StableCategories

Stable categories of additive categories

2023.03-01

1 March 2023

  1.1-1 IsCapCategoryShortSequence
  1.1-2 IsCapCategoryMorphismOfShortSequences
  1.1-3 IsCapCategoryShortExactSequence
  1.1-4 IsCapCategoryConflation

1.2 Exact categories operations

  1.2-1 IsExactCategory
  1.2-2 IsInflation
  1.2-3 IsDeflation
  1.2-4 IsConflationPair
  1.2-5 ExactCokernelObject
  1.2-6 ExactCokernelProjection
  1.2-7 ExactCokernelProjectionWithGivenExactCokernelObject
  1.2-8 ExactCokernelColift
  1.2-9 ExactCokernelObjectFunctorial
  1.2-10 ExactCokernelObjectFunctorialWithGivenExactCokernelObjects
  1.2-11 ColiftAlongDeflation
  1.2-12 ExactKernelObject
  1.2-13 ExactKernelEmbedding
  1.2-14 ExactKernelEmbeddingWithGivenExactKernelObject
  1.2-15 ExactKernelLift
  1.2-16 ExactKernelObjectFunctorial
  1.2-17 ExactKernelObjectFunctorialWithGivenExactKernelObjects
  1.2-18 LiftAlongInflation
  1.2-19 Exact Fiber Product
  1.2-20 ExactFiberProduct
  1.2-21 ProjectionInFirstFactorOfExactFiberProduct
  1.2-22 ProjectionInSecondFactorOfExactFiberProduct
  1.2-23 UniversalMorphismIntoExactFiberProduct
  1.2-24 Exact Pushout
  1.2-25 ExactPushout
  1.2-26 InjectionOfFirstCofactorOfExactPushout
  1.2-27 InjectionOfSecondCofactorOfExactPushout
  1.2-28 UniversalMorphismFromExactPushout
  1.2-29 UniversalMorphismFromExactPushoutWithGivenExactPushout
  1.2-30 Exact Categories With Enough E-projectives
  1.2-31 IsExactCategoryWithEnoughExactProjectives
  1.2-32 IsExactProjectiveObject
  1.2-33 ExactProjectiveLift
  1.2-34 SomeExactProjectiveObject
  1.2-35 DeflationFromSomeExactProjectiveObject
  1.2-36 Exact Categories With Enough E-injecitves
  1.2-37 IsExactCategoryWithEnoughExactInjectives
  1.2-38 IsExactInjectiveObject
  1.2-39 ExactInjectiveColift
  1.2-40 SomeExactInjectiveObject
  1.2-41 InflationIntoSomeExactInjectiveObject
  1.2-42 IsLiftableAlongDeflationFromSomeExactProjectiveObject
  1.2-43 LiftAlongDeflationFromSomeExactProjectiveObject
  1.2-44 IsColiftableAlongInflationIntoSomeExactInjectiveObject
  1.2-45 ColiftAlongInflationIntoSomeExactInjectiveObject
  1.2-46 IsFrobeniusCategory
  1.2-47 SchanuelsIsomorphismByInflationsIntoSomeExactInjectiveObjects
  1.2-48 SchanuelsIsomorphismByDeflationsFromSomeExactProjectiveObjects

2 Lifting and Colifting systems

2.1 Systems of lifting objects

  2.1-1 IsLiftingObject
  2.1-2 LiftingObject
  2.1-3 MorphismFromLiftingObject
  2.1-4 MorphismFromLiftingObjectWithGivenLiftingObject
  2.1-5 SectionOfMorphismFromLiftingObjectWithGivenLiftingObject
  2.1-6 SectionOfMorphismFromLiftingObject
  2.1-7 LiftingMorphismWithGivenLiftingObjects
  2.1-8 LiftingMorphism
  2.1-9 IsLiftableAlongMorphismFromLiftingObject
  2.1-10 WitnessForBeingLiftableAlongMorphismFromLiftingObject

2.2 Systems of colifting objects

  2.2-1 IsColiftingObject
  2.2-2 ColiftingObject
  2.2-3 MorphismToColiftingObject
  2.2-4 MorphismToColiftingObjectWithGivenColiftingObject
  2.2-5 RetractionOfMorphismToColiftingObjectWithGivenColiftingObject
  2.2-6 RetractionOfMorphismToColiftingObject
  2.2-7 ColiftingMorphismWithGivenColiftingObjects
  2.2-8 ColiftingMorphism
  2.2-9 IsColiftableAlongMorphismToColiftingObject
  2.2-10 WitnessForBeingColiftableAlongMorphismToColiftingObject

2.3 Examples for systems of colifting objects

3 Stable Categories

3.1 GAP categories

  3.1-1 IsStableCategory
  3.1-2 IsStableCategoryCell
  3.1-3 IsStableCategoryObject
  3.1-4 IsStableCategoryMorphism

3.2 Constructors

  3.2-1 StableCategory
  3.2-2 CongruencyTestFunction
  3.2-3 ProjectionFunctor
  3.2-4 StableCategoryObject
  3.2-5 UnderlyingCell
  3.2-6 StableCategoryMorphism
  3.2-7 StableCategoryMorphism
  3.2-8 UnderlyingCell

3.3 Stable categories defined by systems of lifting objects

3.3-1 StableCategoryByClassOfLiftingObjects

3.4 Stable categories defined by systems of colifting objects

3.4-1 StableCategoryByClassOfColiftingObjects

4 Bibliography

Index

StableCategories

Stable categories of additive categories

Contents