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FunctorCategories

Categories of functors

2024.12-01

6 December 2024

Mohamed Barakat
Email: mohamed.barakat@uni-siegen.de
Homepage: https://mohamed-barakat.github.io/
Address:
Walter-Flex-Str. 3
57072 Siegen
Germany

Kamal Saleh
Email: kamal.saleh@uni-siegen.de
Homepage: https://github.com/kamalsaleh
Address:
Department Mathematik
Universität Siegen
Walter-Flex-Straße 3
57072 Siegen
Germany

Contents

1 Category of functors of a finitely presented (linear) category

1.1 Constructors

  1.1-1 FunctorCategory
  1.1-2 AsObjectInFunctorCategory
  1.1-3 AsMorphismInFunctorCategoryByValues

  1.2-1 Source
  1.2-2 Target
  1.2-3 Source
  1.2-4 Target
  1.2-5 OppositeOfSource
  1.2-6 ValuesOfFunctor
  1.2-7 ValuesOnAllObjects
  1.2-8 UnderlyingCapTwoCategoryCell
  1.2-9 YonedaEmbeddingInFunctorCategory
  1.2-10 YonedaEmbeddingOfOppositeOfSourceCategory
  1.2-11 YonedaFibration
  1.2-12 YonedaProjection
  1.2-13 YonedaComposition

  1.3-1 DecomposeOnceByRandomEndomorphism
  1.3-2 WeakDirectSumDecomposition
  1.3-3 ApplyObjectInFunctorCategoryToObject
  1.3-4 ApplyObjectInFunctorCategoryToMorphism
  1.3-5 ApplyMorphismInFunctorCategoryToObject

1.4 GAP Categories

  1.4-1 IsFunctorCategory
  1.4-2 IsCellInFunctorCategory
  1.4-3 IsObjectInFunctorCategory
  1.4-4 IsMorphismInFunctorCategory

  1.5-1 Representations of the cyclic groups of order 3 as a category of functors
  1.5-2 Differential modules as a category of functors
  1.5-3 The category of equivariant maps as a category of functors

2 Category of presheaves of a finitely presented (linear) category

2.1 Constructors

  2.1-1 PreSheaves
  2.1-2 CreatePreSheaf
  2.1-3 CreatePreSheafMorphismByValues
  2.1-4 CoYonedaLemmaOnObjects
  2.1-5 CoYonedaLemmaOnMorphisms
  2.1-6 CoequalizerDataOfPreSheafUsingCoYonedaLemma
  2.1-7 CoYonedaLemmaCoequalizerPair
  2.1-8 EmbeddingFunctorOfFiniteStrictCoproductCompletionIntoPreSheavesData
  2.1-9 EmbeddingFunctorOfFiniteStrictCoproductCompletionIntoPreSheaves
  2.1-10 MorphismFromRepresentableByYonedaLemma
  2.1-11 MorphismFromCoproductOfRepresentables
  2.1-12 MaximalMorphismFromRepresentable
  2.1-13 CoveringListOfRepresentables
  2.1-14 CoveringListOfRepresentablesUsingSplits
  2.1-15 EpimorphismFromSomeProjectiveObjectUsingSplits
  2.1-16 SectionAndComplementByCoveringListOfRepresentables
  2.1-17 SectionFromOptimizedCoYonedaProjectiveObjectIntoCoYonedaProjectiveObject
  2.1-18 RetractionByCoveringListOfRepresentables
  2.1-19 RetractionFromCoYonedaProjectiveObjectOntoOptimizedCoYonedaProjectiveObject
  2.1-20 OptimizedCoYonedaLemmaOnObjects
  2.1-21 OptimizedCoYonedaLemmaOnMorphisms
  2.1-22 CoequalizerDataOfPreSheafUsingOptimizedCoYonedaLemma
  2.1-23 AssociatedCoequalizerPairInPreSheaves
  2.1-24 AssociatedCoequalizerPairInPreSheaves
  2.1-25 OptimizedCoYonedaLemmaCoequalizerPair
  2.1-26 ApplyPreSheafToObjectInFiniteStrictCoproductCompletion
  2.1-27 ApplyPreSheafToMorphismInFiniteStrictCoproductCompletion
  2.1-28 CategoryOfInternalCategories

  2.2-1 ValuesOnAllObjects
  2.2-2 UnderlyingCapTwoCategoryCell
  2.2-3 YonedaEmbedding
  2.2-4 YonedaEmbeddingOfSourceCategory
  2.2-5 ImageOfYonedaEmbeddingOfSource
  2.2-6 SimpleObjects
  2.2-7 NerveTruncatedInDegree2
  2.2-8 SievesOfPathsToTruth
  2.2-9 FiniteStrictCoproductCompletionOfSourceCategory
  2.2-10 FiniteColimitCompletionWithStrictCoproductsOfSourceCategory
  2.2-11 CategoryOfColimitQuiversOfSourceCategory

  2.3-1 ApplyObjectInPreSheafCategoryOfFpEnrichedCategoryToObject
  2.3-2 ApplyObjectInPreSheafCategoryOfFpEnrichedCategoryToMorphism
  2.3-3 ApplyObjectInPreSheafCategoryOfFpEnrichedCategoryToGeneratingMorphismOrIdentity
  2.3-4 ApplyMorphismInPreSheafCategoryOfFpEnrichedCategoryToObject

2.4 GAP Categories

  2.4-1 IsPreSheafCategoryOfFpEnrichedCategory
  2.4-2 IsCellInPreSheafCategoryOfFpEnrichedCategory
  2.4-3 IsObjectInPreSheafCategoryOfFpEnrichedCategory
  2.4-4 IsMorphismInPreSheafCategoryOfFpEnrichedCategory

2.5 Global variables

  2.6-1 The subobject classifier
  2.6-2 PreSheaves of non-f.p. categories (experimental)
  2.6-3 The colimit completion of a cyclic monoid
  2.6-4 The linearization of the colimit completion of a cyclic monoid
  2.6-5 The Weierstrass normal form
  2.6-6 Optimized co-Yoneda lemma for posets

3 Finite cocompletion of a finitely presented (linear) category

3.1 Constructors

3.1-1 FiniteCocompletion

3.2-1 UnderlyingCategory
3.2-2 EmbeddingOfUnderlyingCategory

3.3 GAP Categories

  3.3-1 IsFiniteCocompletion
  3.3-2 IsCellInFiniteCocompletion
  3.3-3 IsObjectInFiniteCocompletion
  3.3-4 IsMorphismInFiniteCocompletion

6 Topos operations

6.1 Double-pushout rewriting

9 Finite completion of a finitely presented (linear) category

9.1 Constructors

9.1-1 FiniteCompletion

9.2-1 UnderlyingCategory
9.2-2 EmbeddingOfUnderlyingCategory

9.3 GAP Categories

  9.3-1 IsFiniteCompletion
  9.3-2 IsCellInFiniteCompletion
  9.3-3 IsObjectInFiniteCompletion
  9.3-4 IsMorphismInFiniteCompletion

9.4-1 The terminal category as a finite completion of the initial category

10 Free Abelian category of a finitely presented linear category

10.1 Constructors

10.1-1 AbelianClosure

10.2 Attributes

10.2-1 UnderlyingCategory
10.2-2 EmbeddingOfUnderlyingCategory

10.3 GAP Categories

  10.3-1 IsAbelianClosure
  10.3-2 IsCellInAbelianClosure
  10.3-3 IsObjectInAbelianClosure
  10.3-4 IsMorphismInAbelianClosure

10.4-1 Proof of the snake lemma in a free Abelian category

11 Functors between the various categories

11.1-1 The reflexive completion of the one object category $C 2$

11.2 Attributes

  11.2-1 NakayamaLeftAdjoint
  11.2-2 NakayamaRightAdjoint
  11.2-3 IsbellLeftAdjoint
  11.2-4 IsbellRightAdjoint
  11.2-5 IsbellAdjunctionMonad
  11.2-6 UnitOfIsbellAdjunction
  11.2-7 IsomorphismFromSourceIntoImageOfYonedaEmbeddingOfSource
  11.2-8 IsomorphismFromImageOfYonedaEmbeddingOfSourceIntoSource
  11.2-9 EquivalenceFromFullSubcategoryOfProjectivesObjectsIntoAdditiveClosureOfSource

12 Category of copresheaves of a finitely presented (linear) category

12.1 GAP Categories

  12.1-1 IsCoPreSheafCategory
  12.1-2 IsCellInCoPreSheafCategory
  12.1-3 IsObjectInCoPreSheafCategory
  12.1-4 IsMorphismInCoPreSheafCategory

12.2 Attributes

  12.2-1 Source
  12.2-2 Target
  12.2-3 Source
  12.2-4 Target
  12.2-5 OppositeOfSource
  12.2-6 ValuesOfCoPreSheaf
  12.2-7 ValuesOnAllObjects
  12.2-8 UnderlyingCapTwoCategoryCell
  12.2-9 CoYonedaEmbedding
  12.2-10 CoYonedaEmbeddingOfSourceCategory

12.3 Operations

  12.3-1 ApplyObjectInCoPreSheafCategoryToObject
  12.3-2 ApplyObjectInCoPreSheafCategoryToMorphism
  12.3-3 ApplyMorphismInCoPreSheafCategoryToObject

12.4 Constructors

  12.4-1 CoPreSheaves
  12.4-2 CreateCoPreSheaf
  12.4-3 CreateCoPreSheafMorphismByValues

13 Free distributive completion of a finitely presented (linear) category

13.1 GAP Categories

  13.1-1 IsFreeDistributiveCompletion
  13.1-2 IsCellInFreeDistributiveCompletion
  13.1-3 IsObjectInFreeDistributiveCompletion
  13.1-4 IsMorphismInFreeDistributiveCompletion

13.2 Attributes

13.2-1 UnderlyingCategory
13.2-2 EmbeddingOfUnderlyingCategory

13.3 Constructors

13.3-1 FreeDistributiveCompletion

14 Quotient categories of algebroids from data tables using presheaves categories

14.1 Attributes

14.1-1 AlgebroidAsObjectInPreSheavesCategory
14.1-2 AssociatedMorphismIntoAlgebroidAsObjectInPreSheavesCategory

15.1 Properties

15.1-1 IsReflexive

16 Futher CAP operations

16.1 Add-methods

16.1-1 AddIsReflexive

17 Precompilation

17.1 Precompiling the category of finite quivers

17.2 Precompiling the category of finite reflexive quivers

17.3 Precompiling the presheaf category of an algebroid from data tables with values in a category of rows

17.4 Precompiling the presheaf category of an algebroid with values in a category of rows

17.5 Precompiling the category of presheaves with values in SkeletalFinSets

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