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CAP

Categories, Algorithms, Programming

2022.08-05

16 August 2022

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

Øystein Skartsæterhagen
Email: oysteini@math.ntnu.no
Homepage: http://www.math.ntnu.no/~oysteini/
Address:
NTNU
Institutt for matematiske fag
7491 Trondheim
Norway

Contents

1 CAP Categories
2 Objects
3 Morphisms
4 Category 2-Cells
5 Category of Categories
6 Universal Objects
7 Add Functions
 7.7 Available Add functions

  7.7-1 AddAdditionForMorphisms

  7.7-2 AddAdditiveGenerators

  7.7-3 AddAdditiveInverseForMorphisms

  7.7-4 AddAstrictionToCoimage

  7.7-5 AddAstrictionToCoimageWithGivenCoimageObject

  7.7-6 AddBasisOfExternalHom

  7.7-7 AddCanonicalIdentificationFromCoimageToImageObject

  7.7-8 AddCanonicalIdentificationFromImageObjectToCoimage

  7.7-9 AddCoastrictionToImage

  7.7-10 AddCoastrictionToImageWithGivenImageObject

  7.7-11 AddCoefficientsOfMorphismWithGivenBasisOfExternalHom

  7.7-12 AddCoequalizer

  7.7-13 AddCoequalizerFunctorial

  7.7-14 AddCoequalizerFunctorialWithGivenCoequalizers

  7.7-15 AddCoimageObject

  7.7-16 AddCoimageProjection

  7.7-17 AddCoimageProjectionWithGivenCoimageObject

  7.7-18 AddCokernelColift

  7.7-19 AddCokernelColiftWithGivenCokernelObject

  7.7-20 AddCokernelObject

  7.7-21 AddCokernelObjectFunctorial

  7.7-22 AddCokernelObjectFunctorialWithGivenCokernelObjects

  7.7-23 AddCokernelProjection

  7.7-24 AddCokernelProjectionWithGivenCokernelObject

  7.7-25 AddColift

  7.7-26 AddColiftAlongEpimorphism

  7.7-27 AddColiftOrFail

  7.7-28 AddComponentOfMorphismFromDirectSum

  7.7-29 AddComponentOfMorphismIntoDirectSum

  7.7-30 AddCoproduct

  7.7-31 AddCoproductFunctorial

  7.7-32 AddCoproductFunctorialWithGivenCoproducts

  7.7-33 AddDirectProduct

  7.7-34 AddDirectProductFunctorial

  7.7-35 AddDirectProductFunctorialWithGivenDirectProducts

  7.7-36 AddDirectSum

  7.7-37 AddDirectSumCodiagonalDifference

  7.7-38 AddDirectSumDiagonalDifference

  7.7-39 AddDirectSumFunctorial

  7.7-40 AddDirectSumFunctorialWithGivenDirectSums

  7.7-41 AddDirectSumProjectionInPushout

  7.7-42 AddDistinguishedObjectOfHomomorphismStructure

  7.7-43 AddEmbeddingOfEqualizer

  7.7-44 AddEmbeddingOfEqualizerWithGivenEqualizer

  7.7-45 AddEpimorphismFromSomeProjectiveObject

  7.7-46 AddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject

  7.7-47 AddEqualizer

  7.7-48 AddEqualizerFunctorial

  7.7-49 AddEqualizerFunctorialWithGivenEqualizers

  7.7-50 AddFiberProduct

  7.7-51 AddFiberProductEmbeddingInDirectSum

  7.7-52 AddFiberProductFunctorial

  7.7-53 AddFiberProductFunctorialWithGivenFiberProducts

  7.7-54 AddHomologyObject

  7.7-55 AddHomologyObjectFunctorialWithGivenHomologyObjects

  7.7-56 AddHomomorphismStructureOnMorphisms

  7.7-57 AddHomomorphismStructureOnMorphismsWithGivenObjects

  7.7-58 AddHomomorphismStructureOnObjects

  7.7-59 AddHorizontalPostCompose

  7.7-60 AddHorizontalPreCompose

  7.7-61 AddIdentityMorphism

  7.7-62 AddIdentityTwoCell

  7.7-63 AddImageEmbedding

  7.7-64 AddImageEmbeddingWithGivenImageObject

  7.7-65 AddImageObject

  7.7-66 AddInitialObject

  7.7-67 AddInitialObjectFunctorial

  7.7-68 AddInitialObjectFunctorialWithGivenInitialObjects

  7.7-69 AddInjectionOfCofactorOfCoproduct

  7.7-70 AddInjectionOfCofactorOfCoproductWithGivenCoproduct

  7.7-71 AddInjectionOfCofactorOfDirectSum

  7.7-72 AddInjectionOfCofactorOfDirectSumWithGivenDirectSum

  7.7-73 AddInjectionOfCofactorOfPushout

  7.7-74 AddInjectionOfCofactorOfPushoutWithGivenPushout

  7.7-75 AddInjectiveColift

  7.7-76 AddInjectiveDimension

  7.7-77 AddInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure

  7.7-78 AddInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructureWithGivenObjects

  7.7-79 AddInterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism

  7.7-80 AddInverseForMorphisms

  7.7-81 AddInverseMorphismFromCoimageToImageWithGivenObjects

  7.7-82 AddIsAutomorphism

  7.7-83 AddIsBijectiveObject

  7.7-84 AddIsCodominating

  7.7-85 AddIsColiftable

  7.7-86 AddIsColiftableAlongEpimorphism

  7.7-87 AddIsCongruentForMorphisms

  7.7-88 AddIsDominating

  7.7-89 AddIsEndomorphism

  7.7-90 AddIsEpimorphism

  7.7-91 AddIsEqualAsFactorobjects

  7.7-92 AddIsEqualAsSubobjects

  7.7-93 AddIsEqualForCacheForMorphisms

  7.7-94 AddIsEqualForCacheForObjects

  7.7-95 AddIsEqualForMorphisms

  7.7-96 AddIsEqualForMorphismsOnMor

  7.7-97 AddIsEqualForObjects

  7.7-98 AddIsEqualToIdentityMorphism

  7.7-99 AddIsEqualToZeroMorphism

  7.7-100 AddIsHomSetInhabited

  7.7-101 AddIsIdempotent

  7.7-102 AddIsInitial

  7.7-103 AddIsInjective

  7.7-104 AddIsIsomorphism

  7.7-105 AddIsLiftable

  7.7-106 AddIsLiftableAlongMonomorphism

  7.7-107 AddIsMonomorphism

  7.7-108 AddIsOne

  7.7-109 AddIsProjective

  7.7-110 AddIsSplitEpimorphism

  7.7-111 AddIsSplitMonomorphism

  7.7-112 AddIsTerminal

  7.7-113 AddIsWellDefinedForMorphisms

  7.7-114 AddIsWellDefinedForObjects

  7.7-115 AddIsWellDefinedForTwoCells

  7.7-116 AddIsZeroForMorphisms

  7.7-117 AddIsZeroForObjects

  7.7-118 AddIsomorphismFromCoequalizerOfCoproductDiagramToPushout

  7.7-119 AddIsomorphismFromCoimageToCokernelOfKernel

  7.7-120 AddIsomorphismFromCokernelOfDiagonalDifferenceToPushout

  7.7-121 AddIsomorphismFromCokernelOfKernelToCoimage

  7.7-122 AddIsomorphismFromCoproductToDirectSum

  7.7-123 AddIsomorphismFromDirectProductToDirectSum

  7.7-124 AddIsomorphismFromDirectSumToCoproduct

  7.7-125 AddIsomorphismFromDirectSumToDirectProduct

  7.7-126 AddIsomorphismFromEqualizerOfDirectProductDiagramToFiberProduct

  7.7-127 AddIsomorphismFromFiberProductToEqualizerOfDirectProductDiagram

  7.7-128 AddIsomorphismFromFiberProductToKernelOfDiagonalDifference

  7.7-129 AddIsomorphismFromHomologyObjectToItsConstructionAsAnImageObject

  7.7-130 AddIsomorphismFromImageObjectToKernelOfCokernel

  7.7-131 AddIsomorphismFromInitialObjectToZeroObject

  7.7-132 AddIsomorphismFromItsConstructionAsAnImageObjectToHomologyObject

  7.7-133 AddIsomorphismFromKernelOfCokernelToImageObject

  7.7-134 AddIsomorphismFromKernelOfDiagonalDifferenceToFiberProduct

  7.7-135 AddIsomorphismFromPushoutToCoequalizerOfCoproductDiagram

  7.7-136 AddIsomorphismFromPushoutToCokernelOfDiagonalDifference

  7.7-137 AddIsomorphismFromTerminalObjectToZeroObject

  7.7-138 AddIsomorphismFromZeroObjectToInitialObject

  7.7-139 AddIsomorphismFromZeroObjectToTerminalObject

  7.7-140 AddKernelEmbedding

  7.7-141 AddKernelEmbeddingWithGivenKernelObject

  7.7-142 AddKernelLift

  7.7-143 AddKernelLiftWithGivenKernelObject

  7.7-144 AddKernelObject

  7.7-145 AddKernelObjectFunctorial

  7.7-146 AddKernelObjectFunctorialWithGivenKernelObjects

  7.7-147 AddLift

  7.7-148 AddLiftAlongMonomorphism

  7.7-149 AddLiftOrFail

  7.7-150 AddMereExistenceOfSolutionOfLinearSystemInAbCategory

  7.7-151 AddMonomorphismIntoSomeInjectiveObject

  7.7-152 AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject

  7.7-153 AddMorphismBetweenDirectSums

  7.7-154 AddMorphismBetweenDirectSumsWithGivenDirectSums

  7.7-155 AddMorphismConstructor

  7.7-156 AddMorphismDatum

  7.7-157 AddMorphismFromCoimageToImageWithGivenObjects

  7.7-158 AddMorphismFromEqualizerToSink

  7.7-159 AddMorphismFromEqualizerToSinkWithGivenEqualizer

  7.7-160 AddMorphismFromFiberProductToSink

  7.7-161 AddMorphismFromFiberProductToSinkWithGivenFiberProduct

  7.7-162 AddMorphismFromKernelObjectToSink

  7.7-163 AddMorphismFromKernelObjectToSinkWithGivenKernelObject

  7.7-164 AddMorphismFromSourceToCoequalizer

  7.7-165 AddMorphismFromSourceToCoequalizerWithGivenCoequalizer

  7.7-166 AddMorphismFromSourceToCokernelObject

  7.7-167 AddMorphismFromSourceToCokernelObjectWithGivenCokernelObject

  7.7-168 AddMorphismFromSourceToPushout

  7.7-169 AddMorphismFromSourceToPushoutWithGivenPushout

  7.7-170 AddMultiplyWithElementOfCommutativeRingForMorphisms

  7.7-171 AddObjectConstructor

  7.7-172 AddObjectDatum

  7.7-173 AddPostCompose

  7.7-174 AddPostComposeList

  7.7-175 AddPreCompose

  7.7-176 AddPreComposeList

  7.7-177 AddProjectionInFactorOfDirectProduct

  7.7-178 AddProjectionInFactorOfDirectProductWithGivenDirectProduct

  7.7-179 AddProjectionInFactorOfDirectSum

  7.7-180 AddProjectionInFactorOfDirectSumWithGivenDirectSum

  7.7-181 AddProjectionInFactorOfFiberProduct

  7.7-182 AddProjectionInFactorOfFiberProductWithGivenFiberProduct

  7.7-183 AddProjectionOntoCoequalizer

  7.7-184 AddProjectionOntoCoequalizerWithGivenCoequalizer

  7.7-185 AddProjectiveDimension

  7.7-186 AddProjectiveLift

  7.7-187 AddPushout

  7.7-188 AddPushoutFunctorial

  7.7-189 AddPushoutFunctorialWithGivenPushouts

  7.7-190 AddRandomMorphismByInteger

  7.7-191 AddRandomMorphismByList

  7.7-192 AddRandomMorphismWithFixedRangeByInteger

  7.7-193 AddRandomMorphismWithFixedRangeByList

  7.7-194 AddRandomMorphismWithFixedSourceAndRangeByInteger

  7.7-195 AddRandomMorphismWithFixedSourceAndRangeByList

  7.7-196 AddRandomMorphismWithFixedSourceByInteger

  7.7-197 AddRandomMorphismWithFixedSourceByList

  7.7-198 AddRandomObjectByInteger

  7.7-199 AddRandomObjectByList

  7.7-200 AddSimplifyEndo

  7.7-201 AddSimplifyEndo_IsoFromInputObject

  7.7-202 AddSimplifyEndo_IsoToInputObject

  7.7-203 AddSimplifyMorphism

  7.7-204 AddSimplifyObject

  7.7-205 AddSimplifyObject_IsoFromInputObject

  7.7-206 AddSimplifyObject_IsoToInputObject

  7.7-207 AddSimplifyRange

  7.7-208 AddSimplifyRange_IsoFromInputObject

  7.7-209 AddSimplifyRange_IsoToInputObject

  7.7-210 AddSimplifySource

  7.7-211 AddSimplifySourceAndRange

  7.7-212 AddSimplifySourceAndRange_IsoFromInputRange

  7.7-213 AddSimplifySourceAndRange_IsoFromInputSource

  7.7-214 AddSimplifySourceAndRange_IsoToInputRange

  7.7-215 AddSimplifySourceAndRange_IsoToInputSource

  7.7-216 AddSimplifySource_IsoFromInputObject

  7.7-217 AddSimplifySource_IsoToInputObject

  7.7-218 AddSolveLinearSystemInAbCategory

  7.7-219 AddSolveLinearSystemInAbCategoryOrFail

  7.7-220 AddSomeInjectiveObject

  7.7-221 AddSomeProjectiveObject

  7.7-222 AddSomeReductionBySplitEpiSummand

  7.7-223 AddSomeReductionBySplitEpiSummand_MorphismFromInputRange

  7.7-224 AddSomeReductionBySplitEpiSummand_MorphismToInputRange

  7.7-225 AddSubtractionForMorphisms

  7.7-226 AddTerminalObject

  7.7-227 AddTerminalObjectFunctorial

  7.7-228 AddTerminalObjectFunctorialWithGivenTerminalObjects

  7.7-229 AddUniversalMorphismFromCoequalizer

  7.7-230 AddUniversalMorphismFromCoequalizerWithGivenCoequalizer

  7.7-231 AddUniversalMorphismFromCoproduct

  7.7-232 AddUniversalMorphismFromCoproductWithGivenCoproduct

  7.7-233 AddUniversalMorphismFromDirectSum

  7.7-234 AddUniversalMorphismFromDirectSumWithGivenDirectSum

  7.7-235 AddUniversalMorphismFromImage

  7.7-236 AddUniversalMorphismFromImageWithGivenImageObject

  7.7-237 AddUniversalMorphismFromInitialObject

  7.7-238 AddUniversalMorphismFromInitialObjectWithGivenInitialObject

  7.7-239 AddUniversalMorphismFromPushout

  7.7-240 AddUniversalMorphismFromPushoutWithGivenPushout

  7.7-241 AddUniversalMorphismFromZeroObject

  7.7-242 AddUniversalMorphismFromZeroObjectWithGivenZeroObject

  7.7-243 AddUniversalMorphismIntoCoimage

  7.7-244 AddUniversalMorphismIntoCoimageWithGivenCoimageObject

  7.7-245 AddUniversalMorphismIntoDirectProduct

  7.7-246 AddUniversalMorphismIntoDirectProductWithGivenDirectProduct

  7.7-247 AddUniversalMorphismIntoDirectSum

  7.7-248 AddUniversalMorphismIntoDirectSumWithGivenDirectSum

  7.7-249 AddUniversalMorphismIntoEqualizer

  7.7-250 AddUniversalMorphismIntoEqualizerWithGivenEqualizer

  7.7-251 AddUniversalMorphismIntoFiberProduct

  7.7-252 AddUniversalMorphismIntoFiberProductWithGivenFiberProduct

  7.7-253 AddUniversalMorphismIntoTerminalObject

  7.7-254 AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject

  7.7-255 AddUniversalMorphismIntoZeroObject

  7.7-256 AddUniversalMorphismIntoZeroObjectWithGivenZeroObject

  7.7-257 AddVerticalPostCompose

  7.7-258 AddVerticalPreCompose

  7.7-259 AddZeroMorphism

  7.7-260 AddZeroObject

  7.7-261 AddZeroObjectFunctorial

  7.7-262 AddZeroObjectFunctorialWithGivenZeroObjects
8 Managing Derived Methods
9 Technical Details
 9.2 Install Functions for IsWellDefined

  9.2-1 InstallMethodWithToDoForIsWellDefined

  9.2-2 InstallSetWithToDoForIsWellDefined

  9.2-3 DeclareAttributeWithToDoForIsWellDefined

  9.2-4 DeclareFamilyProperty

  9.2-5 CAP_INTERNAL_REPLACE_STRING_WITH_FILTER

  9.2-6 CAP_INTERNAL_REPLACE_STRINGS_WITH_FILTERS

  9.2-7 CAP_INTERNAL_MERGE_FILTER_LISTS

  9.2-8 CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT

  9.2-9 CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION

  9.2-10 CAP_INTERNAL_MERGE_PRECONDITIONS_LIST

  9.2-11 CAP_INTERNAL_ASSERT_IS_CELL_OF_CATEGORY

  9.2-12 CAP_INTERNAL_ASSERT_IS_OBJECT_OF_CATEGORY

  9.2-13 CAP_INTERNAL_ASSERT_IS_MORPHISM_OF_CATEGORY

  9.2-14 CAP_INTERNAL_ASSERT_IS_TWO_CELL_OF_CATEGORY

  9.2-15 CAP_INTERNAL_ASSERT_IS_LIST_OF_OBJECTS_OF_CATEGORY

  9.2-16 CAP_INTERNAL_ASSERT_IS_LIST_OF_MORPHISMS_OF_CATEGORY

  9.2-17 CAP_INTERNAL_ASSERT_IS_LIST_OF_TWO_CELLS_OF_CATEGORY

  9.2-18 CAP_INTERNAL_ASSERT_IS_NON_NEGATIVE_INTEGER_OR_INFINITY

  9.2-19 CachingStatistic

  9.2-20 BrowseCachingStatistic

  9.2-21 InstallDeprecatedAlias

  9.2-22 IsSpecializationOfFilter

  9.2-23 IsSpecializationOfFilterList

  9.2-24 InstallMethodForCompilerForCAP

  9.2-25 InstallOtherMethodForCompilerForCAP

  9.2-26 CapJitAddKnownMethod

  9.2-27 CapJitAddTypeSignature

  9.2-28 CapJitAddTypeSignatureDeferred

  9.2-29 CapJitDataTypeOfCategory

  9.2-30 CapFixpoint

  9.2-31 TransitivelyNeededOtherPackages

  9.2-32 PackageOfCAPOperation

  9.2-33 SafePosition

  9.2-34 NTuple

  9.2-35 Pair

  9.2-36 Triple

  9.2-37 HandlePrecompiledTowers
10 Limits and Colimits
11 The Category Constructor
12 Create wrapper hulls of a category
13 Dummy categories
14 Examples and Tests
Index

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