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CAP

Categories, Algorithms, Programming

2022.11-28

29 November 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 AddCoefficientsOfMorphism

  7.7-12 AddCoequalizer

  7.7-13 AddCoequalizerFunctorial

  7.7-14 AddCoequalizerFunctorialWithGivenCoequalizers

  7.7-15 AddCoimageObject

  7.7-16 AddCoimageObjectFunctorial

  7.7-17 AddCoimageObjectFunctorialWithGivenCoimageObjects

  7.7-18 AddCoimageProjection

  7.7-19 AddCoimageProjectionWithGivenCoimageObject

  7.7-20 AddCokernelColift

  7.7-21 AddCokernelColiftWithGivenCokernelObject

  7.7-22 AddCokernelObject

  7.7-23 AddCokernelObjectFunctorial

  7.7-24 AddCokernelObjectFunctorialWithGivenCokernelObjects

  7.7-25 AddCokernelProjection

  7.7-26 AddCokernelProjectionWithGivenCokernelObject

  7.7-27 AddColift

  7.7-28 AddColiftAlongEpimorphism

  7.7-29 AddColiftOrFail

  7.7-30 AddComponentOfMorphismFromDirectSum

  7.7-31 AddComponentOfMorphismIntoDirectSum

  7.7-32 AddCoproduct

  7.7-33 AddCoproductFunctorial

  7.7-34 AddCoproductFunctorialWithGivenCoproducts

  7.7-35 AddDirectProduct

  7.7-36 AddDirectProductFunctorial

  7.7-37 AddDirectProductFunctorialWithGivenDirectProducts

  7.7-38 AddDirectSum

  7.7-39 AddDirectSumCodiagonalDifference

  7.7-40 AddDirectSumDiagonalDifference

  7.7-41 AddDirectSumFunctorial

  7.7-42 AddDirectSumFunctorialWithGivenDirectSums

  7.7-43 AddDirectSumProjectionInPushout

  7.7-44 AddDistinguishedObjectOfHomomorphismStructure

  7.7-45 AddEmbeddingOfEqualizer

  7.7-46 AddEmbeddingOfEqualizerWithGivenEqualizer

  7.7-47 AddEpimorphismFromProjectiveCoverObject

  7.7-48 AddEpimorphismFromProjectiveCoverObjectWithGivenProjectiveCoverObject

  7.7-49 AddEpimorphismFromSomeProjectiveObject

  7.7-50 AddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject

  7.7-51 AddEqualizer

  7.7-52 AddEqualizerFunctorial

  7.7-53 AddEqualizerFunctorialWithGivenEqualizers

  7.7-54 AddFiberProduct

  7.7-55 AddFiberProductEmbeddingInDirectSum

  7.7-56 AddFiberProductFunctorial

  7.7-57 AddFiberProductFunctorialWithGivenFiberProducts

  7.7-58 AddHomologyObject

  7.7-59 AddHomologyObjectFunctorialWithGivenHomologyObjects

  7.7-60 AddHomomorphismStructureOnMorphisms

  7.7-61 AddHomomorphismStructureOnMorphismsWithGivenObjects

  7.7-62 AddHomomorphismStructureOnObjects

  7.7-63 AddHorizontalPostCompose

  7.7-64 AddHorizontalPreCompose

  7.7-65 AddIdentityMorphism

  7.7-66 AddIdentityTwoCell

  7.7-67 AddImageEmbedding

  7.7-68 AddImageEmbeddingWithGivenImageObject

  7.7-69 AddImageObject

  7.7-70 AddImageObjectFunctorial

  7.7-71 AddImageObjectFunctorialWithGivenImageObjects

  7.7-72 AddIndecomposableInjectiveObjects

  7.7-73 AddIndecomposableProjectiveObjects

  7.7-74 AddInitialObject

  7.7-75 AddInitialObjectFunctorial

  7.7-76 AddInitialObjectFunctorialWithGivenInitialObjects

  7.7-77 AddInjectionOfCofactorOfCoproduct

  7.7-78 AddInjectionOfCofactorOfCoproductWithGivenCoproduct

  7.7-79 AddInjectionOfCofactorOfDirectSum

  7.7-80 AddInjectionOfCofactorOfDirectSumWithGivenDirectSum

  7.7-81 AddInjectionOfCofactorOfPushout

  7.7-82 AddInjectionOfCofactorOfPushoutWithGivenPushout

  7.7-83 AddInjectiveColift

  7.7-84 AddInjectiveDimension

  7.7-85 AddInjectiveEnvelopeObject

  7.7-86 AddInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructure

  7.7-87 AddInterpretMorphismAsMorphismFromDistinguishedObjectToHomomorphismStructureWithGivenObjects

  7.7-88 AddInterpretMorphismFromDistinguishedObjectToHomomorphismStructureAsMorphism

  7.7-89 AddInverseForMorphisms

  7.7-90 AddInverseMorphismFromCoimageToImageWithGivenObjects

  7.7-91 AddIsAutomorphism

  7.7-92 AddIsBijectiveObject

  7.7-93 AddIsCodominating

  7.7-94 AddIsColiftable

  7.7-95 AddIsColiftableAlongEpimorphism

  7.7-96 AddIsCongruentForMorphisms

  7.7-97 AddIsDominating

  7.7-98 AddIsEndomorphism

  7.7-99 AddIsEpimorphism

  7.7-100 AddIsEqualAsFactorobjects

  7.7-101 AddIsEqualAsSubobjects

  7.7-102 AddIsEqualForCacheForMorphisms

  7.7-103 AddIsEqualForCacheForObjects

  7.7-104 AddIsEqualForMorphisms

  7.7-105 AddIsEqualForMorphismsOnMor

  7.7-106 AddIsEqualForObjects

  7.7-107 AddIsEqualToIdentityMorphism

  7.7-108 AddIsEqualToZeroMorphism

  7.7-109 AddIsHomSetInhabited

  7.7-110 AddIsIdempotent

  7.7-111 AddIsInitial

  7.7-112 AddIsInjective

  7.7-113 AddIsIsomorphism

  7.7-114 AddIsLiftable

  7.7-115 AddIsLiftableAlongMonomorphism

  7.7-116 AddIsMonomorphism

  7.7-117 AddIsOne

  7.7-118 AddIsProjective

  7.7-119 AddIsSplitEpimorphism

  7.7-120 AddIsSplitMonomorphism

  7.7-121 AddIsTerminal

  7.7-122 AddIsWellDefinedForMorphisms

  7.7-123 AddIsWellDefinedForObjects

  7.7-124 AddIsWellDefinedForTwoCells

  7.7-125 AddIsZeroForMorphisms

  7.7-126 AddIsZeroForObjects

  7.7-127 AddIsomorphismFromCoequalizerOfCoproductDiagramToPushout

  7.7-128 AddIsomorphismFromCoimageToCokernelOfKernel

  7.7-129 AddIsomorphismFromCokernelOfDiagonalDifferenceToPushout

  7.7-130 AddIsomorphismFromCokernelOfKernelToCoimage

  7.7-131 AddIsomorphismFromCoproductToDirectSum

  7.7-132 AddIsomorphismFromDirectProductToDirectSum

  7.7-133 AddIsomorphismFromDirectSumToCoproduct

  7.7-134 AddIsomorphismFromDirectSumToDirectProduct

  7.7-135 AddIsomorphismFromEqualizerOfDirectProductDiagramToFiberProduct

  7.7-136 AddIsomorphismFromFiberProductToEqualizerOfDirectProductDiagram

  7.7-137 AddIsomorphismFromFiberProductToKernelOfDiagonalDifference

  7.7-138 AddIsomorphismFromHomologyObjectToItsConstructionAsAnImageObject

  7.7-139 AddIsomorphismFromImageObjectToKernelOfCokernel

  7.7-140 AddIsomorphismFromInitialObjectToZeroObject

  7.7-141 AddIsomorphismFromItsConstructionAsAnImageObjectToHomologyObject

  7.7-142 AddIsomorphismFromKernelOfCokernelToImageObject

  7.7-143 AddIsomorphismFromKernelOfDiagonalDifferenceToFiberProduct

  7.7-144 AddIsomorphismFromPushoutToCoequalizerOfCoproductDiagram

  7.7-145 AddIsomorphismFromPushoutToCokernelOfDiagonalDifference

  7.7-146 AddIsomorphismFromTerminalObjectToZeroObject

  7.7-147 AddIsomorphismFromZeroObjectToInitialObject

  7.7-148 AddIsomorphismFromZeroObjectToTerminalObject

  7.7-149 AddKernelEmbedding

  7.7-150 AddKernelEmbeddingWithGivenKernelObject

  7.7-151 AddKernelLift

  7.7-152 AddKernelLiftWithGivenKernelObject

  7.7-153 AddKernelObject

  7.7-154 AddKernelObjectFunctorial

  7.7-155 AddKernelObjectFunctorialWithGivenKernelObjects

  7.7-156 AddLift

  7.7-157 AddLiftAlongMonomorphism

  7.7-158 AddLiftOrFail

  7.7-159 AddMereExistenceOfSolutionOfLinearSystemInAbCategory

  7.7-160 AddMonomorphismIntoInjectiveEnvelopeObject

  7.7-161 AddMonomorphismIntoInjectiveEnvelopeObjectWithGivenInjectiveEnvelopeObject

  7.7-162 AddMonomorphismIntoSomeInjectiveObject

  7.7-163 AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject

  7.7-164 AddMorphismBetweenDirectSums

  7.7-165 AddMorphismBetweenDirectSumsWithGivenDirectSums

  7.7-166 AddMorphismConstructor

  7.7-167 AddMorphismDatum

  7.7-168 AddMorphismFromCoimageToImageWithGivenObjects

  7.7-169 AddMorphismFromEqualizerToSink

  7.7-170 AddMorphismFromEqualizerToSinkWithGivenEqualizer

  7.7-171 AddMorphismFromFiberProductToSink

  7.7-172 AddMorphismFromFiberProductToSinkWithGivenFiberProduct

  7.7-173 AddMorphismFromKernelObjectToSink

  7.7-174 AddMorphismFromKernelObjectToSinkWithGivenKernelObject

  7.7-175 AddMorphismFromSourceToCoequalizer

  7.7-176 AddMorphismFromSourceToCoequalizerWithGivenCoequalizer

  7.7-177 AddMorphismFromSourceToCokernelObject

  7.7-178 AddMorphismFromSourceToCokernelObjectWithGivenCokernelObject

  7.7-179 AddMorphismFromSourceToPushout

  7.7-180 AddMorphismFromSourceToPushoutWithGivenPushout

  7.7-181 AddMultiplyWithElementOfCommutativeRingForMorphisms

  7.7-182 AddObjectConstructor

  7.7-183 AddObjectDatum

  7.7-184 AddPostCompose

  7.7-185 AddPostComposeList

  7.7-186 AddPostInverseForMorphisms

  7.7-187 AddPreCompose

  7.7-188 AddPreComposeList

  7.7-189 AddPreInverseForMorphisms

  7.7-190 AddProjectionInFactorOfDirectProduct

  7.7-191 AddProjectionInFactorOfDirectProductWithGivenDirectProduct

  7.7-192 AddProjectionInFactorOfDirectSum

  7.7-193 AddProjectionInFactorOfDirectSumWithGivenDirectSum

  7.7-194 AddProjectionInFactorOfFiberProduct

  7.7-195 AddProjectionInFactorOfFiberProductWithGivenFiberProduct

  7.7-196 AddProjectionOntoCoequalizer

  7.7-197 AddProjectionOntoCoequalizerWithGivenCoequalizer

  7.7-198 AddProjectiveCoverObject

  7.7-199 AddProjectiveDimension

  7.7-200 AddProjectiveLift

  7.7-201 AddPushout

  7.7-202 AddPushoutFunctorial

  7.7-203 AddPushoutFunctorialWithGivenPushouts

  7.7-204 AddRandomMorphismByInteger

  7.7-205 AddRandomMorphismByList

  7.7-206 AddRandomMorphismWithFixedRangeByInteger

  7.7-207 AddRandomMorphismWithFixedRangeByList

  7.7-208 AddRandomMorphismWithFixedSourceAndRangeByInteger

  7.7-209 AddRandomMorphismWithFixedSourceAndRangeByList

  7.7-210 AddRandomMorphismWithFixedSourceByInteger

  7.7-211 AddRandomMorphismWithFixedSourceByList

  7.7-212 AddRandomObjectByInteger

  7.7-213 AddRandomObjectByList

  7.7-214 AddSimplifyEndo

  7.7-215 AddSimplifyEndo_IsoFromInputObject

  7.7-216 AddSimplifyEndo_IsoToInputObject

  7.7-217 AddSimplifyMorphism

  7.7-218 AddSimplifyObject

  7.7-219 AddSimplifyObject_IsoFromInputObject

  7.7-220 AddSimplifyObject_IsoToInputObject

  7.7-221 AddSimplifyRange

  7.7-222 AddSimplifyRange_IsoFromInputObject

  7.7-223 AddSimplifyRange_IsoToInputObject

  7.7-224 AddSimplifySource

  7.7-225 AddSimplifySourceAndRange

  7.7-226 AddSimplifySourceAndRange_IsoFromInputRange

  7.7-227 AddSimplifySourceAndRange_IsoFromInputSource

  7.7-228 AddSimplifySourceAndRange_IsoToInputRange

  7.7-229 AddSimplifySourceAndRange_IsoToInputSource

  7.7-230 AddSimplifySource_IsoFromInputObject

  7.7-231 AddSimplifySource_IsoToInputObject

  7.7-232 AddSolveLinearSystemInAbCategory

  7.7-233 AddSolveLinearSystemInAbCategoryOrFail

  7.7-234 AddSomeInjectiveObject

  7.7-235 AddSomeProjectiveObject

  7.7-236 AddSomeReductionBySplitEpiSummand

  7.7-237 AddSomeReductionBySplitEpiSummand_MorphismFromInputRange

  7.7-238 AddSomeReductionBySplitEpiSummand_MorphismToInputRange

  7.7-239 AddSubtractionForMorphisms

  7.7-240 AddSumOfMorphisms

  7.7-241 AddTerminalObject

  7.7-242 AddTerminalObjectFunctorial

  7.7-243 AddTerminalObjectFunctorialWithGivenTerminalObjects

  7.7-244 AddUniversalMorphismFromCoequalizer

  7.7-245 AddUniversalMorphismFromCoequalizerWithGivenCoequalizer

  7.7-246 AddUniversalMorphismFromCoproduct

  7.7-247 AddUniversalMorphismFromCoproductWithGivenCoproduct

  7.7-248 AddUniversalMorphismFromDirectSum

  7.7-249 AddUniversalMorphismFromDirectSumWithGivenDirectSum

  7.7-250 AddUniversalMorphismFromImage

  7.7-251 AddUniversalMorphismFromImageWithGivenImageObject

  7.7-252 AddUniversalMorphismFromInitialObject

  7.7-253 AddUniversalMorphismFromInitialObjectWithGivenInitialObject

  7.7-254 AddUniversalMorphismFromPushout

  7.7-255 AddUniversalMorphismFromPushoutWithGivenPushout

  7.7-256 AddUniversalMorphismFromZeroObject

  7.7-257 AddUniversalMorphismFromZeroObjectWithGivenZeroObject

  7.7-258 AddUniversalMorphismIntoCoimage

  7.7-259 AddUniversalMorphismIntoCoimageWithGivenCoimageObject

  7.7-260 AddUniversalMorphismIntoDirectProduct

  7.7-261 AddUniversalMorphismIntoDirectProductWithGivenDirectProduct

  7.7-262 AddUniversalMorphismIntoDirectSum

  7.7-263 AddUniversalMorphismIntoDirectSumWithGivenDirectSum

  7.7-264 AddUniversalMorphismIntoEqualizer

  7.7-265 AddUniversalMorphismIntoEqualizerWithGivenEqualizer

  7.7-266 AddUniversalMorphismIntoFiberProduct

  7.7-267 AddUniversalMorphismIntoFiberProductWithGivenFiberProduct

  7.7-268 AddUniversalMorphismIntoTerminalObject

  7.7-269 AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject

  7.7-270 AddUniversalMorphismIntoZeroObject

  7.7-271 AddUniversalMorphismIntoZeroObjectWithGivenZeroObject

  7.7-272 AddVerticalPostCompose

  7.7-273 AddVerticalPreCompose

  7.7-274 AddZeroMorphism

  7.7-275 AddZeroObject

  7.7-276 AddZeroObjectFunctorial

  7.7-277 AddZeroObjectFunctorialWithGivenZeroObjects
8 Managing Derived Methods
9 Technical Details
 9.2 Tools

  9.2-1 DeclareFamilyProperty

  9.2-2 CAP_INTERNAL_REPLACE_STRING_WITH_FILTER

  9.2-3 CAP_INTERNAL_REPLACE_STRINGS_WITH_FILTERS

  9.2-4 CAP_INTERNAL_MERGE_FILTER_LISTS

  9.2-5 CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT

  9.2-6 CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION

  9.2-7 CAP_INTERNAL_MERGE_PRECONDITIONS_LIST

  9.2-8 CAP_INTERNAL_ASSERT_IS_CELL_OF_CATEGORY

  9.2-9 CAP_INTERNAL_ASSERT_IS_OBJECT_OF_CATEGORY

  9.2-10 CAP_INTERNAL_ASSERT_IS_MORPHISM_OF_CATEGORY

  9.2-11 CAP_INTERNAL_ASSERT_IS_TWO_CELL_OF_CATEGORY

  9.2-12 CAP_INTERNAL_ASSERT_IS_LIST_OF_OBJECTS_OF_CATEGORY

  9.2-13 CAP_INTERNAL_ASSERT_IS_LIST_OF_MORPHISMS_OF_CATEGORY

  9.2-14 CAP_INTERNAL_ASSERT_IS_LIST_OF_TWO_CELLS_OF_CATEGORY

  9.2-15 CAP_INTERNAL_ASSERT_IS_NON_NEGATIVE_INTEGER_OR_INFINITY

  9.2-16 CachingStatistic

  9.2-17 BrowseCachingStatistic

  9.2-18 InstallDeprecatedAlias

  9.2-19 IsSpecializationOfFilter

  9.2-20 IsSpecializationOfFilterList

  9.2-21 InstallMethodForCompilerForCAP

  9.2-22 InstallOtherMethodForCompilerForCAP

  9.2-23 CapJitAddKnownMethod

  9.2-24 CapJitAddTypeSignature

  9.2-25 CapJitAddTypeSignatureDeferred

  9.2-26 CapJitDataTypeOfCategory

  9.2-27 CapFixpoint

  9.2-28 Iterated

  9.2-29 TransitivelyNeededOtherPackages

  9.2-30 PackageOfCAPOperation

  9.2-31 SafePosition

  9.2-32 SafePositionProperty

  9.2-33 SafeFirst

  9.2-34 NTuple

  9.2-35 Pair

  9.2-36 Triple

  9.2-37 HandlePrecompiledTowers

  9.2-38 CAP_JIT_INCOMPLETE_LOGIC

  9.2-39 ListWithKeys

  9.2-40 SumWithKeys

  9.2-41 ProductWithKeys

  9.2-42 ForAllWithKeys

  9.2-43 ForAnyWithKeys

  9.2-44 NumberWithKeys

  9.2-45 FilteredWithKeys

  9.2-46 FirstWithKeys

  9.2-47 LastWithKeys
10 Limits and Colimits
11 The Category Constructor
12 Create wrapper hulls of a category
13 Dummy categories
14 Examples and Tests
15 Terminal category
Index

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