GAP (CAP) - Chapter 2: Objects

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2 Objects
2.1 Attributes for the Type of Objects
2.1-1 CapCategory
2.2 Equality for Objects
2.2-1 IsEqualForObjects
2.2-2 AddIsEqualForObjects
2.3 Categorical Properties of Objects
2.3-1 AddIsProjective
2.3-2 AddIsInjective
2.3-3 AddIsTerminal
2.3-4 AddIsInitial
2.3-5 IsZeroForObjects
2.3-6 AddIsZeroForObjects
2.4 Random Objects
2.4-1 RandomObjectByInteger
2.4-2 AddRandomObjectByInteger
2.4-3 RandomObjectByList
2.4-4 AddRandomObjectByList
2.4-5 RandomObject
2.5 Tool functions for caches
2.5-1 IsEqualForCacheForObjects
2.5-2 AddIsEqualForCacheForObjects
2.6 Adding Objects to a Category
2.6-1 Add
2.6-2 AddObject
2.6-3 AddObjectRepresentation
2.6-4 ObjectifyObjectForCAPWithAttributes
2.7 Well-Definedness of Objects
2.7-1 IsWellDefinedForObjects
2.7-2 AddIsWellDefinedForObjects
2.8 Projectives
2.8-1 SomeProjectiveObject
2.8-2 EpimorphismFromSomeProjectiveObject
2.8-3 EpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject
2.8-4 ProjectiveLift
2.8-5 AddSomeProjectiveObject
2.8-6 AddEpimorphismFromSomeProjectiveObject
2.8-7 AddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject
2.8-8 AddProjectiveLift
2.9 Injectives
2.9-1 SomeInjectiveObject
2.9-2 MonomorphismIntoSomeInjectiveObject
2.9-3 MonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject
2.9-4 InjectiveColift
2.9-5 AddSomeInjectiveObject
2.9-6 AddMonomorphismIntoSomeInjectiveObject
2.9-7 AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject
2.9-8 AddInjectiveColift

### 2 Objects

Any GAP object which is IsCapCategoryObject can be added to a category and then becomes an object in this category. Any object can belong to one or no category. After a GAP object is added to the category, it knows which things can be computed in its category and to which category it belongs. It knows categorial properties and attributes, and the functions for existential quantifiers can be applied to the object.

#### 2.1 Attributes for the Type of Objects

##### 2.1-1 CapCategory

`‣ CapCategory`( a ) | ( attribute )

Returns: a category

The argument is an object \(a\). The output is the category \(\mathbf{C}\) to which \(a\) was added.

#### 2.2 Equality for Objects

##### 2.2-1 IsEqualForObjects

`‣ IsEqualForObjects`( a, b ) | ( operation )

Returns: a boolean

The arguments are two objects \(a\) and \(b\). The output is `true` if \(a = b\), otherwise the output is `false`.

##### 2.2-2 AddIsEqualForObjects

`‣ AddIsEqualForObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation `IsEqualForObjects`. \(F: (a,b) \mapsto \mathtt{IsEqualForObjects}(a,b)\).

#### 2.3 Categorical Properties of Objects

##### 2.3-1 AddIsProjective

`‣ AddIsProjective`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation `IsProjective`. \(F: a \mapsto \mathtt{IsProjective}(a)\).

##### 2.3-2 AddIsInjective

`‣ AddIsInjective`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation `IsInjective`. \(F: a \mapsto \mathtt{IsInjective}(a)\).

##### 2.3-3 AddIsTerminal

`‣ AddIsTerminal`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation `IsTerminal`. \(F: a \mapsto \mathtt{IsTerminal}(a)\).

##### 2.3-4 AddIsInitial

`‣ AddIsInitial`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation `IsInitial`. \(F: a \mapsto \mathtt{IsInitial}(a)\).

##### 2.3-5 IsZeroForObjects

`‣ IsZeroForObjects`( a ) | ( property )

Returns: a boolean

The argument is an object \(a\) of a category \(\mathbf{C}\). The output is `true` if \(a\) is isomorphic to the zero object of \(\mathbf{C}\), otherwise the output is `false`.

##### 2.3-6 AddIsZeroForObjects

`‣ AddIsZeroForObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation `IsZeroForObjects`. \(F: a \mapsto \mathtt{IsZeroForObjects}(a)\).

#### 2.4 Random Objects

CAP provides two principal methods to generate random objects:

- By integers: The integer is simply a parameter that can be used to create a random object.

- By lists: The list is used when creating a random object would need more than one parameter. Lists offer more flexibility at the expense of the genericity of the methods. This happens because lists that are valid as input in some category may be not valid for other categories. Hence, these operations are not thought to be used in generic categorical algorithms.

##### 2.4-1 RandomObjectByInteger

`‣ RandomObjectByInteger`( C, n ) | ( operation )

Returns: an object or `fail`

The arguments are a category \(C\) and an integer \(n\). The output is a random object in \(C\) or `fail`.

##### 2.4-2 AddRandomObjectByInteger

`‣ AddRandomObjectByInteger`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation `RandomObjectByInteger`. The function \(F\) maps \((C,n)\) to `fail` or to a random object in \(C\).

##### 2.4-3 RandomObjectByList

`‣ RandomObjectByList`( C, L ) | ( operation )

Returns: an object or `fail`

The arguments are a category \(C\) and a list \(L\). The output is a random object in \(C\) or `fail`.

##### 2.4-4 AddRandomObjectByList

`‣ AddRandomObjectByList`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation `RandomObjectByList`. The function \(F\) maps \((C,L)\) to `fail` or to a random object in \(C\).

##### 2.4-5 RandomObject

`‣ RandomObject`( C, n ) | ( operation )

`‣ RandomObject`( C, L ) | ( operation )

These are convenient methods and they, depending on the input, delegate to one of the above methods.

#### 2.5 Tool functions for caches

##### 2.5-1 IsEqualForCacheForObjects

`‣ IsEqualForCacheForObjects`( phi, psi ) | ( operation )

Returns: true or false

Compares two objects in the cache

##### 2.5-2 AddIsEqualForCacheForObjects

`‣ AddIsEqualForCacheForObjects`( c, F ) | ( operation )

Returns: northing

By default, CAP uses caches to store the values of Categorical operations. To get a value out of the cache, one needs to compare the input of a basic operation with its previous input. To compare objects in the category, IsEqualForCacheForObject is used. By default this is an alias for IsEqualForObjects, where fail is substituted by false. If you add a function, this function used instead. A function \(F: a,b \mapsto bool\) is expected here. The output has to be true or false. Fail is not allowed in this context.

#### 2.6 Adding Objects to a Category

##### 2.6-1 Add

`‣ Add`( category, object ) | ( operation )

Adds object as an object to category.

##### 2.6-2 AddObject

`‣ AddObject`( category, object ) | ( operation )

Adds object as an object to category. If object already lies in the filter `IsCapCategoryObject`, the operation `Add` (2.6-1) can be used instead.

##### 2.6-3 AddObjectRepresentation

`‣ AddObjectRepresentation`( category, filter ) | ( operation )

The argument filter is used to create an object type for the category category, which is then used in `ObjectifyObjectForCAPWithAttributes` to objectify objects for this category.

##### 2.6-4 ObjectifyObjectForCAPWithAttributes

`‣ ObjectifyObjectForCAPWithAttributes`( object, category[, attribute1, value1, ...] ) | ( function )

Objectifies the object object with the type created for objects in the category category. The type is created by passing a representation to `AddObjectRepresentation`. Objects which are objectified using this method do not have to be passed to the `AddObject` function.

#### 2.7 Well-Definedness of Objects

##### 2.7-1 IsWellDefinedForObjects

`‣ IsWellDefinedForObjects`( a ) | ( operation )

Returns: a boolean

The argument is an object \(a\). The output is `true` if \(a\) is well-defined, otherwise the output is `false`.

##### 2.7-2 AddIsWellDefinedForObjects

`‣ AddIsWellDefinedForObjects`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation `IsWellDefinedForObjects`. \(F: a \mapsto \mathtt{IsWellDefinedForObjects}( a )\).

#### 2.8 Projectives

For a given object \(A\) in an abelian category having enough projectives, the following commands allow us to compute some projective object \(P\) together with an epimorphism \(\pi: P \rightarrow A\).

##### 2.8-1 SomeProjectiveObject

`‣ SomeProjectiveObject`( A ) | ( attribute )

Returns: an object

The argument is an object \(A\). The output is some projective object \(P\) for which there exists an epimorphism \(\pi: P \rightarrow A\).

##### 2.8-2 EpimorphismFromSomeProjectiveObject

`‣ EpimorphismFromSomeProjectiveObject`( A ) | ( attribute )

Returns: a morphism in \(\mathrm{Hom}(P,A)\)

The argument is an object \(A\). The output is an epimorphism \(\pi: P \rightarrow A\) with \(P\) a projective object that equals the output of \(\mathrm{SomeProjectiveObject}(A)\).

##### 2.8-3 EpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject

`‣ EpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject`( A, P ) | ( operation )

Returns: a morphism in \(\mathrm{Hom}(P,A)\)

The arguments are an object \(A\) and a projective object \(P\) that equals the output of \(\mathrm{SomeProjectiveObject}(A)\). The output is an epimorphism \(\pi: P \rightarrow A\).

##### 2.8-4 ProjectiveLift

`‣ ProjectiveLift`( pi, epsilon ) | ( operation )

Returns: a morphism in \(\mathrm{Hom}(P,B)\)

The arguments are a morphism \(\pi: P \rightarrow A\) with \(P\) a projective, and an epimorphism \(\epsilon: B \rightarrow A\). The output is a morphism \(\lambda: P \rightarrow B\) such that \(\epsilon \circ \lambda = \pi\).

##### 2.8-5 AddSomeProjectiveObject

`‣ AddSomeProjectiveObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation `SomeProjectiveObject`. \(F: A \mapsto P\).

##### 2.8-6 AddEpimorphismFromSomeProjectiveObject

`‣ AddEpimorphismFromSomeProjectiveObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation `EpimorphismFromSomeProjectiveObject`. \(F: A \mapsto \pi\).

##### 2.8-7 AddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject

`‣ AddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation `AddEpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject`. \(F: (A,P) \mapsto \pi\).

##### 2.8-8 AddProjectiveLift

`‣ AddProjectiveLift`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation `ProjectiveLift`. The function \(F\) maps a pair \((\pi, \epsilon)\) to a projective lift \(\lambda\).

#### 2.9 Injectives

For a given object \(A\) in an abelian category having enough injectives, the following commands allow us to compute some injective object \(I\) together with a monomorphism \(\iota: A \rightarrow I\).

##### 2.9-1 SomeInjectiveObject

`‣ SomeInjectiveObject`( A ) | ( attribute )

Returns: an object

The argument is an object \(A\). The output is some injective object \(I\) for which there exists a monomorphism \(\iota: A \rightarrow I\).

##### 2.9-2 MonomorphismIntoSomeInjectiveObject

`‣ MonomorphismIntoSomeInjectiveObject`( A ) | ( attribute )

Returns: a morphism in \(\mathrm{Hom}(I,A)\)

The argument is an object \(A\). The output is a monomorphism \(\iota: A \rightarrow I\) with \(I\) an injective object that equals the output of \(\mathrm{SomeInjectiveObject}(A)\).

##### 2.9-3 MonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject

`‣ MonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject`( A, I ) | ( operation )

Returns: a morphism in \(\mathrm{Hom}(I,A)\)

The arguments are an object \(A\) and an injective object \(I\) that equals the output of \(\mathrm{SomeInjectiveObject}(A)\). The output is a monomorphism \(\iota: A \rightarrow I\).

##### 2.9-4 InjectiveColift

`‣ InjectiveColift`( \iota, \beta ) | ( operation )

Returns: a morphism in \(\mathrm{Hom}(A,I)\)

The arguments are a morphism \(\iota: B \rightarrow A\) and \(\beta: B \rightarrow I\) where \(I\) is an injective object. The output is a morphism \(\lambda: A \rightarrow I\) such that \(\lambda \circ \iota = \beta\).

##### 2.9-5 AddSomeInjectiveObject

`‣ AddSomeInjectiveObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation `SomeInjectiveObject`. \(F: A \mapsto I\).

##### 2.9-6 AddMonomorphismIntoSomeInjectiveObject

`‣ AddMonomorphismIntoSomeInjectiveObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation `MonomorphismIntoSomeInjectiveObject`. \(F: A \mapsto \pi\).

##### 2.9-7 AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject

`‣ AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operation adds the given function \(F\) to the category for the basic operation `AddMonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject`. \(F: (A,I) \mapsto \pi\).

##### 2.9-8 AddInjectiveColift

`‣ AddInjectiveColift`( C, F ) | ( operation )

Returns: nothing

The arguments are a category \(C\) and a function \(F\). This operations adds the given function \(F\) to the category for the basic operation `InjectiveColift`. The function \(F\) maps a pair \((\iota, \beta)\) to an injective colift \(\lambda\) if it exists, and to `fail` otherwise.

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