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### 1 Tools

#### 1.1 Tools

##### 1.1-1 SET_RANGE_CATEGORY_Of_HOMOMORPHISM_STRUCTURE
 ‣ SET_RANGE_CATEGORY_Of_HOMOMORPHISM_STRUCTURE( C, H ) ( operation )

The input are two categories C and H. There is not output but the following side effects are applied to C:

• SetRangeCategoryOfHomomorphismStructure( C, H )

• SetIsEquippedWithHomomorphismStructure( C, true )

Furthermore, if IsCategoryWithDecidableLifts( H ) then

• SetIsCategoryWithDecidableLifts( C, true )

• SetIsCategoryWithDecidableColifts( C, true )

An error is raised if RangeCategoryOfHomomorphismStructure( C ) is already set.

#### 1.2 Properties

##### 1.2-1 IsObjectFiniteCategory
 ‣ IsObjectFiniteCategory( C ) ( property )

Returns: true or false

The (evil) property of C being a category with finitely many objects.

##### 1.2-2 IsFiniteCategory
 ‣ IsFiniteCategory( C ) ( property )

Returns: true or false

The (evil) property of C being a finite category.

##### 1.2-3 IsEquivalentToFiniteCategory
 ‣ IsEquivalentToFiniteCategory( C ) ( property )

Returns: true or false

The property of C being equivalent to a finite category.

#### 1.3 Attributes

##### 1.3-1 SetOfObjectsOfCategory
 ‣ SetOfObjectsOfCategory( C ) ( attribute )

Returns: a list of CAP category objects

Return a duplicate free list of objects of the category C.

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

Returns: a list of CAP category objects

Return a duplicate free list of objects of the category C. The corresponding CAP operation is SetOfObjectsOfCategory.

##### 1.3-3 SetOfMorphismsOfFiniteCategory
 ‣ SetOfMorphismsOfFiniteCategory( C ) ( attribute )

Returns: a list of a CAP category morphisms

Return a duplicate free list of morphisms of the finite category C.

##### 1.3-4 SetOfMorphisms
 ‣ SetOfMorphisms( C ) ( attribute )

Returns: a list of CAP category objects

Return a duplicate free list of morphisms of the finite category C. The corresponding CAP operation is SetOfMorphismsOfFiniteCategory.

#### 1.4 Functors

##### 1.4-1 CovariantHomFunctor
 ‣ CovariantHomFunctor( o ) ( attribute )

Returns: a CAP functor

The input is an object o in a category $$C$$. The output is the covariant Hom functor $$\mathrm{Hom}$$(o,-) from the category $$C$$ to RangeCategoryOfHomomorphismStructure( C ).

##### 1.4-2 GlobalSectionFunctor
 ‣ GlobalSectionFunctor( C ) ( attribute )

Returns: a CAP functor

Returns the global section functor $$\mathrm{Hom}(1,-)$$ from the category C with terminal object $$1$$ to RangeCategoryOfHomomorphismStructure( C ).

#### 1.5 Cell as evaluatable string

##### 1.5-1 DatumOfCellAsEvaluatableString
 ‣ DatumOfCellAsEvaluatableString( c, list_of_evaluatable_strings ) ( operation )

Returns: a string

The arguments is a category cell c and a list list_of_evaluatable_strings of string all which must be evalutable with EvalString. The output is a string str such that

• EvalString( str ) = ObjectDatum( c ) if c is a category object, or

• EvalString( str ) = MorphismDatum( c ) if c is a category morphism.

The output string must, apart from the brackets, only consist of the substrings "ObjectConstructor", "MorphismConstructor" and the string in list_of_evaluatable_strings.

##### 1.5-2 CellAsEvaluatableString
 ‣ CellAsEvaluatableString( c, list_of_evaluatable_strings ) ( operation )

Returns: a string

The arguments is a category cell c and a list list_of_evaluatable_strings of string all which must be evalutable with EvalString. The output is a string str such that

• IsEqualForObjects( EvalString( str ), c ) if c is a category object, or

• IsEqualForMorphismsOnMor( EvalString( str ), c ) if c is a category morphism.

The output string must, apart from the brackets, only consist of the substrings "ObjectConstructor", "MorphismConstructor" and the string in list_of_evaluatable_strings.

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