GeneralizedMorphismsForCAP   Implementations of generalized morphisms for the CAP project  2017.09.09 9 September 2017 Sebastian Gutsche Sebastian Posur Sebastian Gutsche Email: mailto:gutsche@mathematik.uni-siegen.de Homepage: http://www.uni-siegen.de/fb6/rmi/ Address: Department Mathematik Universität Siegen Walter-Flex-Straße 3 57068 Siegen Germany Sebastian Posur Email: mailto:sebastian.posur@uni-siegen.de Homepage: http://www.uni-siegen.de/fb6/rmi/ Address: Department Mathematik Universität Siegen Walter-Flex-Straße 3 57068 Siegen Germany ------------------------------------------------------- Contents (GeneralizedMorphismsForCAP) 1 Generalized Morphism Category 1.1 GAP Categories 1.1-1 IsGeneralizedMorphismCategoryObject 1.1-2 IsGeneralizedMorphism 1.2 Attributes 1.2-1 UnderlyingHonestObject 1.2-2 DomainOfGeneralizedMorphism 1.2-3 Codomain 1.2-4 AssociatedMorphism 1.2-5 DomainAssociatedMorphismCodomainTriple 1.2-6 HonestRepresentative 1.2-7 GeneralizedInverse 1.2-8 IdempotentDefinedBySubobject 1.2-9 IdempotentDefinedByFactorobject 1.2-10 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 IsGeneralizedMorphismCategoryByCospansObject 2.1-2 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 IsGeneralizedMorphismCategoryBySpansObject 3.1-2 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 IsGeneralizedMorphismCategoryByThreeArrowsObject 4.1-2 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 Sweep