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### 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 cateogry C. The output is a filter in which C lies.

##### 1.3-2 CellFilter
 ‣ CellFilter( C ) ( attribute )

Returns: a filter

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

#### 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 )
 ‣ EnableBasicOperationTypeCheck( arg ) ( function )

Most basic operations have a prefunction, which does a (sometimes partial) typecheck at the beginning of the operation. These functions enable or disable this check for a category. (Enabled by default)

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