GAP (CAP) - Chapter 5: Category of Categories

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5 Category of Categories
5.1 The Category Cat
5.1-1 CapCat
5.2 Categories
5.2-1 IsCapCategoryAsCatObject
5.2-2 IsCapFunctor
5.2-3 IsCapNaturalTransformation
5.3 Constructors
5.3-1 AsCatObject
5.3-2 AsCapCategory
5.4 Functors
5.4-1 CapFunctor
5.4-2 CapFunctor
5.4-3 SourceOfFunctor
5.4-4 RangeOfFunctor
5.4-5 AddObjectFunction
5.4-6 FunctorObjectOperation
5.4-7 AddMorphismFunction
5.4-8 FunctorMorphismOperation
5.4-9 ApplyFunctor
5.4-10 InputSignature
5.4-11 InstallFunctor
5.4-12 IdentityFunctor
5.4-13 FunctorCanonicalizeZeroObjects
5.4-14 NaturalIsomorphismFromIdentityToCanonicalizeZeroObjects
5.4-15 FunctorCanonicalizeZeroMorphisms
5.4-16 NaturalIsomorphismFromIdentityToCanonicalizeZeroMorphisms
5.5 Natural transformations
5.5-1 Name
5.5-2 NaturalTransformation
5.5-3 AddNaturalTransformationFunction
5.5-4 ApplyNaturalTransformation
5.5-5 InstallNaturalTransformation
5.5-6 HorizontalPreComposeNaturalTransformationWithFunctor
5.5-7 HorizontalPreComposeFunctorWithNaturalTransformation

### 5 Category of Categories

Categories itself with functors as morphisms form a category Cat. So the data structure of `CapCategory`s is designed to be objects in a category. This category is implemented in `CapCat`. For every category, the corresponding object in Cat can be obtained via `AsCatObject`. The implemetation of the category of categories offers a data structure for functors. Those are implemented as morphisms in this category, so functors can be handled like morphisms in a category. Also convenience functions to install functors as methods are implemented (in order to avoid `ApplyFunctor`).

#### 5.1 The Category Cat

##### 5.1-1 CapCat

`‣ CapCat` | ( global variable )

This variable stores the category of categories. Every category object is constructed as an object in this category, so Cat is constructed when loading the package.

#### 5.2 Categories

##### 5.2-1 IsCapCategoryAsCatObject

`‣ IsCapCategoryAsCatObject`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of CAP categories seen as object in Cat.

##### 5.2-2 IsCapFunctor

`‣ IsCapFunctor`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of functors.

##### 5.2-3 IsCapNaturalTransformation

`‣ IsCapNaturalTransformation`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of natural transformations.

#### 5.3 Constructors

##### 5.3-1 AsCatObject

`‣ AsCatObject`( C ) | ( attribute )

Given a CAP category \(C\), this method returns the corresponding object in Cat. For technical reasons, the filter `IsCapCategory` must not imply the filter `IsCapCategoryObject`. For example, if `InitialObject` is applied to an object, it returns the initial object of its category. If it is applied to a category, it returns the initial object of the category. If a CAP category would be a category object itself, this would be ambiguous. So categories must be wrapped in a CatObject to be an object in Cat. This method returns the wrapper object. The category can be reobtained by `AsCapCategory`.

##### 5.3-2 AsCapCategory

`‣ AsCapCategory`( C ) | ( attribute )

For an object \(C\) in Cat, this method returns the underlying CAP category. This method is inverse to `AsCatObject`, i.e. AsCapCategory( AsCatObject( A ) ) = A.

#### 5.4 Functors

Functors are morphisms in Cat, thus they have source and target which are categories. A multivariate functor can be constructed via a product category as source, a presheaf is constructed via the opposite category as source. However, the user can explicitly decide the arity of a functor (which will only have technical implications). Thus, it is for example possible to consider a functor \(A \times B \rightarrow C\) either as a unary functor with source category \(A \times B\) or as a binary functor. Moreover, an object and a morphism function can be added to a functor, to apply it to objects or morphisms in the source category.

##### 5.4-1 CapFunctor

`‣ CapFunctor`( name, A, B ) | ( operation )

`‣ CapFunctor`( name, A, B ) | ( operation )

`‣ CapFunctor`( name, A, B ) | ( operation )

`‣ CapFunctor`( name, A, B ) | ( operation )

These methods construct a unary CAP functor. The first argument is a string for the functor's name. A and B are the source and target of the functor, and they can be given as objects in `CapCat` or as a CAP-category.

##### 5.4-2 CapFunctor

`‣ CapFunctor`( name, list, B ) | ( operation )

`‣ CapFunctor`( name, list, B ) | ( operation )

These methods construct a possible multivariate CAP functor. The first argument is a string for the functor's name. The second argument is a list encoding the input signature of the functor. It can be given as a list of pairs \([ [ A_1, b_1 ], \dots, [ A_n, b_n ] ]\) where a pair consists of a category \(A_i\) (given as an object in `CapCat` or as a CAP-category) and a boolean \(b_i\) for \(i = 1, \dots, n\). Instead of a pair \([ A_i, b_i ]\), you can also give simply \(A_i\), which will be interpreted as the pair \([ A_i, \mathtt{false} ]\). The third argument is the target B of the functor, and it can be given as an object in `CapCat` or as a CAP-category. The output is a functor with source given by the product category \(D_1 \times ... \times D_n\), where \(D_i = A_i\) if \(b_i = \mathtt{false}\), and \(D_i = A_i^{\mathrm{op}}\) otherwise.

##### 5.4-3 SourceOfFunctor

`‣ SourceOfFunctor`( F ) | ( attribute )

The argument is a functor \(F\). The output is its source as CAP category.

##### 5.4-4 RangeOfFunctor

`‣ RangeOfFunctor`( F ) | ( attribute )

The argument is a functor \(F\). The output is its range as CAP category.

##### 5.4-5 AddObjectFunction

`‣ AddObjectFunction`( F, f ) | ( operation )

This operation adds a function \(f\) to the functor \(F\) which can then be applied to objects in the source. The given function \(f\) has to take arguments according to the `InputSignature` of \(F\), i.e., if the input signature is given by \([ [A_1, b_1], \dots, [A_n,b_n] ]\), then \(f\) must take \(n\) arguments, where the \(i\)-th argument is an object in the category \(A_i\) (the boolean \(b_i\) is ignored). The function should return an object in the range of the functor, except when the automatic call of `AddObject` was enabled via `EnableAddForCategoricalOperations`. In this case the output only has to be a GAP object in `IsAttributeStoringRep`, which will be automatically added as an object to the range of the functor.

##### 5.4-6 FunctorObjectOperation

`‣ FunctorObjectOperation`( F ) | ( attribute )

Returns: a GAP operation

The argument is a functor \(F\). The output is the GAP operation realizing the action of \(F\) on objects.

##### 5.4-7 AddMorphismFunction

`‣ AddMorphismFunction`( F, f ) | ( operation )

This operation adds a function \(f\) to the functor \(F\) which can then be applied to morphisms in the source. The given function \(f\) has to take as its first argument an object \(s\) that is equal (via `IsEqualForObjects`) to the source of the result of applying \(F\) to the input morphisms. The next arguments of \(f\) have to morphisms according to the `InputSignature` of \(F\), i.e., if the input signature is given by \([ [A_1, b_1], \dots, [A_n,b_n] ]\), then \(f\) must take \(n\) arguments, where the \(i\)-th argument is a morphism in the category \(A_i\) (the boolean \(b_i\) is ignored). The last argument of \(f\) must be an object \(r\) that is equal (via `IsEqualForObjects`) to the range of the result of applying \(F\) to the input morphisms. The function should return a morphism in the range of the functor, except when the automatic call of `AddMorphism` was enabled via `EnableAddForCategoricalOperations`. In this case the output only has to be a GAP object in `IsAttributeStoringRep` (with attributes `Source` and `Range` containing also GAP objects in `IsAttributeStoringRep`), which will be automatically added as a morphism to the range of the functor.

##### 5.4-8 FunctorMorphismOperation

`‣ FunctorMorphismOperation`( F ) | ( attribute )

Returns: a GAP operation

The argument is a functor \(F\). The output is the GAP operation realizing the action of \(F\) on morphisms.

##### 5.4-9 ApplyFunctor

`‣ ApplyFunctor`( func, A[, B, ...] ) | ( function )

Returns: IsCapCategoryCell

Applies the functor func either to

- an object or morphism A in the source of func or

- to objects or morphisms belonging to the categories in the input signature of func.

##### 5.4-10 InputSignature

`‣ InputSignature`( F ) | ( attribute )

Returns: IsList

The argument is a functor \(F\). The output is a list of pairs \([ [ A_1, b_1 ], \dots, [ A_n, b_n ] ]\) where a pair consists of a CAP-category \(A_i\) and a boolean \(b_i\) for \(i = 1, \dots, n\). The source of \(F\) is given by the product category \(D_1 \times ... \times D_n\), where \(D_i = A_i\) if \(b_i = \mathtt{false}\), and \(D_i = A_i^{\mathrm{op}}\) otherwise.

##### 5.4-11 InstallFunctor

`‣ InstallFunctor`( F, s ) | ( operation )

Returns: nothing

The arguments are a functor \(F\) and a string \(s\). To simplify the description of this operation, we let \([ [ A_1, b_1 ], \dots, [ A_n, b_n ] ]\) denote the input signature of \(F\). This method tries to install \(3\) operations: an operation \(\omega_1\) with the name \(s\), an operation \(\omega_2\) with the name \(s\mathtt{OnObjects}\), and an operation \(\omega_3\) with the name \(s\mathtt{OnMorphisms}\). The operation \(\omega_1\) takes as input either \(n\)- objects/morphisms in \(A_i\) or a single object/morphism in the source of \(F\), and outputs the result of applying \(F\) to this input. \(\omega_2\) and \(\omega_3\) are the corresponding variants for objects or morphisms only. This function can only be called once for each functor, every further call will be ignored.

##### 5.4-12 IdentityFunctor

`‣ IdentityFunctor`( cat ) | ( attribute )

Returns: a functor

Returns the identity functor of the category cat viewed as an object in the category of categories.

##### 5.4-13 FunctorCanonicalizeZeroObjects

`‣ FunctorCanonicalizeZeroObjects`( cat ) | ( attribute )

Returns: a functor

Returns the endofunctor of the category cat with zero which maps each (object isomorphic to the) zero object to `ZeroObject`(cat) and to itself otherwise. This functor is equivalent to the identity functor.

##### 5.4-14 NaturalIsomorphismFromIdentityToCanonicalizeZeroObjects

`‣ NaturalIsomorphismFromIdentityToCanonicalizeZeroObjects`( cat ) | ( attribute )

Returns: a natural transformation

Returns the natural isomorphism from the identity functor to `FunctorCanonicalizeZeroObjects`(cat).

##### 5.4-15 FunctorCanonicalizeZeroMorphisms

`‣ FunctorCanonicalizeZeroMorphisms`( cat ) | ( attribute )

Returns: a functor

Returns the endofunctor of the category cat with zero which maps each object to itself, each morphism \(\phi\) to itself, unless it is congruent to the zero morphism; in this case it is mapped to `ZeroMorphism`(`Source`(\(\phi\)), `Range`(\(\phi\))). This functor is equivalent to the identity functor.

##### 5.4-16 NaturalIsomorphismFromIdentityToCanonicalizeZeroMorphisms

`‣ NaturalIsomorphismFromIdentityToCanonicalizeZeroMorphisms`( cat ) | ( attribute )

Returns: a natural transformation

Returns the natural isomorphism from the identity functor to `FunctorCanonicalizeZeroMorphisms`(cat).

#### 5.5 Natural transformations

Natural transformations form the \(2\)-cells of Cat. As such, it is possible to compose them vertically and horizontally, see Section 4.2.

##### 5.5-1 Name

`‣ Name`( arg ) | ( attribute )

Returns: a string

As every functor, every natural transformation has a name attribute. It has to be a string and will be set by the Constructor.

##### 5.5-2 NaturalTransformation

`‣ NaturalTransformation`( [name, ]F, G ) | ( operation )

Returns: a natural transformation

Constructs a natural transformation between the functors F\(:A \rightarrow B\) and G\(:A \rightarrow B\). The string name is optional, and, if not given, set automatically from the names of the functors

##### 5.5-3 AddNaturalTransformationFunction

`‣ AddNaturalTransformationFunction`( N, func ) | ( operation )

Adds the function (or list of functions) func to the natural transformation N. The function or each function in the list should take three arguments. If \(N: F \rightarrow G\), the arguments should be \(F(A), A, G(A)\). The ouptput should be a morphism, \(F(A) \rightarrow G(A)\).

##### 5.5-4 ApplyNaturalTransformation

`‣ ApplyNaturalTransformation`( N, A ) | ( function )

Returns: a morphism

Given a natural transformation N\(:F \rightarrow G\) and an object A, this function should return the morphism \(F(A) \rightarrow G(A)\), corresponding to N.

##### 5.5-5 InstallNaturalTransformation

`‣ InstallNaturalTransformation`( N, name ) | ( operation )

Installs the natural transformation N as operation with the name name. Argument for this operation is an object, output is a morphism.

##### 5.5-6 HorizontalPreComposeNaturalTransformationWithFunctor

`‣ HorizontalPreComposeNaturalTransformationWithFunctor`( N, F ) | ( operation )

Returns: a natural transformation

Computes the horizontal composition of the natural transformation N and the functor F.

##### 5.5-7 HorizontalPreComposeFunctorWithNaturalTransformation

`‣ HorizontalPreComposeFunctorWithNaturalTransformation`( F, N ) | ( operation )

Returns: a natural transformation

Computes the horizontal composition of the functor F and the natural transformation N.

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