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                           GeneralizedMorphismsForCAP 

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          Implementations of generalized morphisms for the CAP project 

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                                   2017.09.09

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                                9 September 2017

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                               Sebastian Gutsche

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                                Sebastian Posur

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  Sebastian Gutsche

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      Email:    mailto:[email protected]

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      Homepage: http://www.uni-siegen.de/fb6/rmi/

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      Address:  Department Mathematik

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                Universität Siegen

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                Walter-Flex-Straße 3

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                57068 Siegen

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                Germany

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  Sebastian Posur

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      Email:    mailto:[email protected]

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      Homepage: http://www.uni-siegen.de/fb6/rmi/

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      Address:  Department Mathematik

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                Universität Siegen

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                Walter-Flex-Straße 3

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                57068 Siegen

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                Germany

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  -------------------------------------------------------

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  Contents (GeneralizedMorphismsForCAP)

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  1 Generalized Morphism Category

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    1.1 GAP Categories

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      1.1-1 IsGeneralizedMorphismCategoryObject

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      1.1-2 IsGeneralizedMorphism

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    1.2 Attributes

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      1.2-1 UnderlyingHonestObject

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      1.2-2 DomainOfGeneralizedMorphism

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      1.2-3 Codomain

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      1.2-4 AssociatedMorphism

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      1.2-5 DomainAssociatedMorphismCodomainTriple

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      1.2-6 HonestRepresentative

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      1.2-7 GeneralizedInverse

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      1.2-8 IdempotentDefinedBySubobject

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      1.2-9 IdempotentDefinedByFactorobject

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      1.2-10 UnderlyingHonestCategory

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    1.3 Operations

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      1.3-1 GeneralizedMorphismFromFactorToSubobject

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      1.3-2 CommonRestriction

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      1.3-3 ConcatenationProduct

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    1.4 Properties

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      1.4-1 IsHonest

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      1.4-2 HasFullDomain

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      1.4-3 HasFullCodomain

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      1.4-4 IsSingleValued

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      1.4-5 IsTotal

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    1.5 Convenience methods

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      1.5-1 GeneralizedMorphismCategory

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      1.5-2 GeneralizedMorphismObject

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      1.5-3 AsGeneralizedMorphism

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      1.5-4 GeneralizedMorphism

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      1.5-5 GeneralizedMorphism

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      1.5-6 GeneralizedMorphismWithRangeAid

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      1.5-7 GeneralizedMorphismWithSourceAid

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  2 Generalized Morphism Category by Cospans

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    2.1 GAP Categories

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      2.1-1 IsGeneralizedMorphismCategoryByCospansObject

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      2.1-2 IsGeneralizedMorphismByCospan

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    2.2 Properties

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      2.2-1 HasIdentityAsReversedArrow

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    2.3 Attributes

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      2.3-1 UnderlyingHonestObject

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      2.3-2 Arrow

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      2.3-3 ReversedArrow

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      2.3-4 NormalizedCospanTuple

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      2.3-5 PseudoInverse

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      2.3-6 GeneralizedInverseByCospan

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      2.3-7 IdempotentDefinedBySubobjectByCospan

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      2.3-8 IdempotentDefinedByFactorobjectByCospan

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      2.3-9 NormalizedCospan

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    2.4 Operations

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      2.4-1 GeneralizedMorphismFromFactorToSubobjectByCospan

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    2.5 Constructors

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      2.5-1 GeneralizedMorphismByCospan

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      2.5-2 GeneralizedMorphismByCospan

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      2.5-3 GeneralizedMorphismByCospanWithSourceAid

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      2.5-4 AsGeneralizedMorphismByCospan

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      2.5-5 GeneralizedMorphismCategoryByCospans

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      2.5-6 GeneralizedMorphismByCospansObject

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    2.6 Constructors of lifts of exact functors and natrual (iso)morphisms

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      2.6-1 AsGeneralizedMorphismByCospan

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  3 Generalized Morphism Category by Spans

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    3.1 GAP Categories

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      3.1-1 IsGeneralizedMorphismCategoryBySpansObject

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      3.1-2 IsGeneralizedMorphismBySpan

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    3.2 Properties

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      3.2-1 HasIdentityAsReversedArrow

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    3.3 Attributes

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      3.3-1 UnderlyingHonestObject

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      3.3-2 Arrow

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      3.3-3 ReversedArrow

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      3.3-4 NormalizedSpanTuple

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      3.3-5 PseudoInverse

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      3.3-6 GeneralizedInverseBySpan

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      3.3-7 IdempotentDefinedBySubobjectBySpan

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      3.3-8 IdempotentDefinedByFactorobjectBySpan

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      3.3-9 NormalizedSpan

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    3.4 Operations

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      3.4-1 GeneralizedMorphismFromFactorToSubobjectBySpan

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    3.5 Constructors

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      3.5-1 GeneralizedMorphismBySpan

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      3.5-2 GeneralizedMorphismBySpan

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      3.5-3 GeneralizedMorphismBySpanWithRangeAid

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      3.5-4 AsGeneralizedMorphismBySpan

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      3.5-5 GeneralizedMorphismCategoryBySpans

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      3.5-6 GeneralizedMorphismBySpansObject

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  4 Generalized Morphism Category by Three Arrows

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    4.1 GAP Categories

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      4.1-1 IsGeneralizedMorphismCategoryByThreeArrowsObject

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      4.1-2 IsGeneralizedMorphismByThreeArrows

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    4.2 Properties

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      4.2-1 HasIdentitiesAsReversedArrows

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      4.2-2 HasIdentityAsSourceAid

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      4.2-3 HasIdentityAsRangeAid

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    4.3 Attributes

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      4.3-1 UnderlyingHonestObject

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      4.3-2 SourceAid

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      4.3-3 RangeAid

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      4.3-4 Arrow

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      4.3-5 PseudoInverse

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      4.3-6 GeneralizedInverseByThreeArrows

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      4.3-7 IdempotentDefinedBySubobjectByThreeArrows

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      4.3-8 IdempotentDefinedByFactorobjectByThreeArrows

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    4.4 Operations

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      4.4-1 GeneralizedMorphismFromFactorToSubobjectByThreeArrows

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      4.4-2 CommonCoastriction

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    4.5 Constructors

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      4.5-1 GeneralizedMorphismByThreeArrows

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      4.5-2 GeneralizedMorphismByThreeArrowsWithSourceAid

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      4.5-3 GeneralizedMorphismByThreeArrowsWithRangeAid

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      4.5-4 AsGeneralizedMorphismByThreeArrows

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      4.5-5 GeneralizedMorphismCategoryByThreeArrows

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      4.5-6 GeneralizedMorphismByThreeArrowsObject

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  5 Conversion functors for generalized morphisms

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    5.1 Functors from Cospans

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      5.1-1 FunctorFromCospansToThreeArrows

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      5.1-2 FunctorFromCospansToSpans

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    5.2 Functors from Spans

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      5.2-1 FunctorFromSpansToThreeArrows

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      5.2-2 FunctorFromSpansToCospans

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    5.3 Functors from Three Arrows

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      5.3-1 FunctorFromThreeArrowsToCospans

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      5.3-2 FunctorFromThreeArrowsToSpans

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  6 Serre Quotients Subcategory

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    6.1 implementation

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      6.1-1 FullSubcategoryByMembershipFunction

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  7 Serre Quotients

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    7.1 General operations

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      7.1-1 IsSerreQuotientCategoryObject

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      7.1-2 IsSerreQuotientCategoryMorphism

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      7.1-3 SerreQuotientCategory

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      7.1-4 AsSerreQuotientCategoryObject

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      7.1-5 SerreQuotientCategoryMorphism

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      7.1-6 SerreQuotientCategoryMorphism

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      7.1-7 SerreQuotientCategoryMorphism

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      7.1-8 SerreQuotientCategoryMorphismWithSourceAid

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      7.1-9 SerreQuotientCategoryMorphismWithRangeAid

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      7.1-10 AsSerreQuotientCategoryMorphism

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      7.1-11 SubcategoryMembershipTestFunctionForSerreQuotient

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      7.1-12 UnderlyingHonestCategory

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      7.1-13 UnderlyingGeneralizedMorphismCategory

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      7.1-14 UnderlyingGeneralizedObject

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      7.1-15 UnderlyingHonestObject

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      7.1-16 UnderlyingGeneralizedMorphism

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      7.1-17 CanonicalProjection

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    7.2 Serre quotients by cospans

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      7.2-1 SerreQuotientCategoryByCospans

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      7.2-2 AsSerreQuotientCategoryByCospansObject

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      7.2-3 SerreQuotientCategoryByCospansMorphism

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      7.2-4 SerreQuotientCategoryByCospansMorphism

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      7.2-5 SerreQuotientCategoryByCospansMorphismWithSourceAid

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      7.2-6 SerreQuotientCategoryByCospansMorphism

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      7.2-7 AsSerreQuotientCategoryByCospansMorphism

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    7.3 Serre Quotients by Spans

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      7.3-1 SerreQuotientCategoryBySpans

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      7.3-2 AsSerreQuotientCategoryBySpansObject

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      7.3-3 SerreQuotientCategoryBySpansMorphism

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      7.3-4 SerreQuotientCategoryBySpansMorphism

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      7.3-5 SerreQuotientCategoryBySpansMorphism

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      7.3-6 SerreQuotientCategoryBySpansMorphismWithRangeAid

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      7.3-7 AsSerreQuotientCategoryBySpansMorphism

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    7.4 Serre Quotients modeled by three arrows

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      7.4-1 SerreQuotientCategoryByThreeArrows

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      7.4-2 AsSerreQuotientCategoryByThreeArrowsObject

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      7.4-3 SerreQuotientCategoryByThreeArrowsMorphism

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      7.4-4 SerreQuotientCategoryByThreeArrowsMorphism

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      7.4-5 SerreQuotientCategoryByThreeArrowsMorphismWithSourceAid

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      7.4-6 SerreQuotientCategoryByThreeArrowsMorphismWithRangeAid

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      7.4-7 AsSerreQuotientCategoryByThreeArrowsMorphism

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  8 Serre Quotients Functors

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    8.1 Functors

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      8.1-1 SerreQuotientConversionFunctor

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  9 Examples and Tests

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    9.1 Basic Commands

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    9.2 Intersection of Nodal Curve and Cusp

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    9.3 Sweep

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