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FunctorCategories

Categories of functors

2023.01-15

27 January 2023

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 Range
  1.2-3 Source
  1.2-4 Range
  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 CoYonedaLemma
  2.1-5 SomeDiagramOfRepresentables
  2.1-6 CategoryOfInternalCategories

  2.2-1 Source
  2.2-2 Range
  2.2-3 Source
  2.2-4 Range
  2.2-5 ValuesOnAllObjects
  2.2-6 UnderlyingCapTwoCategoryCell
  2.2-7 YonedaEmbedding
  2.2-8 YonedaEmbeddingOfSourceCategory
  2.2-9 ImageOfYonedaEmbeddingOfSource
  2.2-10 SimpleObjects
  2.2-11 NerveTruncatedInDegree2
  2.2-12 SievesOfPathsToTruth

  2.3-1 ApplyObjectInPreSheafCategoryToObject
  2.3-2 ApplyObjectInPreSheafCategoryToMorphism
  2.3-3 ApplyMorphismInPreSheafCategoryToObject

2.4 GAP Categories

  2.4-1 IsPreSheafCategory
  2.4-2 IsCellInPreSheafCategory
  2.4-3 IsObjectInPreSheafCategory
  2.4-4 IsMorphismInPreSheafCategory
  2.4-5 IsPreSheafCategoryOfFpEnrichedCategory
  2.4-6 IsCellInPreSheafCategoryOfFpEnrichedCategory
  2.4-7 IsObjectInPreSheafCategoryOfFpEnrichedCategory
  2.4-8 IsMorphismInPreSheafCategoryOfFpEnrichedCategory

2.5 Global variables

  2.5-1 CAP_INTERNAL_METHOD_NAME_LIST_FOR_PRESHEAF_CATEGORY
  2.5-2 CAP_INTERNAL_METHOD_NAME_LIST_FOR_MONOIDAL_PRESHEAF_CATEGORY
  2.5-3 CAP_INTERNAL_METHOD_NAME_LIST_FOR_MONOIDAL_PRESHEAF_CATEGORY_WITH_DUALS

2.6-1 The subobject classifier
2.6-2 PreSheaves of non-f.p. categories (experimental)

3 Finite cocompletion of a finitely presented (linear) category

3.1 Constructors

3.1-1 FiniteCocompletion

3.2-1 UnderlyingCategory
3.2-2 YonedaEmbeddingOfUnderlyingCategory

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

10 Finite completion of a finitely presented (linear) category

10.1 Constructors

10.1-1 FiniteCompletion

10.2 Attributes

10.2-1 UnderlyingCategory
10.2-2 CoYonedaEmbeddingOfUnderlyingCategory

10.3 GAP Categories

  10.3-1 IsFiniteCompletion
  10.3-2 IsCellInFiniteCompletion
  10.3-3 IsObjectInFiniteCompletion
  10.3-4 IsMorphismInFiniteCompletion

10.4-1 The terminal category as a finite completion of the initial category

11 Free Abelian category of a finitely presented linear category

11.1 Constructors

11.1-1 AbelianClosure

11.2 Attributes

11.2-1 UnderlyingCategory
11.2-2 YonedaEmbeddingOfUnderlyingCategory

11.3 GAP Categories

  11.3-1 IsAbelianClosure
  11.3-2 IsCellInAbelianClosure
  11.3-3 IsObjectInAbelianClosure
  11.3-4 IsMorphismInAbelianClosure

11.4-1 Proof of the snake lemma in a free Abelian category

12 Functors between the various categories

12.1-1 The reflexive completion of the one object category $C 2$

12.2 Attributes

  12.2-1 NakayamaLeftAdjoint
  12.2-2 NakayamaRightAdjoint
  12.2-3 IsbellLeftAdjoint
  12.2-4 IsbellRightAdjoint
  12.2-5 IsbellAdjunctionMonad
  12.2-6 UnitOfIsbellAdjunction
  12.2-7 IsomorphismFromSourceIntoImageOfYonedaEmbeddingOfSource
  12.2-8 IsomorphismFromImageOfYonedaEmbeddingOfSourceIntoSource
  12.2-9 EquivalenceFromFullSubcategoryOfProjectivesObjectsIntoAdditiveClosureOfSource

13 Futher CAP operations

13.1 Add-methods

13.1-1 AddIsReflexive

14.1 Properties

14.1-1 IsReflexive

15 Precompilation

15.1 Precompiling the category of finite quivers

15.2 Precompiling the category of finite reflexive quivers

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