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### 3 Stable Categories

#### 3.1 GAP categories

##### 3.1-1 IsStableCategory
 ‣ IsStableCategory( seq_obj ) ( filter )

Returns: true or false

The GAP category of stable categories

##### 3.1-2 IsStableCategoryCell
 ‣ IsStableCategoryCell( seq_obj ) ( filter )

Returns: true or false

The GAP category of stable categories cells.

##### 3.1-3 IsStableCategoryObject
 ‣ IsStableCategoryObject( seq_obj ) ( filter )

Returns: true or false

The GAP category of stable categories objects.

##### 3.1-4 IsStableCategoryMorphism
 ‣ IsStableCategoryMorphism( seq_obj ) ( filter )

Returns: true or false

The GAP category of stable categories morphisms.

#### 3.2 Constructors

##### 3.2-1 StableCategory
 ‣ StableCategory( category, f ) ( operation )

Returns: an additive category \CC/I

The arguments are an additive category \CC and a function f which decides whether or a not a morphism \alpha:A \to B in \CC belongs to some two-sided ideal I of morphisms in \CC. The output is the stable category \CC/I.

##### 3.2-2 CongruencyTestFunction
 ‣ CongruencyTestFunction( stable_category ) ( attribute )

Returns: a gap function

The argument is a stable category \CC/I. The output is is a gap function f such that for any \alpha in \CC, [\alpha] \sim 0 if and only if f(\alpha)=\mathrm{true}.

##### 3.2-3 ProjectionFunctor
 ‣ ProjectionFunctor( stable_category ) ( attribute )

Returns: a functor \CC \to \CC/I

The argument is a stable category \CC/I. The output is the natural projection functor \pi: \CC \to \CC/I.

##### 3.2-4 StableCategoryObject
 ‣ StableCategoryObject( stable_category, A ) ( operation )

Returns: [A] in \CC/I

The arguments are a stable category \CC/I and an object A in \CC. The output is [A] in \CC/I.

##### 3.2-5 UnderlyingCell
 ‣ UnderlyingCell( class_A ) ( attribute )

Returns: an object in \CC

The argument is an object [A] in a stable category \CC/I. The output is A.

##### 3.2-6 StableCategoryMorphism
 ‣ StableCategoryMorphism( class_A, alpha, class_B ) ( operation )

Returns: [\alpha]:[A]\to [B] in \CC/I

The arguments are an object [A] in a stable category \CC/I, a morphism \alpha:A\to B in \CC and an object [B] in \CC/I. The output is [\alpha] in \CC/I.

##### 3.2-7 StableCategoryMorphism
 ‣ StableCategoryMorphism( alpha ) ( operation )

Returns: [\alpha]:[A]\to [B] in \CC/I

The arguments are a stable category \CC/I and a morphism \alpha:A\to B in \CC. The output is [\alpha] in \CC/I.

##### 3.2-8 UnderlyingCell
 ‣ UnderlyingCell( class_alpha ) ( attribute )

Returns: an object in \CC

The argument is a morphism [\alpha] in a stable category \CC/I. The output is \alpha.

#### 3.3 Stable categories defined by systems of lifting objects

Let \CC be a category equipped with a system \LL of lifting objects. The set of all morphisms \alpha:A \to B which lift along \ell_B:L_B \to B defines a two-sided ideal I_{\LL} of morphisms in \CC. It can be shown that a morphism \alpha: A \to B belongs to I_{\LL} if and only if it factors through at least on object in \LL. The extra structure which is bundled with the system of lifting objects enables us to lift more than additive structure form \CC to \CC/I_{\LL}. This will be illustrated in the next example. The category \CC/{I_{\LL}} can be constructed by the following operation:

##### 3.3-1 StableCategoryByClassOfLiftingObjects
 ‣ StableCategoryByClassOfLiftingObjects( category ) ( attribute )

Returns: a stable category

The argument is an additive category equipped with a system of lifting objects \LL. The output is the stable category \CC/I_{\LL} where I_{\LL} is the two-sided ideal of morphisms \alpha: A \to B in \CC that lift along \ell_B:L_B \to B.

#### 3.4 Stable categories defined by systems of colifting objects

Let \CC be a category equipped with a system \QQ of colifting objects. The set of all morphisms \alpha:A \to B which colift along q_A:A \to Q_A defines a two-sided ideal I_{\QQ} of morphisms in \CC. It can be shown that a morphism \alpha: A \to B belongs to I_{\QQ} if and only if it factors through at least on object in \QQ. The extra structure which is bundled with the system of colifting objects enables us to lift more than additive structure form \CC to \CC/I_{\LL}. The category \CC/{I_{\QQ}} can be constructed by the following operation:

##### 3.4-1 StableCategoryByClassOfColiftingObjects
 ‣ StableCategoryByClassOfColiftingObjects( category ) ( attribute )

Returns: a stable category

The argument is an additive category equipped with a system of colifting objects \QQ. The output is the stable category \CC/I_{\QQ} where I_{\QQ} is the two-sided ideal of morphisms \alpha: A \to B in \CC that colift along q_A:A \to Q_A.

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