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
2.3 Categorical Properties of Objects
2.3-1 IsBijectiveObject
2.3-2 IsProjective
2.3-3 IsInjective
2.3-4 IsTerminal
2.3-5 IsInitial
2.3-6 IsZeroForObjects
2.3-7 IsZero
2.4 Random Objects
2.4-1 RandomObjectByInteger
2.4-2 RandomObjectByList
2.4-3 RandomObject
2.4-4 RandomObject
2.5 Tool functions for caches
2.5-1 IsEqualForCacheForObjects
2.6 Adding Objects to a Category
2.6-1 Add
2.6-2 AddObject
2.6-3 AddObjectRepresentation
2.6-4 ObjectifyObjectForCAPWithAttributes
2.6-5 CreateCapCategoryObjectWithAttributes
2.7 Object constructors
2.7-1 ObjectConstructor
2.7-2 /
2.7-3 ObjectDatum
2.8 Well-Definedness of Objects
2.8-1 IsWellDefinedForObjects
2.9 Projectives
2.9-1 SomeProjectiveObject
2.9-2 EpimorphismFromSomeProjectiveObject
2.9-3 EpimorphismFromSomeProjectiveObjectWithGivenSomeProjectiveObject
2.9-4 ProjectiveLift
2.10 Injectives
2.10-1 SomeInjectiveObject
2.10-2 MonomorphismIntoSomeInjectiveObject
2.10-3 MonomorphismIntoSomeInjectiveObjectWithGivenSomeInjectiveObject
2.10-4 InjectiveColift
2.11 Simplified Objects
2.11-1 Simplify
2.11-2 SimplifyObject
2.11-3 SimplifyObject_IsoFromInputObject
2.11-4 SimplifyObject_IsoToInputObject
2.12 Dimensions
2.12-1 ProjectiveDimension
2.12-2 InjectiveDimension

### 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`.

#### 2.3 Categorical Properties of Objects

##### 2.3-1 IsBijectiveObject

`‣ IsBijectiveObject`( a ) | ( property )

Returns: a boolean

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

##### 2.3-2 IsProjective

`‣ IsProjective`( a ) | ( property )

Returns: a boolean

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

##### 2.3-3 IsInjective

`‣ IsInjective`( a ) | ( property )

Returns: a boolean

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

##### 2.3-4 IsTerminal

`‣ IsTerminal`( 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 terminal object of \(\mathbf{C}\), otherwise the output is `false`.

##### 2.3-5 IsInitial

`‣ IsInitial`( 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 initial object of \(\mathbf{C}\), otherwise the output is `false`.

##### 2.3-6 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`.

##### 2.3-7 IsZero

`‣ IsZero`( 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`.

#### 2.4 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 in \(C\)

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

##### 2.4-2 RandomObjectByList

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

Returns: an object in \(C\)

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

##### 2.4-3 RandomObject

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

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

##### 2.4-4 RandomObject

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

#### 2.5 Tool functions for caches

##### 2.5-1 IsEqualForCacheForObjects

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

Returns: true or false

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, IsEqualForCacheForObjects is used. By default, IsEqualForCacheForObjects falls back to IsEqualForCache (see ToolsForHomalg), which in turn defaults to recursive comparison for lists and `IsIdenticalObj` in all other cases. If you add a function via `AddIsEqualForCacheForObjects`, that function is used instead. A function \(F: a,b \mapsto bool\) is expected there. 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. filter must imply `IsCapCategoryObject`.

##### 2.6-4 ObjectifyObjectForCAPWithAttributes

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

Returns: an object

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. The optional arguments behave like the corresponding arguments in `ObjectifyWithAttributes`. Also returns the objectified object.

##### 2.6-5 CreateCapCategoryObjectWithAttributes

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

Returns: an object

Shorthand for `ObjectifyObjectForCAPWithAttributes( rec( ), category[, attribute1, value1, ...] )`.

#### 2.7 Object constructors

##### 2.7-1 ObjectConstructor

`‣ ObjectConstructor`( C, a ) | ( operation )

Returns: an object

The arguments are a category \(C\) and an object datum \(a\) (type and semantics of the object datum depend on the category). The output is an object of \(C\) defined by \(a\). Note that by default this CAP operation is not cached. You can change this behaviour by calling `SetCachingToWeak( C, "ObjectConstructor" )` resp. `SetCachingToCrisp( C, "ObjectConstructor" )`.

##### 2.7-2 /

`‣ /`( a, C ) | ( operation )

Returns: an object

Shorthand for `ObjectConstructor( C, a )`.

##### 2.7-3 ObjectDatum

`‣ ObjectDatum`( obj ) | ( attribute )

Returns: depends on the category

The argument is a CAP category object obj. The output is a datum which can be used to construct obj, that is, `IsEqualForObjects(` obj`, ObjectConstructor( CapCategory(` obj`), ObjectDatum(` obj`) ) )`. Note that by default this CAP operation is not cached. You can change this behaviour by calling `SetCachingToWeak( C, "ObjectDatum" )` resp. `SetCachingToCrisp( C, "ObjectDatum" )`.

#### 2.8 Well-Definedness of Objects

##### 2.8-1 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`.

#### 2.9 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 \(\pi: P \rightarrow A\).

##### 2.9-1 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 \(\pi: P \rightarrow A\).

##### 2.9-2 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 \(\mathrm{SomeProjectiveObject}(A)\).

##### 2.9-3 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 \(\pi: P \rightarrow A\).

##### 2.9-4 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 \(\epsilon \circ \lambda = \pi\).

#### 2.10 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 \(\iota: A \rightarrow I\).

##### 2.10-1 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 \(\iota: A \rightarrow I\).

##### 2.10-2 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 \(\mathrm{SomeInjectiveObject}(A)\).

##### 2.10-3 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 \(\iota: A \rightarrow I\).

##### 2.10-4 InjectiveColift

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

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

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

#### 2.11 Simplified Objects

Let \(i\) be a positive integer or \(\infty\). For a given object \(A\), an \(i\)-th simplified object of \(A\) consists of

- an object \(A_i\),

- an isomorphism \(\iota_A^i: A \rightarrow A_i\).

The idea is that the greater the \(i\), the "simpler" the \(A_i\) (but this could mean the harder the computation) with \(\infty\) as a possible value.

##### 2.11-1 Simplify

`‣ Simplify`( A ) | ( attribute )

Returns: an object

The argument is an object \(A\). The output is a simplified object \(A_{\infty}\).

##### 2.11-2 SimplifyObject

`‣ SimplifyObject`( A, i ) | ( operation )

Returns: an object

The arguments are an object \(A\) and a positive integer \(i\) or `infinity`. The output is a simplified object \(A_i\).

##### 2.11-3 SimplifyObject_IsoFromInputObject

`‣ SimplifyObject_IsoFromInputObject`( A, i ) | ( operation )

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

The arguments are an object \(A\) and a positive integer \(i\) or `infinity`. The output is an isomorphism to a simplified object \(\iota_A^i: A \rightarrow A_i\).

##### 2.11-4 SimplifyObject_IsoToInputObject

`‣ SimplifyObject_IsoToInputObject`( A, i ) | ( operation )

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

The arguments are an object \(A\) and a positive integer \(i\) or `infinity`. The output is an isomorphism from a simplified object \((\iota_A^i)^{-1}: A_i \rightarrow A\) which is the inverse of the output of `SimplifyObject_IsoFromInputObject`.

#### 2.12 Dimensions

##### 2.12-1 ProjectiveDimension

`‣ ProjectiveDimension`( A ) | ( attribute )

Returns: a nonnegative integer or infinity

The argument is an object \(A\). The output is a the projective dimension of \(A\).

##### 2.12-2 InjectiveDimension

`‣ InjectiveDimension`( A ) | ( attribute )

Returns: a nonnegative integer or infinity

The argument is an object \(A\). The output is a the injective dimension of \(A\).

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