‣ 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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