2 Category of Categories

2.4 Functors

2.4-1 CapFunctor

2.4-2 AddObjectFunction

2.4-3 FunctorObjectOperation

2.4-4 AddMorphismFunction

2.4-5 FunctorMorphismOperation

2.4-6 ApplyFunctor

2.4-7 InstallFunctor

2.4-8 IdentityFunctor

2.4-9 FunctorCanonicalizeZeroObjects

2.4-10 NaturalIsomorophismFromIdentityToCanonicalizeZeroObjects

2.4-11 FunctorCanonicalizeZeroMorphisms

2.4-12 NaturalIsomorophismFromIdentityToCanonicalizeZeroMorphisms

2.4-1 CapFunctor

2.4-2 AddObjectFunction

2.4-3 FunctorObjectOperation

2.4-4 AddMorphismFunction

2.4-5 FunctorMorphismOperation

2.4-6 ApplyFunctor

2.4-7 InstallFunctor

2.4-8 IdentityFunctor

2.4-9 FunctorCanonicalizeZeroObjects

2.4-10 NaturalIsomorophismFromIdentityToCanonicalizeZeroObjects

2.4-11 FunctorCanonicalizeZeroMorphisms

2.4-12 NaturalIsomorophismFromIdentityToCanonicalizeZeroMorphisms

Categories itself with functors as morphisms form a category. 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`

).

`‣ 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.

`‣ IsCapCategoryAsCatObject` ( object ) | ( filter ) |

Returns: `true`

or `false`

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

`‣ IsCapFunctor` ( object ) | ( filter ) |

Returns: `true`

or `false`

The GAP category of functors.

`‣ IsCapNaturalTransformation` ( object ) | ( filter ) |

Returns: `true`

or `false`

The GAP category of natural transformations.

`‣ 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`

.

`‣ 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.

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. Moreover, an object and a morphism function can be added to a functor, to apply it to objects or morphisms in the source category.

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

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

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

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

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

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

These methods construct a CAP functor, i.e. a morphism in Cat. Name should be an unique name for the functor, it is also used when the functor is installed as a method. `A` and `B` are source and target. Both can be given as object in Cat or as category itself.

`‣ AddObjectFunction` ( functor, function ) | ( operation ) |

This operation adds a function to the functor which can then be applied to objects in the source. The given function `function` has to take one argument which must be an object in the source category and should return a CapCategoryObject. The object is automatically added to the range of the functor when it is applied to the object.

`‣ 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.

`‣ AddMorphismFunction` ( functor, function ) | ( operation ) |

This operation adds a function to the functor which can then be applied to morphisms in the source. The given function `function` has to take three arguments \(A, \tau, B\). When the funtor `functor` is applied to the morphism \(\tau\), \(A\) is the result of `functor` applied to the source of \(\tau\), \(B\) is the result of `functor` applied to the range.

`‣ 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.

`‣ ApplyFunctor` ( func, A ) | ( function ) |

Returns: IsCapCategoryCell

Applies the functor `func` to the object or morphism `A`.

`‣ InstallFunctor` ( functor, method_name ) | ( operation ) |

TODO

`‣ IdentityFunctor` ( category ) | ( attribute ) |

Returns: a functor

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

`‣ FunctorCanonicalizeZeroObjects` ( category ) | ( 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.

`‣ NaturalIsomorophismFromIdentityToCanonicalizeZeroObjects` ( category ) | ( attribute ) |

Returns: a natural transformation

Returns the natural isomorphism from the identity functor to `FunctorCanonicalizeZeroObjects`

(`cat`).

`‣ FunctorCanonicalizeZeroMorphisms` ( category ) | ( 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.

`‣ NaturalIsomorophismFromIdentityToCanonicalizeZeroMorphisms` ( category ) | ( attribute ) |

Returns: a natural transformation

Returns the natural isomorphism from the identity functor to `FunctorCanonicalizeZeroMorphisms`

(`cat`).

`‣ 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.

`‣ 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

`‣ 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, \(F(A) \rightarrow G(A)\).

`‣ 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`.

`‣ 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.

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

Returns: a natural transformation

Computes the horizontal composition of the natural transformation `N` and

`‣ 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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