‣ AbelianClosure( B ) | ( operation ) |
Returns: a CAP category
Construct an Abelian closure category.
‣ UnderlyingCategory( abelian_closure ) | ( attribute ) |
‣ YonedaEmbeddingOfUnderlyingCategory( abelian_closure ) | ( attribute ) |
Returns: a CAP functor
‣ IsAbelianClosure( category ) | ( filter ) |
Returns: true or false
The GAP category of an Abelian closure category.
‣ IsCellInAbelianClosure( cell ) | ( filter ) |
Returns: true or false
The GAP category of cells in an Abelian closure category.
‣ IsObjectInAbelianClosure( obj ) | ( filter ) |
Returns: true or false
The GAP category of objects in an Abelian closure category.
‣ IsMorphismInAbelianClosure( mor ) | ( filter ) |
Returns: true or false
The GAP category of morphisms in an Abelian closure category.
gap> q := RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ); q(4)[a:1->2,b:2->3,c:3->4] gap> Fq := FreeCategory( q ); FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) gap> Q := HomalgFieldOfRationals( ); Q gap> Qq := Q[Fq]; Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) gap> L := Qq / [ Qq.abc ]; Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations gap> A := AbelianClosure( L ); AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) gap> a := A.a; <A morphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> b := A.b; <A morphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> c := A.c; <A morphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> IsZero( PreCompose( [ a, b, c ] ) ); true gap> d := CokernelProjection( a ); <An epimorphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> e := CokernelColift( a, PreCompose( b, c ) ); <A morphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> f := KernelEmbedding( e ); <A monomorphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> g := KernelEmbedding( c ); <A monomorphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> h := KernelLift( c, PreCompose( a, b ) ); <A morphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> i := CokernelProjection( h ); <An epimorphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> ff := AsGeneralizedMorphism( f ); <A morphism in Generalized morphism category of AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> dd := AsGeneralizedMorphism( d ); <A morphism in Generalized morphism category of AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> bb := AsGeneralizedMorphism( b ); <A morphism in Generalized morphism category of AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> gg := AsGeneralizedMorphism( g ); <A morphism in Generalized morphism category of AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> ii := AsGeneralizedMorphism( i ); <A morphism in Generalized morphism category of AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> ss := PreCompose( [ ff, PseudoInverse( dd ), bb, PseudoInverse( gg ), ii ] ); <A morphism in Generalized morphism category of AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> IsHonest( ss ); true gap> s := HonestRepresentative( ss ); <A morphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> j := KernelObjectFunctorial( b, d, e ); <A morphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> k := CokernelObjectFunctorial( h, g, b ); <A morphism in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> HK := HomologyObject( j, s ); <An object in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> IsZero( HK ); true gap> HC := HomologyObject( s, k ); <An object in AbelianClosure( Algebroid( Q, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations )> gap> IsZero( HC ); true
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