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FiniteCocompletions

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

2026.04-02

11 April 2026

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

Marc Talleux
Email: marc.talleux@u-picardie.fr
Homepage: https://mtalleux.pages.math.cnrs.fr/website/
Address:
LAMFA
CNRS
Université de Picardie Jules Verne
33, rue Saint-Leu
80039 Amiens
France

Contents

2 Finite strict product completion

  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

2.4-1 Cartesian braiding
2.4-2 Cartesian diagonal

3 Finite strict coproduct completion of an object-finite category

  3.1-1 PairOfIntAndList
  3.1-2 PairOfLists
  3.1-3 UnderlyingCategory
  3.1-4 NumberOfObjectsOfUnderlyingCategory
  3.1-5 EmbeddingOfUnderlyingCategory
  3.1-6 ExtendFunctorToFiniteStrictCoproductCompletionOfObjectFiniteCategory

3.2 Constructors

3.2-1 FiniteStrictCoproductCompletionOfObjectFiniteCategory

3.3 GAP categories

  3.3-1 IsFiniteStrictCoproductCompletionOfObjectFiniteCategory
  3.3-2 IsCellInFiniteStrictCoproductCompletionOfObjectFiniteCategory
  3.3-3 IsObjectInFiniteStrictCoproductCompletionOfObjectFiniteCategory
  3.3-4 IsMorphismInFiniteStrictCoproductCompletionOfObjectFiniteCategory

3.4-1 Colift along epimorphism
3.4-2 Equalizer

4 Free distributive closure category with strict products and coproducts

4.1-1 UnderlyingCategory

4.2 Constructors

  4.2-1 FreeDistributiveCategoryWithStrictProductAndCoproducts
  4.2-2 EmbeddingOfUnderlyingCategory
  4.2-3 ExtendFunctorToFreeDistributiveCategoryWithStrictProductAndCoproducts

4.3 GAP categories

  4.3-1 IsFreeDistributiveCategoryWithStrictProductAndCoproducts
  4.3-2 IsCellInFreeDistributiveCategoryWithStrictProductAndCoproducts
  4.3-3 IsObjectInFreeDistributiveCategoryWithStrictProductAndCoproducts
  4.3-4 IsMorphismInFreeDistributiveCategoryWithStrictProductAndCoproducts

5 SkeletalFinSets

  5.1-1 Cardinality
  5.1-2 AsList
  5.1-3 AsList

5.2 Constructors

5.2-1 SkeletalCategoryOfFiniteSetsAsFiniteStrictCoproductCompletionOfTerminalCategory
5.2-2 SkeletalFinSetsAsFreeElementaryToposOfInitialCategory

5.3 GAP categories

  5.3-1 IsSkeletalCategoryOfFiniteSetsAsFiniteStrictCoproductCompletionOfTerminalCategory
  5.3-2 IsCellInSkeletalCategoryOfFiniteSetsAsFiniteStrictCoproductCompletionOfTerminalCategory
  5.3-3 IsObjectInSkeletalCategoryOfFiniteSetsAsFiniteStrictCoproductCompletionOfTerminalCategory
  5.3-4 IsMorphismInSkeletalCategoryOfFiniteSetsAsFiniteStrictCoproductCompletionOfTerminalCategory

5.4 Skeletal Examples

  5.4-1 Skeletal WellDefined
  5.4-2 Skeletal PreCompose
  5.4-3 Skeletal Lift
  5.4-4 Morphisms of external Hom
  5.4-5 Skeletal Initial and Terminal Objects
  5.4-6 Skeletal Direct Product
  5.4-7 Skeletal Coproduct
  5.4-8 Skeletal coproduct complement
  5.4-9 Skeletal Equalizer
  5.4-10 Skeletal homomorphism structre

6 The pair of parallel arrows category of a category

6.1 Global variables

6.1-1 QuiverOfCategoryOfQuivers

  6.2-1 PairOfObjectsAndPairOfParallelMorphisms
  6.2-2 DefiningPairOfMorphismBetweenParallelPairs
  6.2-3 UnderlyingCategory

6.3 Constructors

6.3-1 PairOfParallelArrowsCategory

6.4 GAP categories

  6.4-1 IsPairOfParallelArrowsCategory
  6.4-2 IsCellInPairOfParallelArrowsCategory
  6.4-3 IsObjectInPairOfParallelArrowsCategory
  6.4-4 IsMorphismInPairOfParallelArrowsCategory

7 The category of colimit quivers in a category

  7.1-1 UnderlyingCategory
  7.1-2 EmbeddingOfUnderlyingCategory
  7.1-3 DefiningPairOfColimitQuiver
  7.1-4 DefiningPairOfColimitQuiverMorphism
  7.1-5 FiniteColimitCompletionWithStrictCoproductsOfUnderlyingCategory

7.2 Constructors

7.2-1 CategoryOfColimitQuivers
7.2-2 CategoryOfPreSheavesOfUnderlyingCategory

7.3 GAP categories

  7.3-1 IsCategoryOfColimitQuivers
  7.3-2 IsCellInCategoryOfColimitQuivers
  7.3-3 IsObjectInCategoryOfColimitQuivers
  7.3-4 IsMorphismInCategoryOfColimitQuivers

8 The coequalizer completion of a category

  8.1-1 PairOfObjectsAndPairOfParallelMorphisms
  8.1-2 DefiningPairOfMorphismBetweenCoequalizerPairs
  8.1-3 UnderlyingCategory
  8.1-4 EmbeddingOfUnderlyingCategory

8.2 Constructors

8.2-1 CoequalizerCompletion

8.3 GAP categories

  8.3-1 IsCoequalizerCompletion
  8.3-2 IsCellInCoequalizerCompletion
  8.3-3 IsObjectInCoequalizerCompletion
  8.3-4 IsMorphismInCoequalizerCompletion

9 The finite colimit completion of a category

  9.1-1 UnderlyingCategory
  9.1-2 FiniteStrictCoproductCompletionOfUnderlyingCategory
  9.1-3 PairOfObjectsAndPairOfParallelMorphisms
  9.1-4 DefiningPairOfMorphismBetweenCoequalizerPairs
  9.1-5 CategoryOfColimitQuiversOfUnderlyingCategory

9.2 Constructors

  9.2-1 FiniteColimitCompletionWithStrictCoproducts
  9.2-2 CategoryOfPreSheavesOfUnderlyingCategory
  9.2-3 AssociatedColimitQuiver

9.3 GAP categories

  9.3-1 IsFiniteColimitCompletionWithStrictCoproducts
  9.3-2 IsCellInFiniteColimitCompletionWithStrictCoproducts
  9.3-3 IsObjectInFiniteColimitCompletionWithStrictCoproducts
  9.3-4 IsMorphismInFiniteColimitCompletionWithStrictCoproducts

  9.4-1 FinBouquets as finite colimit completion
  9.4-2 FinReflexiveQuivers as finite colimit completion
  9.4-3 Free elementary topos on one object

10 Strict additive closure

10.1 Constructors

10.1-1 EmbeddingOfUnderlyingCategory
10.1-2 ExtendFunctorToStrictAdditiveClosure

11 Cokernel closure

11.1 Constructors

11.1-1 EmbeddingOfUnderlyingCategory
11.1-2 ExtendFunctorToFreydCategory

12 Kernel closure

12.1 Constructors

12.1-1 EmbeddingOfUnderlyingCategory
12.1-2 ExtendFunctorToCoFreydCategory

13 Abelian closure with strict direct sums

13.1 Attributes

13.1-1 UnderlyingCategory

13.2 Constructors

  13.2-1 AbelianClosureAsFreydOfCoFreydOfStrictAdditiveClosure
  13.2-2 EmbeddingOfUnderlyingCategory
  13.2-3 ExtendFunctorToAbelianClosure

13.3 GAP categories

  13.3-1 IsAbelianClosureAsFreydOfCoFreydOfStrictAdditiveClosure
  13.3-2 IsCellInAbelianClosureAsFreydOfCoFreydOfStrictAdditiveClosure
  13.3-3 IsObjectInAbelianClosureAsFreydOfCoFreydOfStrictAdditiveClosure
  13.3-4 IsMorphismInAbelianClosureAsFreydOfCoFreydOfStrictAdditiveClosure

14.1 Operations

  14.1-1 TensorizeObjectWithObjectInRangeCategoryOfHomomorphismStructure
  14.1-2 TensorizeObjectWithMorphismInRangeCategoryOfHomomorphismStructure
  14.1-3 TensorizeMorphismWithObjectInRangeCategoryOfHomomorphismStructure

15 Precompilation

15.1 Precompiling the finite strict coproduct completion

15.2 Precompiling SkeletalFinSets

15.3 Extracting LeftCartesianDistributivityFactoring from ur-algorithms using CategoryFromDataTables

15.4 Extracting LeftCartesianDistributivityFactoring from ur-algorithms using PathCategory

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