gap> LoadPackage( "LinearAlgebraForCAP" ); true gap> LoadPackage( "GeneralizedMorphismsForCAP" ); true gap> SwitchGeneralizedMorphismStandard( "cospan" ); gap> Q := HomalgFieldOfRationals( ); Q gap> id := HomalgIdentityMatrix( 8, Q ); <An unevaluated 8 x 8 identity matrix over an internal ring> gap> a := CertainColumns( CertainRows( id, [ 1, 2, 3 ] ), [ 2, 3, 4, 5 ] ); <An unevaluated non-zero 3 x 4 matrix over an internal ring> gap> b := CertainColumns( CertainRows( id, [ 1, 2, 3, 4 ] ), [ 2, 3, 4, 5, 6 ] ); <An unevaluated non-zero 4 x 5 matrix over an internal ring> gap> c := CertainColumns( > CertainRows( id, [ 1, 2, 3, 4, 5 ] ), [ 3, 4, 5, 6, 7, 8 ] ); <An unevaluated non-zero 5 x 6 matrix over an internal ring> gap> IsZero( a * b ); false gap> IsZero( b * c ); false gap> IsZero( a * b * c ); true gap> Qmat := MatrixCategory( Q ); Category of matrices over Q gap> Lazy := LazyCategory( Qmat : show_evaluation := true ); LazyCategory( Category of matrices over Q ) gap> a := a / Lazy; SetLabel( a, "a" ); <An evaluated morphism in LazyCategory( Category of matrices over Q )> gap> b := b / Lazy; SetLabel( b, "b" ); <An evaluated morphism in LazyCategory( Category of matrices over Q )> gap> c := c / Lazy; SetLabel( c, "c" ); <An evaluated morphism in LazyCategory( Category of matrices over Q )> gap> d := CokernelProjection( a ); <An epimorphism in LazyCategory( Category of matrices over Q )> gap> e := CokernelColift( a, PreCompose( b, c ) ); <A morphism in LazyCategory( Category of matrices over Q )> gap> f := KernelEmbedding( e ); <A monomorphism in LazyCategory( Category of matrices over Q )> gap> g := KernelEmbedding( c ); <A monomorphism in LazyCategory( Category of matrices over Q )> gap> h := KernelLift( c, PreCompose( a, b ) ); <A morphism in LazyCategory( Category of matrices over Q )> gap> i := CokernelProjection( h ); <An epimorphism in LazyCategory( Category of matrices over Q )> gap> ff := AsGeneralizedMorphism( f ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> dd := AsGeneralizedMorphism( d ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> bb := AsGeneralizedMorphism( b ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> gg := AsGeneralizedMorphism( g ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> ii := AsGeneralizedMorphism( i ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> ss := PreCompose( [ ff, PseudoInverse( dd ), bb, PseudoInverse( gg ), ii ] ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> s := HonestRepresentative( ss ); <A morphism in LazyCategory( Category of matrices over Q )> gap> j := KernelObjectFunctorial( b, d, e ); <A morphism in LazyCategory( Category of matrices over Q )> gap> k := CokernelObjectFunctorial( h, g, b ); <A morphism in LazyCategory( Category of matrices over Q )> gap> HK := HomologyObject( j, s ); <An object in LazyCategory( Category of matrices over Q )> gap> HC := HomologyObject( s, k ); <An object in LazyCategory( Category of matrices over Q )>
gap> LoadPackage( "LinearAlgebraForCAP" ); true gap> LoadPackage( "GeneralizedMorphismsForCAP" ); true gap> SwitchGeneralizedMorphismStandard( "cospan" ); gap> Q := HomalgFieldOfRationals( ); Q gap> id := HomalgIdentityMatrix( 8, Q ); <An unevaluated 8 x 8 identity matrix over an internal ring> gap> a := CertainColumns( CertainRows( id, [ 1, 2, 3 ] ), [ 2, 3, 4, 5 ] ); <An unevaluated non-zero 3 x 4 matrix over an internal ring> gap> b := CertainColumns( CertainRows( id, [ 1, 2, 3, 4 ] ), [ 2, 3, 4, 5, 6 ] ); <An unevaluated non-zero 4 x 5 matrix over an internal ring> gap> c := CertainColumns( > CertainRows( id, [ 1, 2, 3, 4, 5 ] ), [ 3, 4, 5, 6, 7, 8 ] ); <An unevaluated non-zero 5 x 6 matrix over an internal ring> gap> IsZero( a * b ); false gap> IsZero( b * c ); false gap> IsZero( a * b * c ); true gap> Qmat := MatrixCategory( Q ); Category of matrices over Q gap> Lazy := LazyCategory( Qmat : > show_evaluation := true, primitive_operations := true ); LazyCategory( Category of matrices over Q ) gap> a := a / Lazy; SetLabel( a, "a" ); <An evaluated morphism in LazyCategory( Category of matrices over Q )> gap> b := b / Lazy; SetLabel( b, "b" ); SetLabel( Target( b ), "C" ); <An evaluated morphism in LazyCategory( Category of matrices over Q )> gap> c := c / Lazy; SetLabel( c, "c" ); <An evaluated morphism in LazyCategory( Category of matrices over Q )> gap> d := CokernelProjection( a ); <An epimorphism in LazyCategory( Category of matrices over Q )> gap> e := CokernelColift( a, PreCompose( b, c ) ); <A morphism in LazyCategory( Category of matrices over Q )> gap> f := KernelEmbedding( e ); <A monomorphism in LazyCategory( Category of matrices over Q )> gap> g := KernelEmbedding( c ); <A monomorphism in LazyCategory( Category of matrices over Q )> gap> h := KernelLift( c, PreCompose( a, b ) ); <A morphism in LazyCategory( Category of matrices over Q )> gap> i := CokernelProjection( h ); <An epimorphism in LazyCategory( Category of matrices over Q )> gap> ff := AsGeneralizedMorphism( f ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> dd := AsGeneralizedMorphism( d ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> bb := AsGeneralizedMorphism( b ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> gg := AsGeneralizedMorphism( g ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> ii := AsGeneralizedMorphism( i ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> ss := PreCompose( [ ff, PseudoInverse( dd ), bb, PseudoInverse( gg ), ii ] ); <A morphism in Generalized morphism category of LazyCategory( Category of matrices over Q ) by cospan> gap> s := HonestRepresentative( ss ); <A morphism in LazyCategory( Category of matrices over Q )> gap> j := KernelObjectFunctorial( b, d, e ); <A morphism in LazyCategory( Category of matrices over Q )> gap> k := CokernelObjectFunctorial( h, g, b ); <A morphism in LazyCategory( Category of matrices over Q )> gap> HK := HomologyObject( j, s ); <An object in LazyCategory( Category of matrices over Q )> gap> HC := HomologyObject( s, k ); <An object in LazyCategory( Category of matrices over Q )>
gap> LoadPackage( "Algebroids", ">= 2022.05-06" ); true 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> zz := HomalgRingOfIntegers( ); Z gap> Zq := zz[Fq]; Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) gap> A := Zq / [ Zq.abc ]; Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations gap> LoadPackage( "FreydCategoriesForCAP" ); true gap> A_add := AdditiveClosure( A ); AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) gap> A_abel := AdelmanCategory( A_add ); Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) gap> LoadPackage( "LazyCategories" ); true gap> Lazy := LazyCategory( A_abel : show_evaluation := true ); LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) gap> a := A.a / A_add / A_abel / Lazy; <An evaluated morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> b := A.b / A_add / A_abel / Lazy; <An evaluated morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> c := A.c / A_add / A_abel / Lazy; <An evaluated morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> SetLabel( a, "a" ); gap> SetLabel( b, "b" ); gap> SetLabel( c, "c" ); gap> SetLabel( Source( a ), "1" ); gap> SetLabel( Source( b ), "2" ); gap> SetLabel( Target( b ), "3" ); gap> SetLabel( Target( c ), "4" ); gap> d := CokernelProjection( a ); <An epimorphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> e := CokernelColift( a, PreCompose( b, c ) ); <A morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> f := KernelEmbedding( e ); <A monomorphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> g := KernelEmbedding( c ); <A monomorphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> h := KernelLift( c, PreCompose( a, b ) ); <A morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> i := CokernelProjection( h ); <An epimorphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, 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 LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> dd := AsGeneralizedMorphism( d ); <A morphism in Generalized morphism category of LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> bb := AsGeneralizedMorphism( b ); <A morphism in Generalized morphism category of LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> gg := AsGeneralizedMorphism( g ); <A morphism in Generalized morphism category of LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> ii := AsGeneralizedMorphism( i ); <A morphism in Generalized morphism category of LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> ss := PreCompose( [ ff, PseudoInverse( dd ), bb, PseudoInverse( gg ), ii ] ); <A morphism in Generalized morphism category of LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> s := HonestRepresentative( ss ); <A morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> j := KernelObjectFunctorial( b, d, e ); <A morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> k := CokernelObjectFunctorial( h, g, b ); <A morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> HK := HomologyObject( j, s ); <An object in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> HC := HomologyObject( s, k ); <An object in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )>
gap> LoadPackage( "Algebroid", ">= 2022.05-02" ); true 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> zz := HomalgRingOfIntegers( ); Z gap> Zq := zz[Fq]; Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) gap> A := Zq / [ Zq.abc ]; Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations gap> LoadPackage( "FreydCategoriesForCAP" ); true gap> A_add := AdditiveClosure( A ); AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) gap> A_abel := AdelmanCategory( A_add ); Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) gap> LoadPackage( "LazyCategories" ); true gap> Lazy := LazyCategory( A_abel : > show_evaluation := true, primitive_operations := true ); LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) gap> a := A.a / A_add / A_abel / Lazy; <An evaluated morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> b := A.b / A_add / A_abel / Lazy; <An evaluated morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> c := A.c / A_add / A_abel / Lazy; <An evaluated morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> SetLabel( a, "a" ); gap> SetLabel( b, "b" ); gap> SetLabel( c, "c" ); gap> SetLabel( Source( a ), "1" ); gap> SetLabel( Source( b ), "2" ); gap> SetLabel( Target( b ), "3" ); gap> SetLabel( Target( c ), "4" ); gap> d := CokernelProjection( a ); <An epimorphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> e := CokernelColift( a, PreCompose( b, c ) ); <A morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> f := KernelEmbedding( e ); <A monomorphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> g := KernelEmbedding( c ); <A monomorphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> h := KernelLift( c, PreCompose( a, b ) ); <A morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> i := CokernelProjection( h ); <An epimorphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, 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 LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> dd := AsGeneralizedMorphism( d ); <A morphism in Generalized morphism category of LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> bb := AsGeneralizedMorphism( b ); <A morphism in Generalized morphism category of LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> gg := AsGeneralizedMorphism( g ); <A morphism in Generalized morphism category of LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> ii := AsGeneralizedMorphism( i ); <A morphism in Generalized morphism category of LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> ss := PreCompose( [ ff, PseudoInverse( dd ), bb, PseudoInverse( gg ), ii ] ); <A morphism in Generalized morphism category of LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> s := HonestRepresentative( ss ); <A morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> j := KernelObjectFunctorial( b, d, e ); <A morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> k := CokernelObjectFunctorial( h, g, b ); <A morphism in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> HK := HomologyObject( j, s ); <An object in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> HC := HomologyObject( s, k ); <An object in LazyCategory( Adelman category( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )>
gap> LoadPackage( "Algebroids", ">= 2022.05-06" ); true 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> zz := HomalgRingOfIntegers( ); Z gap> Zq := zz[Fq]; Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) gap> A := Zq / [ Zq.abc ]; Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations gap> LoadPackage( "FreydCategoriesForCAP" ); true gap> A_add := AdditiveClosure( A ); AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) gap> Lazy := LazyCategory( A_add : show_evaluation := true, > lazify_range_of_hom_structure := true ); LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) gap> a := A.a / A_add / Lazy; SetLabel( a, "a" ); <An evaluated morphism in LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) )> gap> b := A.b / A_add / Lazy; SetLabel( b, "b" ); SetLabel( Target( b ), "3" ); <An evaluated morphism in LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) )> gap> c := A.c / A_add / Lazy; SetLabel( c, "c" ); <An evaluated morphism in LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) )> gap> Adelman := AdelmanCategory( Lazy ); Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) gap> a := a / Adelman; <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> b := b / Adelman; <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> c := c / Adelman; <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> d := CokernelProjection( a ); <An epimorphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> e := CokernelColift( a, PreCompose( b, c ) ); <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> f := KernelEmbedding( e ); <A monomorphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> g := KernelEmbedding( c ); <A monomorphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> h := KernelLift( c, PreCompose( a, b ) ); <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> i := CokernelProjection( h ); <An epimorphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, 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 Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> dd := AsGeneralizedMorphism( d ); <A morphism in Generalized morphism category of Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> bb := AsGeneralizedMorphism( b ); <A morphism in Generalized morphism category of Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> gg := AsGeneralizedMorphism( g ); <A morphism in Generalized morphism category of Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> ii := AsGeneralizedMorphism( i ); <A morphism in Generalized morphism category of Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> ss := PreCompose( [ ff, PseudoInverse( dd ), bb, PseudoInverse( gg ), ii ] ); <A morphism in Generalized morphism category of Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> s := HonestRepresentative( ss ); <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> j := KernelObjectFunctorial( b, d, e ); <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> k := CokernelObjectFunctorial( h, g, b ); <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> HK := HomologyObject( j, s ); <An object in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> HC := HomologyObject( s, k ); <An object in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )>
gap> LoadPackage( "Algebroids", ">= 2022.05-06" ); true 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> zz := HomalgRingOfIntegers( ); Z gap> Zq := zz[Fq]; Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) gap> A := Zq / [ Zq.abc ]; Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations gap> LoadPackage( "FreydCategoriesForCAP" ); true gap> A_add := AdditiveClosure( A ); AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) gap> Lazy := LazyCategory( A_add : show_evaluation := true, > lazify_range_of_hom_structure := true, > primitive_operations := true ); LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) gap> a := A.a / A_add / Lazy; SetLabel( a, "a" ); <An evaluated morphism in LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) )> gap> b := A.b / A_add / Lazy; SetLabel( b, "b" ); SetLabel( Target( b ), "3" ); <An evaluated morphism in LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) )> gap> c := A.c / A_add / Lazy; SetLabel( c, "c" ); <An evaluated morphism in LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) )> gap> Adelman := AdelmanCategory( Lazy ); Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) gap> a := a / Adelman; <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> b := b / Adelman; <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> c := c / Adelman; <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> d := CokernelProjection( a ); <An epimorphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> e := CokernelColift( a, PreCompose( b, c ) ); <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> f := KernelEmbedding( e ); <A monomorphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> g := KernelEmbedding( c ); <A monomorphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> h := KernelLift( c, PreCompose( a, b ) ); <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> i := CokernelProjection( h ); <An epimorphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, 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 Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> dd := AsGeneralizedMorphism( d ); <A morphism in Generalized morphism category of Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> bb := AsGeneralizedMorphism( b ); <A morphism in Generalized morphism category of Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> gg := AsGeneralizedMorphism( g ); <A morphism in Generalized morphism category of Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> ii := AsGeneralizedMorphism( i ); <A morphism in Generalized morphism category of Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> ss := PreCompose( [ ff, PseudoInverse( dd ), bb, PseudoInverse( gg ), ii ] ); <A morphism in Generalized morphism category of Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) ) by cospan> gap> s := HonestRepresentative( ss ); <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> j := KernelObjectFunctorial( b, d, e ); <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> k := CokernelObjectFunctorial( h, g, b ); <A morphism in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> HK := HomologyObject( j, s ); <An object in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )> gap> HC := HomologyObject( s, k ); <An object in Adelman category( LazyCategory( AdditiveClosure( Algebroid( Z, FreeCategory( RightQuiver( "q(4)[a:1->2,b:2->3,c:3->4]" ) ) ) / relations ) ) )>
gap> T := TerminalCategoryWithMultipleObjects( ); TerminalCategoryWithMultipleObjects( ) gap> L := LazyCategory( T : primitive_operations := true, optimize := 0 ); LazyCategory( TerminalCategoryWithMultipleObjects( ) ) gap> a := "a" / T / L; <An evaluated object in LazyCategory( TerminalCategoryWithMultipleObjects( ) )> gap> Display( a ); a An evaluated object given by the above data gap> IsWellDefined( a ); true gap> IsWellDefined( DirectSum( a, a ) ); true gap> IsWellDefined( IdentityMorphism( DirectSum( a, a ) ) ); true gap> aa := ObjectConstructor( T, "a" ) / L; <An evaluated object in LazyCategory( TerminalCategoryWithMultipleObjects( ) )> gap> Display( aa ); a An evaluated object given by the above data gap> a = aa; true gap> b := "b" / T / L; <An evaluated object in LazyCategory( TerminalCategoryWithMultipleObjects( ) )> gap> Display( b ); b An evaluated object given by the above data gap> a = b; false gap> t := TensorProduct( a, b ); <An object in LazyCategory( TerminalCategoryWithMultipleObjects( ) )> gap> Display( t ); TensorProductOnObjects( <An evaluated object in LazyCategory( TerminalCategoryWithMultipleObjects( ) )>, <An evaluated object in LazyCategory( TerminalCategoryWithMultipleObjects( ) )> ) gap> a = t; false gap> TensorProduct( a, a ) = t; false gap> m := MorphismConstructor( EvaluatedCell( a ), "m", EvaluatedCell( b ) ) / L; <An evaluated morphism in LazyCategory( TerminalCategoryWithMultipleObjects( ) )> gap> Display( m ); a | | m v b An evaluated morphism given by the above data gap> IsWellDefined( m ); true gap> n := MorphismConstructor( EvaluatedCell( a ), "n", EvaluatedCell( b ) ) / L; <An evaluated morphism in LazyCategory( TerminalCategoryWithMultipleObjects( ) )> gap> Display( n ); a | | n v b An evaluated morphism given by the above data gap> IsEqualForMorphisms( m, n ); false gap> IsCongruentForMorphisms( m, n ); true gap> m = n; true gap> id := IdentityMorphism( a ); <An identity morphism in LazyCategory( TerminalCategoryWithMultipleObjects( ) )> gap> Display( id ); IdentityMorphism( <An evaluated object in LazyCategory( TerminalCategoryWithMultipleObjects( ) )> ) gap> m = id; false gap> id = MorphismConstructor( EvaluatedCell( a ), "xy", EvaluatedCell( a ) ) / L; true gap> z := ZeroMorphism( a, a ); <A zero morphism in LazyCategory( TerminalCategoryWithMultipleObjects( ) )> gap> Display( z ); ZeroMorphism( <An evaluated object in LazyCategory( TerminalCategoryWithMultipleObjects( ) )>, <An evaluated object in LazyCategory( TerminalCategoryWithMultipleObjects( ) )> ) gap> id = z; true
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