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### 13 Tools

#### 13.1 Operations

##### 13.1-1 TensorizeObjectWithObjectInRangeCategoryOfHomomorphismStructure
 ‣ TensorizeObjectWithObjectInRangeCategoryOfHomomorphismStructure( objC, objH ) ( operation )

Returns: a CAP object

The arguments are an ojbect objC in a category $$C$$ and an object objH in $$H :=$$ RangeCategoryOfHomomorphismStructure( $$C$$ ). The output is an object in EnrichmentSpecificFiniteStrictCoproductCompletion( $$C$$ ), namely the tensor product objC $$\otimes$$ objH, i.e., the formal coproduct of $$l$$ copies of objC, where $$l :=$$ ObjectDatum( objH ) is a nonnegative integer.

##### 13.1-2 TensorizeObjectWithMorphismInRangeCategoryOfHomomorphismStructure
 ‣ TensorizeObjectWithMorphismInRangeCategoryOfHomomorphismStructure( objC, morH ) ( operation )

Returns: a CAP morphism

The arguments are an ojbect objC in a cocartesian category $$C$$ and a morphism morH in $$H :=$$ RangeCategoryOfHomomorphismStructure( $$C$$ ). The output is a morphism in EnrichmentSpecificFiniteStrictCoproductCompletion( $$C$$ ), namely the tensor product objC $$\otimes$$ morH, i.e., the morphism induced by morH from the formal coproduct of $$t$$ copies of objC to the formal coproduct of $$t$$ copies of objC, where $$s :=$$ ObjectDatum( Source( morH ) ) and $$t :=$$ ObjectDatum( Range( morH ) ).

##### 13.1-3 TensorizeMorphismWithObjectInRangeCategoryOfHomomorphismStructure
 ‣ TensorizeMorphismWithObjectInRangeCategoryOfHomomorphismStructure( morC, objH ) ( operation )

Returns: a CAP morphism

The arguments are a morphism morC in a cocartesian category $$C$$ and an object objH in $$H :=$$ RangeCategoryOfHomomorphismStructure( $$C$$ ). The output is a morphism in EnrichmentSpecificFiniteStrictCoproductCompletion( $$C$$ ), namely the tensor product morC $$\otimes$$ objH, i.e., the morphism induced by morC of the formal coproduct of $$l$$ copies of Source( morC ) to the formal coproduct of $$l$$ copies of Range( morC ), where $$l :=$$ ObjectDatum( objH ).

gap> LoadPackage( "FunctorCategories", ">= 2023.11-07", false );
true
gap> q := RightQuiver( "q(o)[x:o->o]" );
q(o)[x:o->o]
gap> F := FreeCategory( q );
FreeCategory( RightQuiver( "q(o)[x:o->o]" ) )
gap> Q := HomalgFieldOfRationals( );
Q
gap> QF := Q[F];
Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) )
gap> A := QF / [ QF.xxx ];
Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) ) / relations
gap> o := A.o;
<(o)>
gap> x := A.x;
(o)-[{ 1*(x) }]->(o)
gap> Qmat := RangeCategoryOfHomomorphismStructure( A );
Rows( Q )
gap> U := 3 / Qmat;
<A row module over Q of rank 3>
gap> phi := HomalgMatrix( [ 0,1,0, 0,0,1, 0,0,0 ], 3, 3, Q ) / Qmat;
<A morphism in Rows( Q )>
gap> Display( phi );
Source:
A row module over Q of rank 3

Matrix:
[ [  0,  1,  0 ],
[  0,  0,  1 ],
[  0,  0,  0 ] ]

Range:
A row module over Q of rank 3

A morphism in Rows( Q )
gap> PSh := PreSheaves( A );
PreSheaves( Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) )
/ relations, Rows( Q ) )
gap> Mphi := CreatePreSheafByValues( PSh, Pair( [ U ], [ phi ] ) );
<(o)->3; (x)->3x3>
gap> IsWellDefined( Mphi );
true
gap> Display( Mphi );
Image of <(o)>:
A row module over Q of rank 3

Image of (o)-[{ 1*(x) }]->(o):
Source:
A row module over Q of rank 3

Matrix:
[ [  0,  1,  0 ],
[  0,  0,  1 ],
[  0,  0,  0 ] ]

Range:
A row module over Q of rank 3

A morphism in Rows( Q )

An object in PreSheaves(
Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) ) / relations,
Rows( Q ) ) given by the above data
gap> Mphi_as_coequqlizer_pair := CoYonedaLemmaOnObjects( Mphi );
<An object in FiniteColimitCompletionWithStrictCoproducts(
Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) ) / relations )>
gap> Display( Mphi_as_coequqlizer_pair );
Image of <(V)>:
A formal direct sum consisting of 3 objects.
<(o)>
<(o)>
<(o)>

Image of <(A)>:
A formal direct sum consisting of 3 objects.
<(o)>
<(o)>
<(o)>

Image of (V)-[(s)]->(A):
A 3 x 3 matrix with entries in
Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) ) / relations

[1,1]: (o)-[{ 0 }]->(o)
[1,2]: (o)-[{ 1*(o) }]->(o)
[1,3]: (o)-[{ 0 }]->(o)
[2,1]: (o)-[{ 0 }]->(o)
[2,2]: (o)-[{ 0 }]->(o)
[2,3]: (o)-[{ 1*(o) }]->(o)
[3,1]: (o)-[{ 0 }]->(o)
[3,2]: (o)-[{ 0 }]->(o)
[3,3]: (o)-[{ 0 }]->(o)
Image of (V)-[(t)]->(A):
A 3 x 3 matrix with entries in
Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) ) / relations

[1,1]: (o)-[{ 1*(x) }]->(o)
[1,2]: (o)-[{ 0 }]->(o)
[1,3]: (o)-[{ 0 }]->(o)
[2,1]: (o)-[{ 0 }]->(o)
[2,2]: (o)-[{ 1*(x) }]->(o)
[2,3]: (o)-[{ 0 }]->(o)
[3,1]: (o)-[{ 0 }]->(o)
[3,2]: (o)-[{ 0 }]->(o)
[3,3]: (o)-[{ 1*(x) }]->(o)
An object in PreSheaves( FreeCategory( RightQuiver( "q(V,A)[s:V->A,t:V->A]" ) ),
AdditiveClosure( Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) ) /
relations ) ) given by the above data

An object in FiniteColimitCompletionWithStrictCoproducts(
Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) ) / relations )
given by the above data
>   TensorizeObjectWithMorphismInRangeCategoryOfHomomorphismStructure( o, phi );
Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) ) / relations )
defined by a 3 x 3 matrix of underlying morphisms>