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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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                                      CAP 

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                      Categories, Algorithms, Programming 

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                                   2017.09.25

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                               25 September 2017

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                               Sebastian Gutsche

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                                Sebastian Posur

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                            Øystein Skartsæterhagen

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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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  Øystein Skartsæterhagen

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      Email:    mailto:[email protected]

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      Homepage: http://www.math.ntnu.no/~oysteini/

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      Address:  NTNU

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                Institutt for matematiske fag

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                7491 Trondheim

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                Norway

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

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  Contents (CAP)

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  1 CAP Categories

59
    1.1 Categories

60
      1.1-1 IsCapCategory

61
      1.1-2 IsCapCategoryCell

62
      1.1-3 IsCapCategoryObject

63
      1.1-4 IsCapCategoryMorphism

64
      1.1-5 IsCapCategoryTwoCell

65
    1.2 Constructor

66
      1.2-1 CreateCapCategory

67
      1.2-2 CreateCapCategory

68
    1.3 Internal Attributes

69
      1.3-1 CategoryFilter

70
      1.3-2 CellFilter

71
      1.3-3 ObjectFilter

72
      1.3-4 MorphismFilter

73
      1.3-5 TwoCellFilter

74
    1.4 Logic switcher

75
      1.4-1 CapCategorySwitchLogicOn

76
      1.4-2 CapCategorySwitchLogicOff

77
    1.5 Tool functions

78
      1.5-1 CanCompute

79
      1.5-2 CheckConstructivenessOfCategory

80
    1.6 Well-Definedness of Cells

81
      1.6-1 IsWellDefined

82
    1.7 Type check

83
      1.7-1 DisableBasicOperationTypeCheck

84
  2 Category of Categories

85
    2.1 The Category Cat

86
      2.1-1 CapCat

87
    2.2 Categories

88
      2.2-1 IsCapCategoryAsCatObject

89
      2.2-2 IsCapFunctor

90
      2.2-3 IsCapNaturalTransformation

91
    2.3 Constructors

92
      2.3-1 AsCatObject

93
      2.3-2 AsCapCategory

94
    2.4 Functors

95
      2.4-1 CapFunctor

96
      2.4-2 AddObjectFunction

97
      2.4-3 FunctorObjectOperation

98
      2.4-4 AddMorphismFunction

99
      2.4-5 FunctorMorphismOperation

100
      2.4-6 ApplyFunctor

101
      2.4-7 InstallFunctor

102
      2.4-8 IdentityFunctor

103
      2.4-9 FunctorCanonicalizeZeroObjects

104
      2.4-10 NaturalIsomorophismFromIdentityToCanonicalizeZeroObjects

105
      2.4-11 FunctorCanonicalizeZeroMorphisms

106
      2.4-12 NaturalIsomorophismFromIdentityToCanonicalizeZeroMorphisms

107
    2.5 Natural transformations

108
      2.5-1 Name

109
      2.5-2 NaturalTransformation

110
      2.5-3 AddNaturalTransformationFunction

111
      2.5-4 ApplyNaturalTransformation

112
      2.5-5 InstallNaturalTransformation

113
      2.5-6 HorizontalPreComposeNaturalTransformationWithFunctor

114
      2.5-7 HorizontalPreComposeFunctorWithNaturalTransformation

115
  3 Morphisms

116
    3.1 Attributes for the Type of Morphisms

117
      3.1-1 CapCategory

118
      3.1-2 Source

119
      3.1-3 Range

120
    3.2 Categorical Properties of Morphisms

121
      3.2-1 AddIsMonomorphism

122
      3.2-2 AddIsEpimorphism

123
      3.2-3 AddIsIsomorphism

124
      3.2-4 AddIsSplitMonomorphism

125
      3.2-5 AddIsSplitEpimorphism

126
      3.2-6 AddIsOne

127
      3.2-7 AddIsIdempotent

128
    3.3 Non-Categorical Properties of Morphisms

129
      3.3-1 IsIdenticalToIdentityMorphism

130
      3.3-2 AddIsIdenticalToIdentityMorphism

131
      3.3-3 IsIdenticalToZeroMorphism

132
      3.3-4 AddIsIdenticalToZeroMorphism

133
      3.3-5 AddIsEndomorphism

134
      3.3-6 AddIsAutomorphism

135
    3.4 Equality and Congruence for Morphisms

136
      3.4-1 IsCongruentForMorphisms

137
      3.4-2 AddIsCongruentForMorphisms

138
      3.4-3 IsEqualForMorphisms

139
      3.4-4 AddIsEqualForMorphisms

140
      3.4-5 IsEqualForMorphismsOnMor

141
      3.4-6 AddIsEqualForMorphismsOnMor

142
    3.5 Basic Operations for Morphisms in Ab-Categories

143
      3.5-1 IsZeroForMorphisms

144
      3.5-2 AddIsZeroForMorphisms

145
      3.5-3 AdditionForMorphisms

146
      3.5-4 AddAdditionForMorphisms

147
      3.5-5 SubtractionForMorphisms

148
      3.5-6 AddSubtractionForMorphisms

149
      3.5-7 AdditiveInverseForMorphisms

150
      3.5-8 AddAdditiveInverseForMorphisms

151
      3.5-9 ZeroMorphism

152
      3.5-10 AddZeroMorphism

153
    3.6 Subobject and Factorobject Operations

154
      3.6-1 IsEqualAsSubobjects

155
      3.6-2 AddIsEqualAsSubobjects

156
      3.6-3 IsEqualAsFactorobjects

157
      3.6-4 AddIsEqualAsFactorobjects

158
      3.6-5 IsDominating

159
      3.6-6 AddIsDominating

160
      3.6-7 IsCodominating

161
      3.6-8 AddIsCodominating

162
    3.7 Identity Morphism and Composition of Morphisms

163
      3.7-1 IdentityMorphism

164
      3.7-2 AddIdentityMorphism

165
      3.7-3 PreCompose

166
      3.7-4 PreCompose

167
      3.7-5 AddPreCompose

168
      3.7-6 PostCompose

169
      3.7-7 PostCompose

170
      3.7-8 AddPostCompose

171
    3.8 Well-Definedness of Morphisms

172
      3.8-1 IsWellDefinedForMorphisms

173
      3.8-2 AddIsWellDefinedForMorphisms

174
    3.9 Basic Operations for Morphisms in Abelian Categories

175
      3.9-1 LiftAlongMonomorphism

176
      3.9-2 AddLiftAlongMonomorphism

177
      3.9-3 ColiftAlongEpimorphism

178
      3.9-4 AddColiftAlongEpimorphism

179
    3.10 Lift/ Colift

180
      3.10-1 Lift

181
      3.10-2 AddLift

182
      3.10-3 Colift

183
      3.10-4 AddColift

184
    3.11 Inverses

185
      3.11-1 AddInverse

186
    3.12 Tool functions for caches

187
      3.12-1 IsEqualForCacheForMorphisms

188
      3.12-2 AddIsEqualForCacheForMorphisms

189
  4 Objects

190
    4.1 Attributes for the Type of Objects

191
      4.1-1 CapCategory

192
    4.2 Equality for Objects

193
      4.2-1 IsEqualForObjects

194
      4.2-2 AddIsEqualForObjects

195
    4.3 Categorical Properties of Objects

196
      4.3-1 AddIsProjective

197
      4.3-2 AddIsInjective

198
      4.3-3 AddIsTerminal

199
      4.3-4 AddIsInitial

200
      4.3-5 IsZeroForObjects

201
      4.3-6 AddIsZeroForObjects

202
    4.4 Tool functions for caches

203
      4.4-1 IsEqualForCacheForObjects

204
      4.4-2 AddIsEqualForCacheForObjects

205
    4.5 Well-Definedness of Objects

206
      4.5-1 IsWellDefinedForObjects

207
      4.5-2 AddIsWellDefinedForObjects

208
    4.6 Projectives

209
      4.6-1 SomeProjectiveObject

210
      4.6-2 EpimorphismFromSomeProjectiveObject

211
      4.6-3 EpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject

212
      4.6-4 ProjectiveLift

213
      4.6-5 AddSomeProjectiveObject

214
      4.6-6 AddEpimorphismFromSomeProjectiveObject

215
      4.6-7 AddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject

216
      4.6-8 AddProjectiveLift

217
    4.7 Injectives

218
      4.7-1 SomeInjectiveObject

219
      4.7-2 MonomorphismIntoSomeInjectiveObject

220
      4.7-3 MonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject

221
      4.7-4 InjectiveColift

222
      4.7-5 AddSomeInjectiveObject

223
      4.7-6 AddMonomorphismIntoSomeInjectiveObject

224
      4.7-7 AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject

225
      4.7-8 AddInjectiveColift

226
  5 Category 2-Cells

227
    5.1 Attributes for the Type of 2-Cells

228
      5.1-1 Source

229
      5.1-2 Range

230
    5.2 Identity 2-Cell and Composition of 2-Cells

231
      5.2-1 IdentityTwoCell

232
      5.2-2 AddIdentityTwoCell

233
      5.2-3 HorizontalPreCompose

234
      5.2-4 AddHorizontalPreCompose

235
      5.2-5 HorizontalPostCompose

236
      5.2-6 AddHorizontalPostCompose

237
      5.2-7 VerticalPreCompose

238
      5.2-8 AddVerticalPreCompose

239
      5.2-9 VerticalPostCompose

240
      5.2-10 AddVerticalPostCompose

241
    5.3 Well-Definedness for 2-Cells

242
      5.3-1 IsWellDefinedForTwoCells

243
      5.3-2 AddIsWellDefinedForTwoCells

244
  6 Universal Objects

245
    6.1 Kernel

246
      6.1-1 KernelObject

247
      6.1-2 KernelEmbedding

248
      6.1-3 KernelEmbeddingWithGivenKernelObject

249
      6.1-4 KernelLift

250
      6.1-5 KernelLiftWithGivenKernelObject

251
      6.1-6 AddKernelObject

252
      6.1-7 AddKernelEmbedding

253
      6.1-8 AddKernelEmbeddingWithGivenKernelObject

254
      6.1-9 AddKernelLift

255
      6.1-10 AddKernelLiftWithGivenKernelObject

256
      6.1-11 KernelObjectFunctorial

257
      6.1-12 KernelObjectFunctorial

258
      6.1-13 KernelObjectFunctorialWithGivenKernelObjects

259
      6.1-14 AddKernelObjectFunctorialWithGivenKernelObjects

260
    6.2 Cokernel

261
      6.2-1 CokernelObject

262
      6.2-2 CokernelProjection

263
      6.2-3 CokernelProjectionWithGivenCokernelObject

264
      6.2-4 CokernelColift

265
      6.2-5 CokernelColiftWithGivenCokernelObject

266
      6.2-6 AddCokernelObject

267
      6.2-7 AddCokernelProjection

268
      6.2-8 AddCokernelProjectionWithGivenCokernelObject

269
      6.2-9 AddCokernelColift

270
      6.2-10 AddCokernelColiftWithGivenCokernelObject

271
      6.2-11 CokernelObjectFunctorial

272
      6.2-12 CokernelObjectFunctorial

273
      6.2-13 CokernelObjectFunctorialWithGivenCokernelObjects

274
      6.2-14 AddCokernelObjectFunctorialWithGivenCokernelObjects

275
    6.3 Zero Object

276
      6.3-1 ZeroObject

277
      6.3-2 ZeroObject

278
      6.3-3 MorphismFromZeroObject

279
      6.3-4 MorphismIntoZeroObject

280
      6.3-5 UniversalMorphismFromZeroObject

281
      6.3-6 UniversalMorphismFromZeroObjectWithGivenZeroObject

282
      6.3-7 UniversalMorphismIntoZeroObject

283
      6.3-8 UniversalMorphismIntoZeroObjectWithGivenZeroObject

284
      6.3-9 IsomorphismFromZeroObjectToInitialObject

285
      6.3-10 IsomorphismFromInitialObjectToZeroObject

286
      6.3-11 IsomorphismFromZeroObjectToTerminalObject

287
      6.3-12 IsomorphismFromTerminalObjectToZeroObject

288
      6.3-13 AddZeroObject

289
      6.3-14 AddUniversalMorphismIntoZeroObject

290
      6.3-15 AddUniversalMorphismIntoZeroObjectWithGivenZeroObject

291
      6.3-16 AddUniversalMorphismFromZeroObject

292
      6.3-17 AddUniversalMorphismFromZeroObjectWithGivenZeroObject

293
      6.3-18 AddIsomorphismFromZeroObjectToInitialObject

294
      6.3-19 AddIsomorphismFromInitialObjectToZeroObject

295
      6.3-20 AddIsomorphismFromZeroObjectToTerminalObject

296
      6.3-21 AddIsomorphismFromTerminalObjectToZeroObject

297
      6.3-22 ZeroObjectFunctorial

298
      6.3-23 AddZeroObjectFunctorial

299
    6.4 Terminal Object

300
      6.4-1 TerminalObject

301
      6.4-2 TerminalObject

302
      6.4-3 UniversalMorphismIntoTerminalObject

303
      6.4-4 UniversalMorphismIntoTerminalObjectWithGivenTerminalObject

304
      6.4-5 AddTerminalObject

305
      6.4-6 AddUniversalMorphismIntoTerminalObject

306
      6.4-7 AddUniversalMorphismIntoTerminalObjectWithGivenTerminalObject

307
      6.4-8 TerminalObjectFunctorial

308
      6.4-9 AddTerminalObjectFunctorial

309
    6.5 Initial Object

310
      6.5-1 InitialObject

311
      6.5-2 InitialObject

312
      6.5-3 UniversalMorphismFromInitialObject

313
      6.5-4 UniversalMorphismFromInitialObjectWithGivenInitialObject

314
      6.5-5 AddInitialObject

315
      6.5-6 AddUniversalMorphismFromInitialObject

316
      6.5-7 AddUniversalMorphismFromInitialObjectWithGivenInitialObject

317
      6.5-8 InitialObjectFunctorial

318
      6.5-9 AddInitialObjectFunctorial

319
    6.6 Direct Sum

320
      6.6-1 DirectSumOp

321
      6.6-2 ProjectionInFactorOfDirectSum

322
      6.6-3 ProjectionInFactorOfDirectSumOp

323
      6.6-4 ProjectionInFactorOfDirectSumWithGivenDirectSum

324
      6.6-5 InjectionOfCofactorOfDirectSum

325
      6.6-6 InjectionOfCofactorOfDirectSumOp

326
      6.6-7 InjectionOfCofactorOfDirectSumWithGivenDirectSum

327
      6.6-8 UniversalMorphismIntoDirectSum

328
      6.6-9 UniversalMorphismIntoDirectSumOp

329
      6.6-10 UniversalMorphismIntoDirectSumWithGivenDirectSum

330
      6.6-11 UniversalMorphismFromDirectSum

331
      6.6-12 UniversalMorphismFromDirectSumOp

332
      6.6-13 UniversalMorphismFromDirectSumWithGivenDirectSum

333
      6.6-14 IsomorphismFromDirectSumToDirectProduct

334
      6.6-15 IsomorphismFromDirectSumToDirectProductOp

335
      6.6-16 IsomorphismFromDirectProductToDirectSum

336
      6.6-17 IsomorphismFromDirectProductToDirectSumOp

337
      6.6-18 IsomorphismFromDirectSumToCoproduct

338
      6.6-19 IsomorphismFromDirectSumToCoproductOp

339
      6.6-20 IsomorphismFromCoproductToDirectSum

340
      6.6-21 IsomorphismFromCoproductToDirectSumOp

341
      6.6-22 MorphismBetweenDirectSums

342
      6.6-23 MorphismBetweenDirectSumsOp

343
      6.6-24 AddProjectionInFactorOfDirectSum

344
      6.6-25 AddProjectionInFactorOfDirectSumWithGivenDirectSum

345
      6.6-26 AddInjectionOfCofactorOfDirectSum

346
      6.6-27 AddInjectionOfCofactorOfDirectSumWithGivenDirectSum

347
      6.6-28 AddUniversalMorphismIntoDirectSum

348
      6.6-29 AddUniversalMorphismIntoDirectSumWithGivenDirectSum

349
      6.6-30 AddUniversalMorphismFromDirectSum

350
      6.6-31 AddUniversalMorphismFromDirectSumWithGivenDirectSum

351
      6.6-32 AddIsomorphismFromDirectSumToDirectProduct

352
      6.6-33 AddIsomorphismFromDirectProductToDirectSum

353
      6.6-34 AddIsomorphismFromDirectSumToCoproduct

354
      6.6-35 AddIsomorphismFromCoproductToDirectSum

355
      6.6-36 AddDirectSum

356
      6.6-37 DirectSumFunctorial

357
      6.6-38 DirectSumFunctorialWithGivenDirectSums

358
      6.6-39 AddDirectSumFunctorialWithGivenDirectSums

359
    6.7 Coproduct

360
      6.7-1 Coproduct

361
      6.7-2 Coproduct

362
      6.7-3 Coproduct

363
      6.7-4 CoproductOp

364
      6.7-5 InjectionOfCofactorOfCoproduct

365
      6.7-6 InjectionOfCofactorOfCoproductOp

366
      6.7-7 InjectionOfCofactorOfCoproductWithGivenCoproduct

367
      6.7-8 UniversalMorphismFromCoproduct

368
      6.7-9 UniversalMorphismFromCoproductOp

369
      6.7-10 UniversalMorphismFromCoproductWithGivenCoproduct

370
      6.7-11 AddCoproduct

371
      6.7-12 AddInjectionOfCofactorOfCoproduct

372
      6.7-13 AddInjectionOfCofactorOfCoproductWithGivenCoproduct

373
      6.7-14 AddUniversalMorphismFromCoproduct

374
      6.7-15 AddUniversalMorphismFromCoproductWithGivenCoproduct

375
      6.7-16 CoproductFunctorial

376
      6.7-17 CoproductFunctorialWithGivenCoproducts

377
      6.7-18 AddCoproductFunctorialWithGivenCoproducts

378
    6.8 Direct Product

379
      6.8-1 DirectProductOp

380
      6.8-2 ProjectionInFactorOfDirectProduct

381
      6.8-3 ProjectionInFactorOfDirectProductOp

382
      6.8-4 ProjectionInFactorOfDirectProductWithGivenDirectProduct

383
      6.8-5 UniversalMorphismIntoDirectProduct

384
      6.8-6 UniversalMorphismIntoDirectProductOp

385
      6.8-7 UniversalMorphismIntoDirectProductWithGivenDirectProduct

386
      6.8-8 AddDirectProduct

387
      6.8-9 AddProjectionInFactorOfDirectProduct

388
      6.8-10 AddProjectionInFactorOfDirectProductWithGivenDirectProduct

389
      6.8-11 AddUniversalMorphismIntoDirectProduct

390
      6.8-12 AddUniversalMorphismIntoDirectProductWithGivenDirectProduct

391
      6.8-13 DirectProductFunctorial

392
      6.8-14 DirectProductFunctorialWithGivenDirectProducts

393
      6.8-15 AddDirectProductFunctorialWithGivenDirectProducts

394
    6.9 Fiber Product

395
      6.9-1 IsomorphismFromFiberProductToKernelOfDiagonalDifference

396
      6.9-2 IsomorphismFromFiberProductToKernelOfDiagonalDifferenceOp

397
      6.9-3 AddIsomorphismFromFiberProductToKernelOfDiagonalDifference

398
      6.9-4 IsomorphismFromKernelOfDiagonalDifferenceToFiberProduct

399
      6.9-5 IsomorphismFromKernelOfDiagonalDifferenceToFiberProductOp

400
      6.9-6 AddIsomorphismFromKernelOfDiagonalDifferenceToFiberProduct

401
      6.9-7 DirectSumDiagonalDifference

402
      6.9-8 DirectSumDiagonalDifferenceOp

403
      6.9-9 AddDirectSumDiagonalDifference

404
      6.9-10 FiberProductEmbeddingInDirectSum

405
      6.9-11 FiberProductEmbeddingInDirectSumOp

406
      6.9-12 AddFiberProductEmbeddingInDirectSum

407
      6.9-13 FiberProduct

408
      6.9-14 FiberProductOp

409
      6.9-15 ProjectionInFactorOfFiberProduct

410
      6.9-16 ProjectionInFactorOfFiberProductOp

411
      6.9-17 ProjectionInFactorOfFiberProductWithGivenFiberProduct

412
      6.9-18 UniversalMorphismIntoFiberProduct

413
      6.9-19 UniversalMorphismIntoFiberProductOp

414
      6.9-20 UniversalMorphismIntoFiberProductWithGivenFiberProduct

415
      6.9-21 AddFiberProduct

416
      6.9-22 AddProjectionInFactorOfFiberProduct

417
      6.9-23 AddProjectionInFactorOfFiberProductWithGivenFiberProduct

418
      6.9-24 AddUniversalMorphismIntoFiberProduct

419
      6.9-25 AddUniversalMorphismIntoFiberProductWithGivenFiberProduct

420
      6.9-26 FiberProductFunctorial

421
      6.9-27 FiberProductFunctorialWithGivenFiberProducts

422
      6.9-28 AddFiberProductFunctorialWithGivenFiberProducts

423
    6.10 Pushout

424
      6.10-1 IsomorphismFromPushoutToCokernelOfDiagonalDifference

425
      6.10-2 IsomorphismFromPushoutToCokernelOfDiagonalDifferenceOp

426
      6.10-3 AddIsomorphismFromPushoutToCokernelOfDiagonalDifference

427
      6.10-4 IsomorphismFromCokernelOfDiagonalDifferenceToPushout

428
      6.10-5 IsomorphismFromCokernelOfDiagonalDifferenceToPushoutOp

429
      6.10-6 AddIsomorphismFromCokernelOfDiagonalDifferenceToPushout

430
      6.10-7 DirectSumCodiagonalDifference

431
      6.10-8 DirectSumCodiagonalDifferenceOp

432
      6.10-9 AddDirectSumCodiagonalDifference

433
      6.10-10 DirectSumProjectionInPushout

434
      6.10-11 DirectSumProjectionInPushoutOp

435
      6.10-12 AddDirectSumProjectionInPushout

436
      6.10-13 Pushout

437
      6.10-14 Pushout

438
      6.10-15 PushoutOp

439
      6.10-16 InjectionOfCofactorOfPushout

440
      6.10-17 InjectionOfCofactorOfPushoutOp

441
      6.10-18 InjectionOfCofactorOfPushoutWithGivenPushout

442
      6.10-19 UniversalMorphismFromPushout

443
      6.10-20 UniversalMorphismFromPushoutOp

444
      6.10-21 UniversalMorphismFromPushoutWithGivenPushout

445
      6.10-22 AddPushout

446
      6.10-23 AddInjectionOfCofactorOfPushout

447
      6.10-24 AddInjectionOfCofactorOfPushoutWithGivenPushout

448
      6.10-25 AddUniversalMorphismFromPushout

449
      6.10-26 AddUniversalMorphismFromPushoutWithGivenPushout

450
      6.10-27 PushoutFunctorial

451
      6.10-28 PushoutFunctorialWithGivenPushouts

452
      6.10-29 AddPushoutFunctorialWithGivenPushouts

453
    6.11 Image

454
      6.11-1 IsomorphismFromImageObjectToKernelOfCokernel

455
      6.11-2 AddIsomorphismFromImageObjectToKernelOfCokernel

456
      6.11-3 IsomorphismFromKernelOfCokernelToImageObject

457
      6.11-4 AddIsomorphismFromKernelOfCokernelToImageObject

458
      6.11-5 ImageObject

459
      6.11-6 ImageEmbedding

460
      6.11-7 ImageEmbeddingWithGivenImageObject

461
      6.11-8 CoastrictionToImage

462
      6.11-9 CoastrictionToImageWithGivenImageObject

463
      6.11-10 UniversalMorphismFromImage

464
      6.11-11 UniversalMorphismFromImageWithGivenImageObject

465
      6.11-12 AddImageObject

466
      6.11-13 AddImageEmbedding

467
      6.11-14 AddImageEmbeddingWithGivenImageObject

468
      6.11-15 AddCoastrictionToImage

469
      6.11-16 AddCoastrictionToImageWithGivenImageObject

470
      6.11-17 AddUniversalMorphismFromImage

471
      6.11-18 AddUniversalMorphismFromImageWithGivenImageObject

472
    6.12 Coimage

473
      6.12-1 MorphismFromCoimageToImage

474
      6.12-2 MorphismFromCoimageToImageWithGivenObjects

475
      6.12-3 AddMorphismFromCoimageToImageWithGivenObjects

476
      6.12-4 InverseMorphismFromCoimageToImage

477
      6.12-5 InverseMorphismFromCoimageToImageWithGivenObjects

478
      6.12-6 AddInverseMorphismFromCoimageToImageWithGivenObjects

479
      6.12-7 IsomorphismFromCoimageToCokernelOfKernel

480
      6.12-8 AddIsomorphismFromCoimageToCokernelOfKernel

481
      6.12-9 IsomorphismFromCokernelOfKernelToCoimage

482
      6.12-10 AddIsomorphismFromCokernelOfKernelToCoimage

483
      6.12-11 Coimage

484
      6.12-12 CoimageProjection

485
      6.12-13 CoimageProjection

486
      6.12-14 CoimageProjectionWithGivenCoimage

487
      6.12-15 AstrictionToCoimage

488
      6.12-16 AstrictionToCoimage

489
      6.12-17 AstrictionToCoimageWithGivenCoimage

490
      6.12-18 UniversalMorphismIntoCoimage

491
      6.12-19 UniversalMorphismIntoCoimageWithGivenCoimage

492
      6.12-20 AddCoimage

493
      6.12-21 AddCoimageProjection

494
      6.12-22 AddCoimageProjectionWithGivenCoimage

495
      6.12-23 AddAstrictionToCoimage

496
      6.12-24 AddAstrictionToCoimageWithGivenCoimage

497
      6.12-25 AddUniversalMorphismIntoCoimage

498
      6.12-26 AddUniversalMorphismIntoCoimageWithGivenCoimage

499
  7 Tensor Product and Internal Hom

500
    7.1 Monoidal Categories

501
      7.1-1 TensorProductOnObjects

502
      7.1-2 AddTensorProductOnObjects

503
      7.1-3 TensorProductOnMorphisms

504
      7.1-4 TensorProductOnMorphismsWithGivenTensorProducts

505
      7.1-5 AddTensorProductOnMorphismsWithGivenTensorProducts

506
      7.1-6 AssociatorRightToLeft

507
      7.1-7 AssociatorRightToLeftWithGivenTensorProducts

508
      7.1-8 AddAssociatorRightToLeftWithGivenTensorProducts

509
      7.1-9 AssociatorLeftToRight

510
      7.1-10 AssociatorLeftToRightWithGivenTensorProducts

511
      7.1-11 AddAssociatorLeftToRightWithGivenTensorProducts

512
      7.1-12 TensorUnit

513
      7.1-13 AddTensorUnit

514
      7.1-14 LeftUnitor

515
      7.1-15 LeftUnitorWithGivenTensorProduct

516
      7.1-16 AddLeftUnitorWithGivenTensorProduct

517
      7.1-17 LeftUnitorInverse

518
      7.1-18 LeftUnitorInverseWithGivenTensorProduct

519
      7.1-19 AddLeftUnitorInverseWithGivenTensorProduct

520
      7.1-20 RightUnitor

521
      7.1-21 RightUnitorWithGivenTensorProduct

522
      7.1-22 AddRightUnitorWithGivenTensorProduct

523
      7.1-23 RightUnitorInverse

524
      7.1-24 RightUnitorInverseWithGivenTensorProduct

525
      7.1-25 AddRightUnitorInverseWithGivenTensorProduct

526
      7.1-26 LeftDistributivityExpanding

527
      7.1-27 LeftDistributivityExpandingWithGivenObjects

528
      7.1-28 AddLeftDistributivityExpandingWithGivenObjects

529
      7.1-29 LeftDistributivityFactoring

530
      7.1-30 LeftDistributivityFactoringWithGivenObjects

531
      7.1-31 AddLeftDistributivityFactoringWithGivenObjects

532
      7.1-32 RightDistributivityExpanding

533
      7.1-33 RightDistributivityExpandingWithGivenObjects

534
      7.1-34 AddRightDistributivityExpandingWithGivenObjects

535
      7.1-35 RightDistributivityFactoring

536
      7.1-36 RightDistributivityFactoringWithGivenObjects

537
      7.1-37 AddRightDistributivityFactoringWithGivenObjects

538
    7.2 Braided Monoidal Categories

539
      7.2-1 Braiding

540
      7.2-2 BraidingWithGivenTensorProducts

541
      7.2-3 AddBraidingWithGivenTensorProducts

542
      7.2-4 BraidingInverse

543
      7.2-5 BraidingInverseWithGivenTensorProducts

544
      7.2-6 AddBraidingInverseWithGivenTensorProducts

545
    7.3 Symmetric Monoidal Categories

546
    7.4 Symmetric Closed Monoidal Categories

547
      7.4-1 InternalHomOnObjects

548
      7.4-2 AddInternalHomOnObjects

549
      7.4-3 InternalHomOnMorphisms

550
      7.4-4 InternalHomOnMorphismsWithGivenInternalHoms

551
      7.4-5 AddInternalHomOnMorphismsWithGivenInternalHoms

552
      7.4-6 EvaluationMorphism

553
      7.4-7 EvaluationMorphismWithGivenSource

554
      7.4-8 AddEvaluationMorphismWithGivenSource

555
      7.4-9 CoevaluationMorphism

556
      7.4-10 CoevaluationMorphismWithGivenRange

557
      7.4-11 AddCoevaluationMorphismWithGivenRange

558
      7.4-12 TensorProductToInternalHomAdjunctionMap

559
      7.4-13 AddTensorProductToInternalHomAdjunctionMap

560
      7.4-14 InternalHomToTensorProductAdjunctionMap

561
      7.4-15 AddInternalHomToTensorProductAdjunctionMap

562
      7.4-16 MonoidalPreComposeMorphism

563
      7.4-17 MonoidalPreComposeMorphismWithGivenObjects

564
      7.4-18 AddMonoidalPreComposeMorphismWithGivenObjects

565
      7.4-19 MonoidalPostComposeMorphism

566
      7.4-20 MonoidalPostComposeMorphismWithGivenObjects

567
      7.4-21 AddMonoidalPostComposeMorphismWithGivenObjects

568
      7.4-22 DualOnObjects

569
      7.4-23 AddDualOnObjects

570
      7.4-24 DualOnMorphisms

571
      7.4-25 DualOnMorphismsWithGivenDuals

572
      7.4-26 AddDualOnMorphismsWithGivenDuals

573
      7.4-27 EvaluationForDual

574
      7.4-28 EvaluationForDualWithGivenTensorProduct

575
      7.4-29 AddEvaluationForDualWithGivenTensorProduct

576
      7.4-30 CoevaluationForDual

577
      7.4-31 CoevaluationForDualWithGivenTensorProduct

578
      7.4-32 AddCoevaluationForDualWithGivenTensorProduct

579
      7.4-33 MorphismToBidual

580
      7.4-34 MorphismToBidualWithGivenBidual

581
      7.4-35 AddMorphismToBidualWithGivenBidual

582
      7.4-36 TensorProductInternalHomCompatibilityMorphism

583
      7.4-37 TensorProductInternalHomCompatibilityMorphismWithGivenObjects

584
      7.4-38 AddTensorProductInternalHomCompatibilityMorphismWithGivenObjects

585
      7.4-39 TensorProductDualityCompatibilityMorphism

586
      7.4-40 TensorProductDualityCompatibilityMorphismWithGivenObjects

587
      7.4-41 AddTensorProductDualityCompatibilityMorphismWithGivenObjects

588
      7.4-42 MorphismFromTensorProductToInternalHom

589
      7.4-43 MorphismFromTensorProductToInternalHomWithGivenObjects

590
      7.4-44 AddMorphismFromTensorProductToInternalHomWithGivenObjects

591
      7.4-45 IsomorphismFromTensorProductToInternalHom

592
      7.4-46 AddIsomorphismFromTensorProductToInternalHom

593
      7.4-47 MorphismFromInternalHomToTensorProduct

594
      7.4-48 MorphismFromInternalHomToTensorProductWithGivenObjects

595
      7.4-49 AddMorphismFromInternalHomToTensorProductWithGivenObjects

596
      7.4-50 IsomorphismFromInternalHomToTensorProduct

597
      7.4-51 AddIsomorphismFromInternalHomToTensorProduct

598
      7.4-52 TraceMap

599
      7.4-53 AddTraceMap

600
      7.4-54 RankMorphism

601
      7.4-55 AddRankMorphism

602
      7.4-56 IsomorphismFromDualToInternalHom

603
      7.4-57 AddIsomorphismFromDualToInternalHom

604
      7.4-58 IsomorphismFromInternalHomToDual

605
      7.4-59 AddIsomorphismFromInternalHomToDual

606
      7.4-60 UniversalPropertyOfDual

607
      7.4-61 AddUniversalPropertyOfDual

608
      7.4-62 LambdaIntroduction

609
      7.4-63 AddLambdaIntroduction

610
      7.4-64 LambdaElimination

611
      7.4-65 AddLambdaElimination

612
      7.4-66 IsomorphismFromObjectToInternalHom

613
      7.4-67 IsomorphismFromObjectToInternalHomWithGivenInternalHom

614
      7.4-68 AddIsomorphismFromObjectToInternalHomWithGivenInternalHom

615
      7.4-69 IsomorphismFromInternalHomToObject

616
      7.4-70 IsomorphismFromInternalHomToObjectWithGivenInternalHom

617
      7.4-71 AddIsomorphismFromInternalHomToObjectWithGivenInternalHom

618
    7.5 Rigid Symmetric Closed Monoidal Categories

619
      7.5-1 TensorProductInternalHomCompatibilityMorphismInverse

620
      7.5-2 TensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects

621
      7.5-3 AddTensorProductInternalHomCompatibilityMorphismInverseWithGivenObjects

622
      7.5-4 MorphismFromBidual

623
      7.5-5 MorphismFromBidualWithGivenBidual

624
      7.5-6 AddMorphismFromBidualWithGivenBidual

625
  8 Managing Derived Methods

626
    8.1 Info Class

627
      8.1-1 DerivationInfo

628
      8.1-2 ActivateDerivationInfo

629
      8.1-3 DeactivateDerivationInfo

630
    8.2 Derivation Objects

631
      8.2-1 IsDerivedMethod

632
      8.2-2 MakeDerivation

633
      8.2-3 DerivationName

634
      8.2-4 DerivationWeight

635
      8.2-5 DerivationFunctionsWithExtraFilters

636
      8.2-6 CategoryFilter

637
      8.2-7 IsApplicableToCategory

638
      8.2-8 TargetOperation

639
      8.2-9 UsedOperations

640
      8.2-10 UsedOperationMultiples

641
      8.2-11 UsedOperationsWithMultiples

642
      8.2-12 InstallDerivationForCategory

643
      8.2-13 DerivationResultWeight

644
    8.3 Derivation Graphs

645
      8.3-1 IsDerivedMethodGraph

646
      8.3-2 MakeDerivationGraph

647
      8.3-3 AddOperationsToDerivationGraph

648
      8.3-4 AddDerivation

649
      8.3-5 AddDerivation

650
      8.3-6 AddDerivation

651
      8.3-7 AddDerivation

652
      8.3-8 AddDerivationPair

653
      8.3-9 AddDerivationPair

654
      8.3-10 AddDerivationPair

655
      8.3-11 AddDerivationPair

656
      8.3-12 AddDerivationToCAP

657
      8.3-13 AddDerivationPairToCAP

658
      8.3-14 AddWithGivenDerivationPairToCAP

659
      8.3-15 Operations

660
      8.3-16 DerivationsUsingOperation

661
      8.3-17 DerivationsOfOperation

662
    8.4 Managing Derivations in a Category

663
      8.4-1 IsOperationWeightList

664
      8.4-2 MakeOperationWeightList

665
      8.4-3 DerivationGraph

666
      8.4-4 CategoryOfOperationWeightList

667
      8.4-5 CurrentOperationWeight

668
      8.4-6 OperationWeightUsingDerivation

669
      8.4-7 DerivationOfOperation

670
      8.4-8 InstallDerivationsUsingOperation

671
      8.4-9 Reevaluate

672
      8.4-10 AddPrimitiveOperation

673
      8.4-11 PrintDerivationTree

674
      8.4-12 PrintTree

675
      8.4-13 PrintTreeRec

676
    8.5 Min Heaps for Strings

677
      8.5-1 IsStringMinHeap

678
      8.5-2 StringMinHeap

679
      8.5-3 Add

680
      8.5-4 ExtractMin

681
      8.5-5 DecreaseKey

682
      8.5-6 IsEmptyHeap

683
      8.5-7 HeapSize

684
      8.5-8 Contains

685
      8.5-9 Swap

686
      8.5-10 Heapify

687
  9 Add Functions

688
    9.1 Functions Installed by Add

689
    9.2 Add Method

690
    9.3 InstallAdd Function

691
    9.4 Install All Adds

692
  10 Technical Details

693
    10.1 The Category Cat

694
      10.1-1 ObjectCache

695
      10.1-2 MorphismCache

696
    10.2 Install Functions for IsWellDefined

697
      10.2-1 InstallMethodWithToDoForIsWellDefined

698
      10.2-2 InstallSetWithToDoForIsWellDefined

699
      10.2-3 DeclareAttributeWithToDoForIsWellDefined

700
      10.2-4 DeclareFamilyProperty

701
      10.2-5 CAP_INTERNAL_REPLACE_STRINGS_WITH_FILTERS

702
      10.2-6 CAP_INTERNAL_MERGE_FILTER_LISTS

703
      10.2-7 CAP_INTERNAL_RETURN_OPTION_OR_DEFAULT

704
      10.2-8 CAP_INTERNAL_FIND_APPEARANCE_OF_SYMBOL_IN_FUNCTION

705
      10.2-9 CAP_INTERNAL_MERGE_PRECONDITIONS_LIST

706
  11 Monoidal Categories

707
  12 Examples and Tests

708
    12.1 Spectral Sequences

709
    12.2 Monoidal Categories

710
    12.3 Generalized Morphisms Category

711
    12.4 IsWellDefined

712
    12.5 Kernel

713
    12.6 FiberProduct

714
  

715
  

716
  

717


718