GAP (CAP) - Chapter 1: CAP Categories

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1 CAP Categories
1.1 Categories
1.1-1 IsCapCategory
1.1-2 IsCapCategoryCell
1.1-3 IsCapCategoryObject
1.1-4 IsCapCategoryMorphism
1.1-5 IsCapCategoryTwoCell
1.2 Constructor
1.2-1 CreateCapCategory
1.2-2 CreateCapCategory
1.3 Internal Attributes
1.3-1 CategoryFilter
1.3-2 CellFilter
1.3-3 ObjectFilter
1.3-4 MorphismFilter
1.3-5 TwoCellFilter
1.3-6 CommutativeRingOfLinearCategory
1.4 Logic switcher
1.4-1 CapCategorySwitchLogicOn
1.4-2 CapCategorySwitchLogicOff
1.5 Tool functions
1.5-1 CanCompute
1.5-2 CheckConstructivenessOfCategory
1.6 Well-Definedness of Cells
1.6-1 IsWellDefined
1.7 Type check
1.7-1 DisableBasicOperationTypeCheck
1.8 Disable add in functions
1.8-1 DisableAddForCategoricalOperations

### 1 CAP Categories

Categories are the main GAP objects in CAP. They are used to associate GAP objects which represent objects and morphisms with their category. By associating a GAP object to the category, one of two filters belonging to the category (ObjectFilter/MorphismFilter) are set to true. Via Add methods, functions for specific existential quantifiers can be associated to the category and after that can be applied to GAP objects in the category. A GAP category object also knows which constructions are currently possible in this category.

#### 1.1 Categories

##### 1.1-1 IsCapCategory

`‣ IsCapCategory`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of CAP categories. Objects of this type handle the CAP category information, the caching, and filters for objects in the CAP category. Please note that the object itself is not related to methods, you only need it as a handler and a presentation of the CAP category.

##### 1.1-2 IsCapCategoryCell

`‣ IsCapCategoryCell`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of CAP category cells. Every object, morphism, and 2-cell of a CAP category lies in this GAP category.

##### 1.1-3 IsCapCategoryObject

`‣ IsCapCategoryObject`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of CAP category objects. Every object of a CAP category lies in this GAP category.

##### 1.1-4 IsCapCategoryMorphism

`‣ IsCapCategoryMorphism`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of CAP category morphisms. Every morphism of a CAP category lies in this GAP category.

##### 1.1-5 IsCapCategoryTwoCell

`‣ IsCapCategoryTwoCell`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of CAP category 2-cells. Every 2-cell of a CAP category lies in this GAP category.

#### 1.2 Constructor

##### 1.2-1 CreateCapCategory

`‣ CreateCapCategory`( ) | ( operation )

Returns: a category

Creates a new CAP category from scratch. It gets a generic name.

##### 1.2-2 CreateCapCategory

`‣ CreateCapCategory`( s ) | ( operation )

Returns: a category

The argument is a string s. This operation creates a new CAP category from scratch. Its name is set to s.

#### 1.3 Internal Attributes

Each category C stores various filters. They are used to apply the right functions in the method selection.

##### 1.3-1 CategoryFilter

`‣ CategoryFilter`( C ) | ( attribute )

Returns: a filter

The argument is a category C. The output is a filter in which C lies.

##### 1.3-2 CellFilter

`‣ CellFilter`( C ) | ( attribute )

Returns: a filter

The argument is a category C. The output is a filter in which all cells of C shall lie.

##### 1.3-3 ObjectFilter

`‣ ObjectFilter`( C ) | ( attribute )

Returns: a filter

The argument is a category C. The output is a filter in which all objects of C shall lie.

##### 1.3-4 MorphismFilter

`‣ MorphismFilter`( C ) | ( attribute )

Returns: a filter

The argument is a category C. The output is a filter in which all morphisms of C shall lie.

##### 1.3-5 TwoCellFilter

`‣ TwoCellFilter`( C ) | ( attribute )

Returns: a filter

The argument is a category C. The output is a filter in which all 2-cells of C shall lie.

##### 1.3-6 CommutativeRingOfLinearCategory

`‣ CommutativeRingOfLinearCategory`( C ) | ( attribute )

Returns: a ring

The argument is a category C which is expected to lie in the filter `IsLinearCategoryOverCommutativeRing`. The output is a commutative ring over which the category is linear.

#### 1.4 Logic switcher

##### 1.4-1 CapCategorySwitchLogicOn

`‣ CapCategorySwitchLogicOn`( C ) | ( function )

Activates the predicate implication logic for the category C.

##### 1.4-2 CapCategorySwitchLogicOff

`‣ CapCategorySwitchLogicOff`( C ) | ( function )

Deactivates the predicate implication logic for the category C.

#### 1.5 Tool functions

##### 1.5-1 CanCompute

`‣ CanCompute`( C, s ) | ( operation )

Returns: `true` or `false`

The argument is a category C and a string s, which should be the name of a primitive operation, e.g., PreCompose. If applying this method is possible in C, the method returns `true`, `false` otherwise. If the string is not the name of a primitive operation, an error is raised.

##### 1.5-2 CheckConstructivenessOfCategory

`‣ CheckConstructivenessOfCategory`( C, s ) | ( operation )

Returns: a list

The arguments are a category C and a string s. If s is a categorical property (e.g. `"IsAbelianCategory"`), the output is a list of strings with basic operations which are missing in C to have the categorical property constructively. If s is not a categorical property, an error is raised.

#### 1.6 Well-Definedness of Cells

##### 1.6-1 IsWellDefined

`‣ IsWellDefined`( c ) | ( property )

Returns: a boolean

The argument is a cell c. The output is `true` if c is well-defined, otherwise the output is `false`.

#### 1.7 Type check

##### 1.7-1 DisableBasicOperationTypeCheck

`‣ DisableBasicOperationTypeCheck`( category ) | ( function )

`‣ EnablePartialBasicOperationTypeCheck`( arg ) | ( function )

`‣ EnableFullBasicOperationTypeCheck`( arg ) | ( function )

Most operations have prefunctions, that perform additional checks on the arguments. The checks can either be partial (enabled by default), full, or disabled. With the following commands you can either enable the full checks, just the basic checks, or, for performance, disable the checks alltogether.

#### 1.8 Disable add in functions

##### 1.8-1 DisableAddForCategoricalOperations

`‣ DisableAddForCategoricalOperations`( C ) | ( function )

`‣ EnableAddForCategoricalOperations`( arg ) | ( function )

Enables/disables the call of add for the output of primitively added functions for the category C. This can be savely done if the all objects and morphisms are added to the category by their constructors.

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