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CAP

Categories, Algorithms, Programming

2024.03-02

13 March 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

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

Fabian Zickgraf
Email: fabian.zickgraf@uni-siegen.de
Homepage: https://github.com/zickgraf/
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

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

  7.6-1 AddAdditionForMorphisms

  7.6-2 AddAdditiveGenerators

  7.6-3 AddAdditiveInverseForMorphisms

  7.6-4 AddAstrictionToCoimage

  7.6-5 AddAstrictionToCoimageWithGivenCoimageObject

  7.6-6 AddBasisOfExternalHom

  7.6-7 AddCoastrictionToImage

  7.6-8 AddCoastrictionToImageWithGivenImageObject

  7.6-9 AddCoefficientsOfMorphism

  7.6-10 AddCoequalizer

  7.6-11 AddCoequalizerFunctorial

  7.6-12 AddCoequalizerFunctorialWithGivenCoequalizers

  7.6-13 AddCoimageObject

  7.6-14 AddCoimageObjectFunctorial

  7.6-15 AddCoimageObjectFunctorialWithGivenCoimageObjects

  7.6-16 AddCoimageProjection

  7.6-17 AddCoimageProjectionWithGivenCoimageObject

  7.6-18 AddCokernelColift

  7.6-19 AddCokernelColiftWithGivenCokernelObject

  7.6-20 AddCokernelObject

  7.6-21 AddCokernelObjectFunctorial

  7.6-22 AddCokernelObjectFunctorialWithGivenCokernelObjects

  7.6-23 AddCokernelProjection

  7.6-24 AddCokernelProjectionWithGivenCokernelObject

  7.6-25 AddColift

  7.6-26 AddColiftAlongEpimorphism

  7.6-27 AddColiftOrFail

  7.6-28 AddComponentOfMorphismFromCoproduct

  7.6-29 AddComponentOfMorphismFromDirectSum

  7.6-30 AddComponentOfMorphismIntoDirectProduct

  7.6-31 AddComponentOfMorphismIntoDirectSum

  7.6-32 AddCoproduct

  7.6-33 AddCoproductFunctorial

  7.6-34 AddCoproductFunctorialWithGivenCoproducts

  7.6-35 AddDirectProduct

  7.6-36 AddDirectProductFunctorial

  7.6-37 AddDirectProductFunctorialWithGivenDirectProducts

  7.6-38 AddDirectSum

  7.6-39 AddDirectSumFunctorial

  7.6-40 AddDirectSumFunctorialWithGivenDirectSums

  7.6-41 AddDistinguishedObjectOfHomomorphismStructure

  7.6-42 AddEmbeddingOfEqualizer

  7.6-43 AddEmbeddingOfEqualizerWithGivenEqualizer

  7.6-44 AddEpimorphismFromProjectiveCoverObject

  7.6-45 AddEpimorphismFromProjectiveCoverObjectWithGivenProjectiveCoverObject

  7.6-46 AddEpimorphismFromSomeProjectiveObject

  7.6-47 AddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject

  7.6-48 AddEqualizer

  7.6-49 AddEqualizerFunctorial

  7.6-50 AddEqualizerFunctorialWithGivenEqualizers

  7.6-51 AddFiberProduct

  7.6-52 AddFiberProductFunctorial

  7.6-53 AddFiberProductFunctorialWithGivenFiberProducts

  7.6-54 AddHomologyObject

  7.6-55 AddHomologyObjectFunctorialWithGivenHomologyObjects

  7.6-56 AddHomomorphismStructureOnMorphisms

  7.6-57 AddHomomorphismStructureOnMorphismsWithGivenObjects

  7.6-58 AddHomomorphismStructureOnObjects

  7.6-59 AddHorizontalPostCompose

  7.6-60 AddHorizontalPreCompose

  7.6-61 AddIdentityMorphism

  7.6-62 AddIdentityTwoCell

  7.6-63 AddImageEmbedding

  7.6-64 AddImageEmbeddingWithGivenImageObject

  7.6-65 AddImageObject

  7.6-66 AddImageObjectFunctorial

  7.6-67 AddImageObjectFunctorialWithGivenImageObjects

  7.6-68 AddIndecomposableInjectiveObjects

  7.6-69 AddIndecomposableProjectiveObjects

  7.6-70 AddInitialObject

  7.6-71 AddInitialObjectFunctorial

  7.6-72 AddInitialObjectFunctorialWithGivenInitialObjects

  7.6-73 AddInjectionOfCofactorOfCoproduct

  7.6-74 AddInjectionOfCofactorOfCoproductWithGivenCoproduct

  7.6-75 AddInjectionOfCofactorOfDirectSum

  7.6-76 AddInjectionOfCofactorOfDirectSumWithGivenDirectSum

  7.6-77 AddInjectionOfCofactorOfPushout

  7.6-78 AddInjectionOfCofactorOfPushoutWithGivenPushout

  7.6-79 AddInjectiveColift

  7.6-80 AddInjectiveDimension

  7.6-81 AddInjectiveEnvelopeObject

  7.6-82 AddInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure

  7.6-83 AddInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructureWithGivenObjects

  7.6-84 AddInterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism

  7.6-85 AddInverseForMorphisms

  7.6-86 AddInverseOfMorphismFromCoimageToImage

  7.6-87 AddInverseOfMorphismFromCoimageToImageWithGivenObjects

  7.6-88 AddIsAutomorphism

  7.6-89 AddIsBijectiveObject

  7.6-90 AddIsCodominating

  7.6-91 AddIsColiftable

  7.6-92 AddIsColiftableAlongEpimorphism

  7.6-93 AddIsCongruentForMorphisms

  7.6-94 AddIsDominating

  7.6-95 AddIsEndomorphism

  7.6-96 AddIsEpimorphism

  7.6-97 AddIsEqualAsFactorobjects

  7.6-98 AddIsEqualAsSubobjects

  7.6-99 AddIsEqualForCacheForMorphisms

  7.6-100 AddIsEqualForCacheForObjects

  7.6-101 AddIsEqualForMorphisms

  7.6-102 AddIsEqualForMorphismsOnMor

  7.6-103 AddIsEqualForObjects

  7.6-104 AddIsEqualToIdentityMorphism

  7.6-105 AddIsEqualToZeroMorphism

  7.6-106 AddIsHomSetInhabited

  7.6-107 AddIsIdempotent

  7.6-108 AddIsInitial

  7.6-109 AddIsInjective

  7.6-110 AddIsIsomorphicForObjects

  7.6-111 AddIsIsomorphism

  7.6-112 AddIsLiftable

  7.6-113 AddIsLiftableAlongMonomorphism

  7.6-114 AddIsMonomorphism

  7.6-115 AddIsOne

  7.6-116 AddIsProjective

  7.6-117 AddIsSplitEpimorphism

  7.6-118 AddIsSplitMonomorphism

  7.6-119 AddIsTerminal

  7.6-120 AddIsWellDefinedForMorphisms

  7.6-121 AddIsWellDefinedForMorphismsWithGivenSourceAndRange

  7.6-122 AddIsWellDefinedForObjects

  7.6-123 AddIsWellDefinedForTwoCells

  7.6-124 AddIsZeroForMorphisms

  7.6-125 AddIsZeroForObjects

  7.6-126 AddIsomorphismFromCoequalizerOfCoproductDiagramToPushout

  7.6-127 AddIsomorphismFromCoequalizerToCokernelOfJointPairwiseDifferencesOfMorphismsFromCoproduct

  7.6-128 AddIsomorphismFromCoimageToCokernelOfKernel

  7.6-129 AddIsomorphismFromCokernelOfJointPairwiseDifferencesOfMorphismsFromCoproductToCoequalizer

  7.6-130 AddIsomorphismFromCokernelOfKernelToCoimage

  7.6-131 AddIsomorphismFromCoproductToDirectSum

  7.6-132 AddIsomorphismFromDirectProductToDirectSum

  7.6-133 AddIsomorphismFromDirectSumToCoproduct

  7.6-134 AddIsomorphismFromDirectSumToDirectProduct

  7.6-135 AddIsomorphismFromEqualizerOfDirectProductDiagramToFiberProduct

  7.6-136 AddIsomorphismFromEqualizerToKernelOfJointPairwiseDifferencesOfMorphismsIntoDirectProduct

  7.6-137 AddIsomorphismFromFiberProductToEqualizerOfDirectProductDiagram

  7.6-138 AddIsomorphismFromHomologyObjectToItsConstructionAsAnImageObject

  7.6-139 AddIsomorphismFromImageObjectToKernelOfCokernel

  7.6-140 AddIsomorphismFromInitialObjectToZeroObject

  7.6-141 AddIsomorphismFromItsConstructionAsAnImageObjectToHomologyObject

  7.6-142 AddIsomorphismFromKernelOfCokernelToImageObject

  7.6-143 AddIsomorphismFromKernelOfJointPairwiseDifferencesOfMorphismsIntoDirectProductToEqualizer

  7.6-144 AddIsomorphismFromPushoutToCoequalizerOfCoproductDiagram

  7.6-145 AddIsomorphismFromTerminalObjectToZeroObject

  7.6-146 AddIsomorphismFromZeroObjectToInitialObject

  7.6-147 AddIsomorphismFromZeroObjectToTerminalObject

  7.6-148 AddJointPairwiseDifferencesOfMorphismsFromCoproduct

  7.6-149 AddJointPairwiseDifferencesOfMorphismsIntoDirectProduct

  7.6-150 AddKernelEmbedding

  7.6-151 AddKernelEmbeddingWithGivenKernelObject

  7.6-152 AddKernelLift

  7.6-153 AddKernelLiftWithGivenKernelObject

  7.6-154 AddKernelObject

  7.6-155 AddKernelObjectFunctorial

  7.6-156 AddKernelObjectFunctorialWithGivenKernelObjects

  7.6-157 AddLift

  7.6-158 AddLiftAlongMonomorphism

  7.6-159 AddLiftOrFail

  7.6-160 AddLinearCombinationOfMorphisms

  7.6-161 AddMereExistenceOfSolutionOfLinearSystemInAbCategory

  7.6-162 AddMonomorphismIntoInjectiveEnvelopeObject

  7.6-163 AddMonomorphismIntoInjectiveEnvelopeObjectWithGivenInjectiveEnvelopeObject

  7.6-164 AddMonomorphismIntoSomeInjectiveObject

  7.6-165 AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject

  7.6-166 AddMorphismBetweenDirectSums

  7.6-167 AddMorphismBetweenDirectSumsWithGivenDirectSums

  7.6-168 AddMorphismConstructor

  7.6-169 AddMorphismDatum

  7.6-170 AddMorphismFromCoimageToImage

  7.6-171 AddMorphismFromCoimageToImageWithGivenObjects

  7.6-172 AddMorphismFromEqualizerToSink

  7.6-173 AddMorphismFromEqualizerToSinkWithGivenEqualizer

  7.6-174 AddMorphismFromFiberProductToSink

  7.6-175 AddMorphismFromFiberProductToSinkWithGivenFiberProduct

  7.6-176 AddMorphismFromKernelObjectToSink

  7.6-177 AddMorphismFromKernelObjectToSinkWithGivenKernelObject

  7.6-178 AddMorphismFromSourceToCoequalizer

  7.6-179 AddMorphismFromSourceToCoequalizerWithGivenCoequalizer

  7.6-180 AddMorphismFromSourceToCokernelObject

  7.6-181 AddMorphismFromSourceToCokernelObjectWithGivenCokernelObject

  7.6-182 AddMorphismFromSourceToPushout

  7.6-183 AddMorphismFromSourceToPushoutWithGivenPushout

  7.6-184 AddMorphismsOfExternalHom

  7.6-185 AddMultiplyWithElementOfCommutativeRingForMorphisms

  7.6-186 AddObjectConstructor

  7.6-187 AddObjectDatum

  7.6-188 AddPostCompose

  7.6-189 AddPostComposeList

  7.6-190 AddPostInverseForMorphisms

  7.6-191 AddPreCompose

  7.6-192 AddPreComposeList

  7.6-193 AddPreInverseForMorphisms

  7.6-194 AddProjectionInFactorOfDirectProduct

  7.6-195 AddProjectionInFactorOfDirectProductWithGivenDirectProduct

  7.6-196 AddProjectionInFactorOfDirectSum

  7.6-197 AddProjectionInFactorOfDirectSumWithGivenDirectSum

  7.6-198 AddProjectionInFactorOfFiberProduct

  7.6-199 AddProjectionInFactorOfFiberProductWithGivenFiberProduct

  7.6-200 AddProjectionOntoCoequalizer

  7.6-201 AddProjectionOntoCoequalizerWithGivenCoequalizer

  7.6-202 AddProjectiveCoverObject

  7.6-203 AddProjectiveDimension

  7.6-204 AddProjectiveLift

  7.6-205 AddPushout

  7.6-206 AddPushoutFunctorial

  7.6-207 AddPushoutFunctorialWithGivenPushouts

  7.6-208 AddRandomMorphismByInteger

  7.6-209 AddRandomMorphismByList

  7.6-210 AddRandomMorphismWithFixedRangeByInteger

  7.6-211 AddRandomMorphismWithFixedRangeByList

  7.6-212 AddRandomMorphismWithFixedSourceAndRangeByInteger

  7.6-213 AddRandomMorphismWithFixedSourceAndRangeByList

  7.6-214 AddRandomMorphismWithFixedSourceByInteger

  7.6-215 AddRandomMorphismWithFixedSourceByList

  7.6-216 AddRandomObjectByInteger

  7.6-217 AddRandomObjectByList

  7.6-218 AddSimplifyEndo

  7.6-219 AddSimplifyEndo_IsoFromInputObject

  7.6-220 AddSimplifyEndo_IsoToInputObject

  7.6-221 AddSimplifyMorphism

  7.6-222 AddSimplifyObject

  7.6-223 AddSimplifyObject_IsoFromInputObject

  7.6-224 AddSimplifyObject_IsoToInputObject

  7.6-225 AddSimplifyRange

  7.6-226 AddSimplifyRange_IsoFromInputObject

  7.6-227 AddSimplifyRange_IsoToInputObject

  7.6-228 AddSimplifySource

  7.6-229 AddSimplifySourceAndRange

  7.6-230 AddSimplifySourceAndRange_IsoFromInputRange

  7.6-231 AddSimplifySourceAndRange_IsoFromInputSource

  7.6-232 AddSimplifySourceAndRange_IsoToInputRange

  7.6-233 AddSimplifySourceAndRange_IsoToInputSource

  7.6-234 AddSimplifySource_IsoFromInputObject

  7.6-235 AddSimplifySource_IsoToInputObject

  7.6-236 AddSolveLinearSystemInAbCategory

  7.6-237 AddSolveLinearSystemInAbCategoryOrFail

  7.6-238 AddSomeInjectiveObject

  7.6-239 AddSomeIsomorphismBetweenObjects

  7.6-240 AddSomeProjectiveObject

  7.6-241 AddSomeReductionBySplitEpiSummand

  7.6-242 AddSomeReductionBySplitEpiSummand_MorphismFromInputRange

  7.6-243 AddSomeReductionBySplitEpiSummand_MorphismToInputRange

  7.6-244 AddSubtractionForMorphisms

  7.6-245 AddSumOfMorphisms

  7.6-246 AddTerminalObject

  7.6-247 AddTerminalObjectFunctorial

  7.6-248 AddTerminalObjectFunctorialWithGivenTerminalObjects

  7.6-249 AddUniversalMorphismFromCoequalizer

  7.6-250 AddUniversalMorphismFromCoequalizerWithGivenCoequalizer

  7.6-251 AddUniversalMorphismFromCoproduct

  7.6-252 AddUniversalMorphismFromCoproductWithGivenCoproduct

  7.6-253 AddUniversalMorphismFromDirectSum

  7.6-254 AddUniversalMorphismFromDirectSumWithGivenDirectSum

  7.6-255 AddUniversalMorphismFromImage

  7.6-256 AddUniversalMorphismFromImageWithGivenImageObject

  7.6-257 AddUniversalMorphismFromInitialObject

  7.6-258 AddUniversalMorphismFromInitialObjectWithGivenInitialObject

  7.6-259 AddUniversalMorphismFromPushout

  7.6-260 AddUniversalMorphismFromPushoutWithGivenPushout

  7.6-261 AddUniversalMorphismFromZeroObject

  7.6-262 AddUniversalMorphismFromZeroObjectWithGivenZeroObject

  7.6-263 AddUniversalMorphismIntoCoimage

  7.6-264 AddUniversalMorphismIntoCoimageWithGivenCoimageObject

  7.6-265 AddUniversalMorphismIntoDirectProduct

  7.6-266 AddUniversalMorphismIntoDirectProductWithGivenDirectProduct

  7.6-267 AddUniversalMorphismIntoDirectSum

  7.6-268 AddUniversalMorphismIntoDirectSumWithGivenDirectSum

  7.6-269 AddUniversalMorphismIntoEqualizer

  7.6-270 AddUniversalMorphismIntoEqualizerWithGivenEqualizer

  7.6-271 AddUniversalMorphismIntoFiberProduct

  7.6-272 AddUniversalMorphismIntoFiberProductWithGivenFiberProduct

  7.6-273 AddUniversalMorphismIntoTerminalObject

  7.6-274 AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject

  7.6-275 AddUniversalMorphismIntoZeroObject

  7.6-276 AddUniversalMorphismIntoZeroObjectWithGivenZeroObject

  7.6-277 AddVerticalPostCompose

  7.6-278 AddVerticalPreCompose

  7.6-279 AddZeroMorphism

  7.6-280 AddZeroObject

  7.6-281 AddZeroObjectFunctorial

  7.6-282 AddZeroObjectFunctorialWithGivenZeroObjects
8 Managing Derived Methods
9 Technical Details
 9.2 Tools

  9.2-1 FunctionWithNamedArguments

  9.2-2 CAP_INTERNAL_GET_DATA_TYPE_FROM_STRING

  9.2-3 CAP_INTERNAL_GET_DATA_TYPES_FROM_STRINGS

  9.2-4 CAP_INTERNAL_REPLACED_STRING_WITH_FILTER

  9.2-5 CAP_INTERNAL_REPLACED_STRINGS_WITH_FILTERS

  9.2-6 CAP_INTERNAL_MERGE_FILTER_LISTS

  9.2-7 CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT

  9.2-8 CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION

  9.2-9 CAP_INTERNAL_MERGE_PRECONDITIONS_LIST

  9.2-10 CAP_INTERNAL_ASSERT_VALUE_IS_OF_TYPE_GETTER

  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 CachingStatistic

  9.2-16 BrowseCachingStatistic

  9.2-17 InstallDeprecatedAlias

  9.2-18 IsSpecializationOfFilter

  9.2-19 IsSpecializationOfFilterList

  9.2-20 InstallMethodForCompilerForCAP

  9.2-21 InstallOtherMethodForCompilerForCAP

  9.2-22 CapJitAddKnownMethod

  9.2-23 CapJitAddTypeSignature

  9.2-24 CapJitAddTypeSignatureDeferred

  9.2-25 CapJitDataTypeOfListOf

  9.2-26 CapJitDataTypeOfNTupleOf

  9.2-27 CapJitDataTypeOfCategory

  9.2-28 CapJitTypedExpression

  9.2-29 CapFixpoint

  9.2-30 Iterated

  9.2-31 Iterated

  9.2-32 TransitivelyNeededOtherPackages

  9.2-33 PackageOfCAPOperation

  9.2-34 SafePosition

  9.2-35 SafeUniquePosition

  9.2-36 SafePositionProperty

  9.2-37 SafeUniquePositionProperty

  9.2-38 SafeFirst

  9.2-39 SafeUniqueEntry

  9.2-40 NTuple

  9.2-41 Pair

  9.2-42 Triple

  9.2-43 TransposedMatWithGivenDimensions

  9.2-44 HandlePrecompiledTowers

  9.2-45 CAP_JIT_INCOMPLETE_LOGIC

  9.2-46 CAP_JIT_EXPR_CASE_WRAPPER

  9.2-47 ListWithKeys

  9.2-48 SumWithKeys

  9.2-49 ProductWithKeys

  9.2-50 ForAllWithKeys

  9.2-51 ForAnyWithKeys

  9.2-52 NumberWithKeys

  9.2-53 FilteredWithKeys

  9.2-54 FirstWithKeys

  9.2-55 LastWithKeys
10 Limits and Colimits
11 The Category Constructor
12 Reinterpretations of categories
13 Create wrapper hulls of a category
14 Dummy implementations
15 Examples and Tests
16 Terminal category
Index

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