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### 7 The elementary topos of finite reflexive quivers

#### 7.1 Constructors

##### 7.1-1 CategoryOfReflexiveQuiversEnrichedOver
 ‣ CategoryOfReflexiveQuiversEnrichedOver( V ) ( attribute )

Returns: a CAP category

Construct the category of reflexive quivers enriched over the category V.

##### 7.1-2 CreateReflexiveQuiver
 ‣ CreateReflexiveQuiver( arg1, arg2, arg3, arg4 ) ( operation )

##### 7.1-3 CreateReflexiveQuiver
 ‣ CreateReflexiveQuiver( arg1, arg2, arg3 ) ( operation )

##### 7.1-4 CreateReflexiveQuiverMorphism
 ‣ CreateReflexiveQuiverMorphism( arg1, arg2, arg3, arg4 ) ( operation )

##### 7.1-5 Subobject
 ‣ Subobject( arg1, arg2, arg3 ) ( operation )

##### 7.1-6 Subobject
 ‣ Subobject( arg1, arg2 ) ( operation )

#### 7.2 Attributes

##### 7.2-1 UnderlyingCategory
 ‣ UnderlyingCategory( fin_reflexive_quivers ) ( attribute )

##### 7.2-2 EmbeddingOfUnderlyingCategory
 ‣ EmbeddingOfUnderlyingCategory( fin_reflexive_quivers ) ( attribute )

##### 7.2-3 Loops
 ‣ Loops( quiver ) ( attribute )

##### 7.2-4 Arrows
 ‣ Arrows( quiver ) ( attribute )

##### 7.2-5 SvgString
 ‣ SvgString( cell ) ( attribute )

##### 7.2-6 DotVertexLabelledDigraph
 ‣ DotVertexLabelledDigraph( cell ) ( operation )

#### 7.4 Global variables

The quiver generating the category of reflexive quivers

##### 7.4-1 QuiverOfCategoryOfReflexiveQuivers
 ‣ QuiverOfCategoryOfReflexiveQuivers ( global variable )

The category of reflexive quivers as a category of presheaves with values in SkeletalFinSets.

##### 7.4-2 FinReflexiveQuivers
 ‣ FinReflexiveQuivers ( global variable )

#### 7.5 GAP Categories

##### 7.5-1 IsCategoryOfReflexiveQuivers
 ‣ IsCategoryOfReflexiveQuivers( category ) ( filter )

Returns: true or false

The GAP category of the category of reflexive quivers.

##### 7.5-2 IsCellInCategoryOfReflexiveQuivers
 ‣ IsCellInCategoryOfReflexiveQuivers( cell ) ( filter )

Returns: true or false

The GAP category of cells in the category of reflexive quivers.

##### 7.5-3 IsObjectInCategoryOfReflexiveQuivers
 ‣ IsObjectInCategoryOfReflexiveQuivers( obj ) ( filter )

Returns: true or false

The GAP category of objects in the category of reflexive quivers.

##### 7.5-4 IsMorphismInCategoryOfReflexiveQuivers
 ‣ IsMorphismInCategoryOfReflexiveQuivers( mor ) ( filter )

Returns: true or false

The GAP category of morphisms in the category of reflexive quivers.

#### 7.6 Example

In the following we construct the category of finite reflexive quivers:

gap> LoadPackage( "FunctorCategories" );
true
gap> FinReflexiveQuivers;
FinReflexiveQuivers
gap> V := FinReflexiveQuivers.V;
<A projective object in FinReflexiveQuivers>
gap> Display( V );
( { 0 }, { 0 := [ 0 ] } )
gap> A := FinReflexiveQuivers.A;
<A projective object in FinReflexiveQuivers>
gap> Display( A );
( { 0, 1 }, { 0 := [ 0, 1 ], 1 := [ 0 ], 2 := [ 1 ] } )
gap> T := TerminalObject( FinReflexiveQuivers );
<An object in FinReflexiveQuivers>
gap> Display( T );
( { 0 }, { 0 := [ 0 ] } )
gap> T = V;
true
gap> G := CreateReflexiveQuiver( 2,
>              [ 1, 4 ],
>              [ 0,1, 0,0, 0,0, 1,0, 1,1 ] );
<An object in FinReflexiveQuivers>
gap> IsWellDefined( G );
true
gap> Display( G );
( { 0, 1 },
{ 0 := [ 0, 1 ], 1 := [ 0 ], 2 := [ 0, 0 ], 3 := [ 1, 0 ], 4 := [ 1 ] } )
gap> global_G := HomStructure( T, G );
|2|
gap> Display( global_G );
{ 0, 1 }
gap> DirectProduct( G, V ) = G;
true
gap> iso := CartesianLeftEvaluationMorphism( T, G );
<A morphism in FinReflexiveQuivers>
gap> IsIsomorphism( iso );
true
gap> Display( iso );
Image of <(C0)>:
{ 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1 }

Image of <(C1)>:
{ 0,..., 4 } ⱶ[ 1, 2, 3, 0, 4 ]→ { 0,..., 4 }

A morphism in FinReflexiveQuivers
given by the above data
gap> s := FinReflexiveQuivers.s;
<A split monomorphism in
FinReflexiveQuivers>
gap> Display( s );
Image of <(C0)>:
{ 0 } ⱶ[ 0 ]→ { 0, 1 }

Image of <(C1)>:
{ 0 } ⱶ[ 1 ]→ { 0, 1, 2 }

A morphism in FinReflexiveQuivers
given by the above data
gap> t := FinReflexiveQuivers.t;
<A split monomorphism in
FinReflexiveQuivers>
gap> Display( t );
Image of <(C0)>:
{ 0 } ⱶ[ 1 ]→ { 0, 1 }

Image of <(C1)>:
{ 0 } ⱶ[ 2 ]→ { 0, 1, 2 }

A morphism in FinReflexiveQuivers
given by the above data
gap> l := FinReflexiveQuivers.l;
<A split epimorphism in
FinReflexiveQuivers>
gap> Display( l );
Image of <(C0)>:
{ 0, 1 } ⱶ[ 0, 0 ]→ { 0 }

Image of <(C1)>:
{ 0, 1, 2 } ⱶ[ 0, 0, 0 ]→ { 0 }

A morphism in FinReflexiveQuivers
given by the above data
gap> omega := SubobjectClassifier( FinReflexiveQuivers );
<An object in FinReflexiveQuivers>
gap> Display( omega );
( { 0, 1 },
{ 0 := [ 0 ], 1 := [ 1, 0 ], 2 := [ 0, 1 ],
3 := [ 1, 1 ], 4 := [ 1 ] } )
gap> HomStructure( A, omega );
|5|
gap> subsA := ListOfSubobjects( A );
[ <A monomorphism in FinReflexiveQuivers>,
<A monomorphism in FinReflexiveQuivers>,
<A monomorphism in FinReflexiveQuivers>,
<A monomorphism in FinReflexiveQuivers>,
<A monomorphism in FinReflexiveQuivers> ]
gap> Perform( subsA, Display );
Image of <(C0)>:
∅ ⱶ[  ]→ { 0, 1 }

Image of <(C1)>:
∅ ⱶ[  ]→ { 0, 1, 2 }

A morphism in FinReflexiveQuivers
given by the above data
Image of <(C0)>:
{ 0 } ⱶ[ 0 ]→ { 0, 1 }

Image of <(C1)>:
{ 0 } ⱶ[ 1 ]→ { 0, 1, 2 }

A morphism in FinReflexiveQuivers
given by the above data
Image of <(C0)>:
{ 0 } ⱶ[ 1 ]→ { 0, 1 }

Image of <(C1)>:
{ 0 } ⱶ[ 2 ]→ { 0, 1, 2 }

A morphism in FinReflexiveQuivers
given by the above data
Image of <(C0)>:
{ 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1 }

Image of <(C1)>:
{ 0, 1 } ⱶ[ 1, 2 ]→ { 0, 1, 2 }

A morphism in FinReflexiveQuivers
given by the above data
Image of <(C0)>:
{ 0, 1 } ⱶ[ 0, 1 ]→ { 0, 1 }

Image of <(C1)>:
{ 0, 1, 2 } ⱶ[ 0, 1, 2 ]→ { 0, 1, 2 }

A morphism in FinReflexiveQuivers
given by the above data

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