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### 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

 ‣ Add( category, object ) ( operation )

Adds object as an object to category.

 ‣ 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.

 ‣ 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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