‣ 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.
‣ 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 ) ).
‣ 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 gap> phi_in_additive_closure := > TensorizeObjectWithMorphismInRangeCategoryOfHomomorphismStructure( o, phi ); <A morphism in AdditiveClosure( Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) ) / relations ) defined by a 3 x 3 matrix of underlying morphisms> gap> x_in_additive_closure := > TensorizeMorphismWithObjectInRangeCategoryOfHomomorphismStructure( x, U ); <A morphism in AdditiveClosure( Algebra( Q, FreeCategory( RightQuiver( "q(o)[x:o->o]" ) ) ) / relations ) defined by a 3 x 3 matrix of underlying morphisms> gap> Mphi_as_coequqlizer_pair.s = phi_in_additive_closure; true gap> Mphi_as_coequqlizer_pair.t = x_in_additive_closure; true
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