThreeArrows
  4.3-7 IdempotentDefinedBySubobjectByThreeArrows
  4.3-8 IdempotentDefinedByFactorobjectByThreeArrows
 4.4 Operations

  4.4-1 GeneralizedMorphismFromFactorToSubobjectByThreeArrows
  4.4-2 CommonCoastriction
 4.5 Constructors

  4.5-1 GeneralizedMorphismByThreeArrows
  4.5-2 GeneralizedMorphismByThreeArrowsWithSourceAid
  4.5-3 GeneralizedMorphismByThreeArrowsWithRangeAid
  4.5-4 AsGeneralizedMorphismByThreeArrows
  4.5-5 GeneralizedMorphismCategoryByThreeArrows
  4.5-6 GeneralizedMorphismByThreeArrowsObject
5 Conversion functors for generalized morphisms
 5.1 Functors from Cospans

  5.1-1 FunctorFromCospansToThreeArrows
  5.1-2 FunctorFromCospansToSpans
 5.2 Functors from Spans

  5.2-1 FunctorFromSpansToThreeArrows
  5.2-2 FunctorFromSpansToCospans
 5.3 Functors from Three Arrows

  5.3-1 FunctorFromThreeArrowsToCospans
  5.3-2 FunctorFromThreeArrowsToSpans
6 Serre Quotients Subcategory
 6.1 implementation

  6.1-1 FullSubcategoryByMembershipFunction
7 Serre Quotients
 7.1 General operations

  7.1-1 IsSerreQuotientCategoryObject
  7.1-2 IsSerreQuotientCategoryMorphism
  7.1-3 SerreQuotientCategory
  7.1-4 AsSerreQuotientCategoryObject
  7.1-5 SerreQuotientCategoryMorphism
  7.1-6 SerreQuotientCategoryMorphism
  7.1-7 SerreQuotientCategoryMorphism
  7.1-8 SerreQuotientCategoryMorphismWithSourceAid
  7.1-9 SerreQuotientCategoryMorphismWithRangeAid
  7.1-10 AsSerreQuotientCategoryMorphism
  7.1-11 SubcategoryMembershipTestFunctionForSerreQuotient
  7.1-12 UnderlyingHonestCategory
  7.1-13 UnderlyingGeneralizedMorphismCategory
  7.1-14 UnderlyingGeneralizedObject
  7.1-15 UnderlyingHonestObject
  7.1-16 UnderlyingGeneralizedMorphism
  7.1-17 CanonicalProjection
 7.2 Serre quotients by cospans

  7.2-1 SerreQuotientCategoryByCospans
  7.2-2 AsSerreQuotientCategoryByCospansObject
  7.2-3 SerreQuotientCategoryByCospansMorphism
  7.2-4 SerreQuotientCategoryByCospansMorphism
  7.2-5 SerreQuotientCategoryByCospansMorphismWithSourceAid
  7.2-6 SerreQuotientCategoryByCospansMorphism
  7.2-7 AsSerreQuotientCategoryByCospansMorphism
 7.3 Serre Quotients by Spans

  7.3-1 SerreQuotientCategoryBySpans
  7.3-2 AsSerreQuotientCategoryBySpansObject
  7.3-3 SerreQuotientCategoryBySpansMorphism
  7.3-4 SerreQuotientCategoryBySpansMorphism
  7.3-5 SerreQuotientCategoryBySpansMorphism
  7.3-6 SerreQuotientCategoryBySpansMorphismWithRangeAid
  7.3-7 AsSerreQuotientCategoryBySpansMorphism
 7.4 Serre Quotients modeled by three arrows

  7.4-1 SerreQuotientCategoryByThreeArrows
  7.4-2 AsSerreQuotientCategoryByThreeArrowsObject
  7.4-3 SerreQuotientCategoryByThreeArrowsMorphism
  7.4-4 SerreQuotientCategoryByThreeArrowsMorphism
  7.4-5 SerreQuotientCategoryByThreeArrowsMorphismWithSourceAid
  7.4-6 SerreQuotientCategoryByThreeArrowsMorphismWithRangeAid
  7.4-7 AsSerreQuotientCategoryByThreeArrowsMorphism
8 Serre Quotients Functors
 8.1 Functors

  8.1-1 SerreQuotientConversionFunctor
9 Examples and Tests
 9.1 Basic Commands
 9.2 Intersection of Nodal Curve and Cusp
 9.3 WrapperCategory
 9.4 Sweep
 9.5 Generalized Morphisms Category
 9.6 IsWellDefined
Index

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