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### 1 Prosets (preordered sets)

#### 1.1 Info Class

##### 1.1-1 InfoLocales
 ‣ InfoLocales ( info class )

#### 1.2 Constructors

##### 1.2-1 UniqueMorphism
 ‣ UniqueMorphism( A, B ) ( operation )

Returns: a CAP morphism

Construct the unique morphism with source A and target B.

#### 1.3 Properties

##### 1.3-1 IsThinCategory
 ‣ IsThinCategory( C ) ( property )

Returns: true or false

The property of C being a thin CAP category.

##### 1.3-2 IsDiscreteCategory
 ‣ IsDiscreteCategory( C ) ( property )

Returns: true or false

The property of C being a discrete CAP category, i.e., equivalent to a category in which any morphism is an identity.

##### 1.3-3 IsMonoidalProset
 ‣ IsMonoidalProset( C ) ( property )

Returns: true or false

The property of C being a monoidal thin category.

##### 1.3-4 IsClosedMonoidalProset
 ‣ IsClosedMonoidalProset( C ) ( property )

Returns: true or false

The property of C being a closed monoidal thin category.

##### 1.3-5 IsCoclosedMonoidalProset
 ‣ IsCoclosedMonoidalProset( C ) ( property )

Returns: true or false

The property of C being a coclosed monoidal thin category.

##### 1.3-6 IsSymmetricMonoidalProset
 ‣ IsSymmetricMonoidalProset( C ) ( property )

Returns: true or false

The property of C being a symmetric monoidal thin category.

##### 1.3-7 IsSymmetricClosedMonoidalProset
 ‣ IsSymmetricClosedMonoidalProset( C ) ( property )

Returns: true or false

The property of C being a symmetric closed monoidal thin category.

##### 1.3-8 IsSymmetricCoclosedMonoidalProset
 ‣ IsSymmetricCoclosedMonoidalProset( C ) ( property )

Returns: true or false

The property of C being a symmetric coclosed monoidal thin category.

#### 1.4 Operations

##### 1.4-1 AreIsomorphicForObjectsIfIsHomSetInhabited
 ‣ AreIsomorphicForObjectsIfIsHomSetInhabited( A, B ) ( operation )

Returns: true or false

Check if A is isomorphic to B under the assumption that there exists a morphism from A to B, i.e., if A is known to be less or equal to B w.r.t. the preorder.

#### 1.5 GAP categories

##### 1.5-1 IsObjectInThinCategory
 ‣ IsObjectInThinCategory( object ) ( filter )

Returns: true or false

The GAP category of objects in a thin CAP category.

##### 1.5-2 IsMorphismInThinCategory
 ‣ IsMorphismInThinCategory( morphism ) ( filter )

Returns: true or false

The GAP category of morphisms in a thin CAP category.

 ‣ AddAreIsomorphicForObjectsIfIsHomSetInhabited( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation AreIsomorphicForObjectsIfIsHomSetInhabited. F: ( arg2, arg3 ) \mapsto \mathtt{AreIsomorphicForObjectsIfIsHomSetInhabited}(arg2, arg3).

 ‣ AddUniqueMorphism( C, F ) ( operation )

Returns: nothing

The arguments are a category C and a function F. This operation adds the given function F to the category for the basic operation UniqueMorphism. F: ( A, B ) \mapsto \mathtt{UniqueMorphism}(A, B).

Prosets are thin categories, i.e., each Hom-set is either a singleton or empty.

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