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FiniteCocompletions

Finite (co)product/(co)limit (co)completions

2025.06-01

25 June 2025

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io
Address:
Walter-Flex-Str. 3
57068 Siegen
Germany

Contents

1 Finite strict coproduct cocompletion
 1.1 Attributes

  1.1-1 PairOfIntAndList
  1.1-2 PairOfLists
  1.1-3 UnderlyingCategory
  1.1-4 EmbeddingOfUnderlyingCategory
  1.1-5 ExtendFunctorToFiniteStrictCoproductCompletion
  1.1-6 ExtendEmbeddingToFiniteStrictCoproductCompletion
 1.2 Constructors

  1.2-1 FiniteStrictCoproductCompletion
 1.3 GAP categories

  1.3-1 IsFiniteStrictCoproductCompletion
  1.3-2 IsCellInFiniteStrictCoproductCompletion
  1.3-3 IsObjectInFiniteStrictCoproductCompletion
  1.3-4 IsMorphismInFiniteStrictCoproductCompletion
 1.4 Examples

  1.4-1 Cocartesian associator
  1.4-2 Cocartesian braiding
  1.4-3 Cocartesian codiagonal
  1.4-4 Homomorphism structure
  1.4-5 Reflexive pair
2 Finite strict product completion
 2.1 Attributes

  2.1-1 PairOfIntAndList
  2.1-2 PairOfLists
  2.1-3 UnderlyingCategory
  2.1-4 EmbeddingOfUnderlyingCategory
  2.1-5 ExtendFunctorToFiniteStrictProductCompletion
 2.2 Constructors

  2.2-1 FiniteStrictProductCompletion
 2.3 GAP categories

  2.3-1 IsFiniteStrictProductCompletion
  2.3-2 IsCellInFiniteStrictProductCompletion
  2.3-3 IsObjectInFiniteStrictProductCompletion
  2.3-4 IsMorphismInFiniteStrictProductCompletion
 2.4 Examples

  2.4-1 Cartesian braiding
  2.4-2 Cartesian diagonal
3 Free distributive closure category with strict products and coproducts
 3.1 Attributes

  3.1-1 UnderlyingCategory
 3.2 Constructors

  3.2-1 FreeDistributiveCategoryWithStrictProductAndCoproducts
  3.2-2 EmbeddingOfUnderlyingCategory
  3.2-3 ExtendFunctorToFreeDistributiveCategoryWithStrictProductAndCoproducts
 3.3 GAP categories

  3.3-1 IsFreeDistributiveCategoryWithStrictProductAndCoproducts
  3.3-2 IsCellInFreeDistributiveCategoryWithStrictProductAndCoproducts
  3.3-3 IsObjectInFreeDistributiveCategoryWithStrictProductAndCoproducts
  3.3-4 IsMorphismInFreeDistributiveCategoryWithStrictProductAndCoproducts
4 SkeletalFinSets
 4.1 Attributes

  4.1-1 Length
  4.1-2 AsList
  4.1-3 AsList
 4.2 Constructors

  4.2-1 SkeletalCategoryOfFiniteSetsAsFiniteStrictCoproductCompletionOfTerminalCategory
  4.2-2 SkeletalFinSetsAsFreeElementaryToposOfInitialCategory
 4.3 GAP categories

  4.3-1 IsSkeletalCategoryOfFiniteSetsAsFiniteStrictCoproductCompletionOfTerminalCategory
  4.3-2 IsCellInSkeletalCategoryOfFiniteSetsAsFiniteStrictCoproductCompletionOfTerminalCategory
  4.3-3 IsObjectInSkeletalCategoryOfFiniteSetsAsFiniteStrictCoproductCompletionOfTerminalCategory
  4.3-4 IsMorphismInSkeletalCategoryOfFiniteSetsAsFiniteStrictCoproductCompletionOfTerminalCategory
 4.4 Skeletal Examples

  4.4-1 Skeletal WellDefined
  4.4-2 Skeletal PreCompose
  4.4-3 Skeletal Lift
  4.4-4 Morphisms of external Hom
  4.4-5 Skeletal Initial and Terminal Objects
  4.4-6 Skeletal Direct Product
  4.4-7 Skeletal Coproduct
  4.4-8 Skeletal coproduct complement
  4.4-9 Skeletal Equalizer
  4.4-10 Skeletal homomorphism structre
5 The pair of parallel arrows category of a category
 5.1 Global variables

  5.1-1 QuiverOfCategoryOfQuivers
 5.2 Attributes

  5.2-1 PairOfObjectsAndPairOfParallelMorphisms
  5.2-2 DefiningPairOfMorphismBetweenParallelPairs
  5.2-3 UnderlyingCategory
 5.3 Constructors

  5.3-1 PairOfParallelArrowsCategory
 5.4 GAP categories

  5.4-1 IsPairOfParallelArrowsCategory
  5.4-2 IsCellInPairOfParallelArrowsCategory
  5.4-3 IsObjectInPairOfParallelArrowsCategory
  5.4-4 IsMorphismInPairOfParallelArrowsCategory
6 The category of colimit quivers in a category
 6.1 Attributes

  6.1-1 UnderlyingCategory
  6.1-2 EmbeddingOfUnderlyingCategory
  6.1-3 DefiningPairOfColimitQuiver
  6.1-4 DefiningPairOfColimitQuiverMorphism
  6.1-5 FiniteColimitCompletionWithStrictCoproductsOfUnderlyingCategory
 6.2 Constructors

  6.2-1 CategoryOfColimitQuivers
  6.2-2 CreateColimitQuiver
  6.2-3 CreateMorphismOfColimitQuivers
  6.2-4 CategoryOfPreSheavesOfUnderlyingCategory
 6.3 GAP categories

  6.3-1 IsCategoryOfColimitQuivers
  6.3-2 IsCellInCategoryOfColimitQuivers
  6.3-3 IsObjectInCategoryOfColimitQuivers
  6.3-4 IsMorphismInCategoryOfColimitQuivers
7 The coequalizer completion of a category
 7.1 Attributes

  7.1-1 PairOfObjectsAndPairOfParallelMorphisms
  7.1-2 DefiningPairOfMorphismBetweenCoequalizerPairs
  7.1-3 UnderlyingCategory
  7.1-4 EmbeddingOfUnderlyingCategory
 7.2 Constructors

  7.2-1 CoequalizerCompletion
 7.3 GAP categories

  7.3-1 IsCoequalizerCompletion
  7.3-2 IsCellInCoequalizerCompletion
  7.3-3 IsObjectInCoequalizerCompletion
  7.3-4 IsMorphismInCoequalizerCompletion
8 The finite colimit completion of a category
 8.1 Attributes

  8.1-1 UnderlyingCategory
  8.1-2 FiniteStrictCoproductCompletionOfUnderlyingCategory
  8.1-3 PairOfObjectsAndPairOfParallelMorphisms
  8.1-4 DefiningPairOfMorphismBetweenCoequalizerPairs
  8.1-5 CategoryOfColimitQuiversOfUnderlyingCategory
 8.2 Constructors

  8.2-1 FiniteColimitCompletionWithStrictCoproducts
  8.2-2 CategoryOfPreSheavesOfUnderlyingCategory
  8.2-3 AssociatedColimitQuiver
 8.3 GAP categories

  8.3-1 IsFiniteColimitCompletionWithStrictCoproducts
  8.3-2 IsCellInFiniteColimitCompletionWithStrictCoproducts
  8.3-3 IsObjectInFiniteColimitCompletionWithStrictCoproducts
  8.3-4 IsMorphismInFiniteColimitCompletionWithStrictCoproducts
 8.4 Examples

  8.4-1 FinBouquets as finite colimit completion
  8.4-2 FinReflexiveQuivers as finite colimit completion
  8.4-3 Free elementary topos on one object
9 Strict additive closure
 9.1 Constructors

  9.1-1 EmbeddingOfUnderlyingCategory
  9.1-2 ExtendFunctorToStrictAdditiveClosure
10 Cokernel closure
 10.1 Constructors

  10.1-1 EmbeddingOfUnderlyingCategory
  10.1-2 ExtendFunctorToFreydCategory
11 Kernel closure
 11.1 Constructors

  11.1-1 EmbeddingOfUnderlyingCategory
  11.1-2 ExtendFunctorToCoFreydCategory
12 Abelian closure with strict direct sums
 12.1 Attributes

  12.1-1 UnderlyingCategory
 12.2 Constructors

  12.2-1 AbelianClosureWithStrictDirectSumsAsFreydOfCoFreydOfStrictAdditiveClosure
  12.2-2 EmbeddingOfUnderlyingCategory
  12.2-3 ExtendFunctorToAbelianClosureWithStrictDirectSums
 12.3 GAP categories

  12.3-1 IsAbelianClosureWithStrictDirectSumsAsFreydOfCoFreydOfStrictAdditiveClosure
  12.3-2 IsCellInAbelianClosureWithStrictDirectSumsAsFreydOfCoFreydOfStrictAdditiveClosure
  12.3-3 IsObjectInAbelianClosureWithStrictDirectSumsAsFreydOfCoFreydOfStrictAdditiveClosure
  12.3-4 IsMorphismInAbelianClosureWithStrictDirectSumsAsFreydOfCoFreydOfStrictAdditiveClosure
13 Tools
 13.1 Operations

  13.1-1 TensorizeObjectWithObjectInRangeCategoryOfHomomorphismStructure
  13.1-2 TensorizeObjectWithMorphismInRangeCategoryOfHomomorphismStructure
  13.1-3 TensorizeMorphismWithObjectInRangeCategoryOfHomomorphismStructure
14 Precompilation
 14.1 Precompiling the finite strict coproduct completion
 14.2 Precompiling SkeletalFinSets
 14.3 Extracting LeftCartesianDistributivityFactoring from ur-algorithms using CategoryFromDataTables
 14.4 Extracting LeftCartesianDistributivityFactoring from ur-algorithms using PathCategory
Index

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