gap> LoadPackage( "ModulePresentations", false );
true
gap> LoadPackage( "HomotopyCategories", false );
true
gap> HOMALG_IO.show_banners := false;;
gap> QQ := HomalgFieldOfRationalsInSingular( );
Q
gap> QQ_xy := QQ["x, y"];
Q[x,y]
gap> QQ_xy_mod := LeftPresentations( QQ_xy );
Category of left presentations of Q[x,y]
gap> K_QQ_xy_mod := HomotopyCategoryByCochains( QQ_xy_mod );
Homotopy category by cochains( Category of left presentations of Q[x,y] )
gap> m := HomalgMatrix("[[0,0,25*y+9,30,0,0,7*y], [0,0,0,0,24*x,0,20*y], [0,0,0,0,0,24,0], [0,0,29*x+27*y+10,0,0,0,0], [0,0,0,0,34,0,0], [0,0,0,0,0,0,0], [0,0,0,33*y,4*y+12,0,42], [0,20*x+34,0,0,0,0,0], [0,0,0,43*x,0,0,24*y], [0,0,0,0,0,0,0]]", 10, 7, QQ_xy );
<A 10 x 7 matrix over an external ring>
gap> n := HomalgMatrix("[[0,0,25*y+9], [0,0,725*x+7], [0,10*x+17,0]]", 3, 3, QQ_xy );
<A 3 x 3 matrix over an external ring>
gap> M := AsLeftPresentation( QQ_xy_mod, m );
<An object in Category of left presentations of Q[x,y]>
gap> Display( M );
0,0,      25*y+9,      30,  0,     0, 7*y, 
0,0,      0,           0,   24*x,  0, 20*y,
0,0,      0,           0,   0,     24,0,   
0,0,      29*x+27*y+10,0,   0,     0, 0,   
0,0,      0,           0,   34,    0, 0,   
0,0,      0,           0,   0,     0, 0,   
0,0,      0,           33*y,4*y+12,0, 42,  
0,20*x+34,0,           0,   0,     0, 0,   
0,0,      0,           43*x,0,     0, 24*y,
0,0,      0,           0,   0,     0, 0    

An object in Category of left presentations of Q[x,y]
gap> IsZero( M );
false
gap> IsProjective( M );
false
gap> N := AsLeftPresentation( QQ_xy_mod, n );
<An object in Category of left presentations of Q[x,y]>
gap> Display( N );
0,0,      25*y+9, 
0,0,      725*x+7,
0,10*x+17,0       

An object in Category of left presentations of Q[x,y]
gap> a := HomalgMatrix( "[[1,0,0], [0,1,0], [0,0,1], [0,0,-5/6*y-3/10], [0,0,0], [0,0,0], [0,0,0]]", 7, 3, QQ_xy );
<A 7 x 3 matrix over an external ring>
gap> alpha := PresentationMorphism( M, a, N );
<A morphism in Category of left presentations of Q[x,y]>
gap> Display( alpha );
1,0,0,
0,1,0,
0,0,1,
0,0,-5/6*y-3/10,
0,0,0,
0,0,0,
0,0,0

A morphism in Category of left presentations of Q[x,y]
gap> IsIsomorphism( alpha );
true
gap> M := CreateComplex( K_QQ_xy_mod, [ MorphismIntoZeroObject( M ) ], 0 );
<An object in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ 0 .. 1 ]>
gap> ObjectsSupport( M );
[ 0 ]
gap> ObjectAt( M, 0 );
<An object in Category of left presentations of Q[x,y]>
gap> DifferentialAt( M, 0 );
<A zero, split epimorphism in Category of left presentations of Q[x,y]>
gap> N := CreateComplex( K_QQ_xy_mod, [ MorphismIntoZeroObject( N ) ], 0 );
<An object in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ 0 .. 1 ]>
gap> ObjectsSupport( N );
[ 0 ]
gap> alpha := CreateComplexMorphism( K_QQ_xy_mod, M, [ alpha ], 0, N );
<A morphism in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ 0 .. 1 ]>
gap> p_M := ProjectiveResolution( M, true );
<An object in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ -3 .. 1 ]>
gap> q_M := QuasiIsomorphismFromProjectiveResolution( M, true );
<A morphism in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ -3 .. 1 ]>
gap> p_N := ProjectiveResolution( N, true );
<An object in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ -2 .. 1 ]>
gap> q_N := QuasiIsomorphismFromProjectiveResolution( N, true );
<A morphism in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ -2 .. 1 ]>
gap> p_alpha := MorphismBetweenProjectiveResolutions( alpha, true );
<A morphism in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ -3 .. 1 ]>
gap> IsIsomorphism( p_alpha );
true
gap> MorphismAt( p_alpha, 0 );
<A morphism in Category of left presentations of Q[x,y]>
gap> PreCompose( p_alpha, q_N ) = PreCompose(q_M, alpha );
true
gap> cone_q_M := StandardConeObject( q_M );
<An object in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ -4 .. 1 ]>
gap> CohomologySupport( cone_q_M ); # quasi-iso <-> cone is acyclic
[ ]
gap> iota_q_M := MorphismIntoStandardConeObject( q_M );
<A morphism in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ -4 .. 1 ]>
gap> Display( iota_q_M );

== 1 =======================
(an empty 0 x 0 matrix)

A zero, isomorphism in Category of left presentations of Q[x,y]

== 0 =======================
1,0,0,0,0,0,0,
0,1,0,0,0,0,0,
0,0,1,0,0,0,0,
0,0,0,1,0,0,0,
0,0,0,0,1,0,0,
0,0,0,0,0,1,0,
0,0,0,0,0,0,1 

A morphism in Category of left presentations of Q[x,y]

== -1 =======================
(an empty 0 x 7 matrix)

A zero, split monomorphism in Category of left presentations of Q[x,y]

== -2 =======================
(an empty 0 x 8 matrix)

A zero, split monomorphism in Category of left presentations of Q[x,y]

== -3 =======================
(an empty 0 x 3 matrix)

A zero, split monomorphism in Category of left presentations of Q[x,y]

== -4 =======================
(an empty 0 x 1 matrix)

A zero, split monomorphism in Category of left presentations of Q[x,y]


A morphism in Homotopy category by cochains( Category of left presentations of Q[x,y] ) defined by the above data

gap> IsWellDefined( iota_q_M );
true
gap> pi_q_M := MorphismFromStandardConeObject( q_M );
<A morphism in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ -4 .. 1 ]>
gap> Display( pi_q_M );
== 1 =======================
(an empty 0 x 0 matrix)

A zero, isomorphism in Category of left presentations of Q[x,y]

== 0 =======================
(an empty 7 x 0 matrix)

A morphism in Category of left presentations of Q[x,y]

== -1 =======================
1,0,0,0,0,0,0,
0,1,0,0,0,0,0,
0,0,1,0,0,0,0,
0,0,0,1,0,0,0,
0,0,0,0,1,0,0,
0,0,0,0,0,1,0,
0,0,0,0,0,0,1 

A morphism in Category of left presentations of Q[x,y]

== -2 =======================
1,0,0,0,0,0,0,0,
0,1,0,0,0,0,0,0,
0,0,1,0,0,0,0,0,
0,0,0,1,0,0,0,0,
0,0,0,0,1,0,0,0,
0,0,0,0,0,1,0,0,
0,0,0,0,0,0,1,0,
0,0,0,0,0,0,0,1 

A morphism in Category of left presentations of Q[x,y]

== -3 =======================
1,0,0,
0,1,0,
0,0,1 

A morphism in Category of left presentations of Q[x,y]

== -4 =======================
1

A morphism in Category of left presentations of Q[x,y]


A morphism in Homotopy category by cochains( Category of left presentations of Q[x,y] ) defined by the above data
gap> nu := MorphismBetweenStandardConeObjects(q_M, p_alpha, alpha, q_N);
<A morphism in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ -4 .. 1 ]>
gap> IsWellDefined( nu );
true
gap> IsZero( nu );
false
gap> IsIsomorphism( nu );
true
gap> ShiftOfObjectByInteger( p_M, 2 );
<An object in Homotopy category by cochains( Category of left presentations of Q[x,y] ) supported on the interval [ -5 .. -1 ]>

gap> LoadPackage( "Algebroids", false );
true
gap> LoadPackage( "HomotopyCategories", false );
true
gap> q_O := FinQuiver( "q_O(O0,O1,O2)[x0:O0->O1,x1:O0->O1,x2:O0->O1,y0:O1->O2,y1:O1->O2,y2:O1->O2]" );
FinQuiver( "q_O(O0,O1,O2)[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2]" )
gap> SetLaTeXStringsOfObjects( q_O, [ "\\mathcal{O}_{0}", "\\mathcal{O}_{1}", "\\mathcal{O}_{2}" ] );
gap> SetLaTeXStringsOfMorphisms( q_O, [ "x_0", "x_1", "x_2", "y_0", "y_1", "y_2" ] );
gap> P_O := PathCategory( q_O );
PathCategory( FinQuiver( "q_O(O0,O1,O2)[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2, y2:O1-≻O2]" ) )
gap> rho_O := [ [ P_O.x0y1, P_O.x1y0 ], [ P_O.x0y2, P_O.x2y0 ], [ P_O.x1y2, P_O.x2y1 ] ];
[ [ x0⋅y1:(O0) -≻ (O2), x1⋅y0:(O0) -≻ (O2) ], [ x0⋅y2:(O0) -≻ (O2), x2⋅y0:(O0) -≻ (O2) ], [ x1⋅y2:(O0) -≻ (O2), x2⋅y1:(O0) -≻ (O2) ] ]
gap> quotient_P_O := P_O / rho_O;
PathCategory( FinQuiver( "q_O(O0,O1,O2)[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2]" ) ) / [ x0⋅y1 = x1⋅y0, x0⋅y2 = x2⋅y0, x1⋅y2 = x2⋅y1 ]
gap> QQ := HomalgFieldOfRationals( );
Q
gap> k := QQ;
Q
gap> k_quotient_P_O := k[quotient_P_O];
Q-LinearClosure( PathCategory( FinQuiver( "q_O(O0,O1,O2)[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2]" ) ) / [ x0⋅y1 = x1⋅y0, x0⋅y2 = x2⋅y0, x1⋅y2 = x2⋅y1 ] )
gap> A_O := AlgebroidFromDataTables( k_quotient_P_O );
Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms
gap> phi := 2 * A_O.x0 + 3 * A_O.x1 - A_O.x2;
<2*x0 + 3*x1 - 1*x2:(O0) -≻ (O1)>
gap> A_Oadd := AdditiveClosure( A_O );
AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2, y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms )
gap> KA_Oadd := HomotopyCategoryByCochains( A_Oadd );
Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) )
gap> E10 := [ A_O.O0, A_O.O0, A_O.O0 ] / A_Oadd;
<An object in AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2, y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) defined by 3 underlying objects>
gap> E11 := [ A_O.O1, A_O.O1, A_O.O1 ] / A_Oadd;
<An object in AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2, y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) defined by 3 underlying objects>
gap> E12 := [ A_O.O2 ] / A_Oadd;
<An object in AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2, y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) defined by 1 underlying objects>
gap> delta_0 := AdditiveClosureMorphism(
>         E10,
>         [ [ A_O.x1, -A_O.x0, ZeroMorphism(A_O.O0, A_O.O1) ],
>           [ A_O.x2, ZeroMorphism(A_O.O0, A_O.O1), -A_O.x0 ],
>           [ ZeroMorphism(A_O.O0, A_O.O1), A_O.x2, -A_O.x1 ] ],
>         E11 );
<A morphism in AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2, y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) defined by a 3 x 3 matrix of underlying morphisms>
gap> delta_1 := AdditiveClosureMorphism(
>         E11,
>         [ [ A_O.y0 ],
>           [ A_O.y1 ],
>           [ A_O.y2 ] ],
>         E12 );
<A morphism in AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) defined by a 3 x 1 matrix of underlying morphisms>
gap> E1 := CreateComplex( KA_Oadd, [ delta_0, delta_1 ], 0 );
<An object in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ 0 .. 2 ]>
gap> E20 := [ A_O.O0, A_O.O0, A_O.O0 ] / A_Oadd;
<An object in AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) defined by 3 underlying objects>
gap> E21 := [ A_O.O1] / A_Oadd;
<An object in AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) defined by 1 underlying objects>
gap> delta_0 := AdditiveClosureMorphism(
>       E20,
>       [ [ A_O.x0 ],
>         [ A_O.x1 ],
>         [ A_O.x2 ] ],
>       E21 );
<A morphism in AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) defined by a 3 x 1 matrix of underlying morphisms>
gap> E2 := CreateComplex( KA_Oadd, [ delta_0 ], 0 );
<An object in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ 0 .. 1 ]>
gap> E3 := CreateComplex( KA_Oadd, A_O.O0 / A_Oadd, 0 );
<An object in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ 0 ]>
gap> seq := CreateStrongExceptionalSequence( [ E1, E2, E3 ] );
A strong exceptional sequence in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[ x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) )
gap> T := DirectSum( [ E1, E2, E3 ] );
<An object in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ 0 .. 2 ]>
gap> RankOfObject( HomStructure( E1, E1 ) ) = 1 and
>   RankOfObject( HomStructure( E2, E2 ) ) = 1 and
>     RankOfObject( HomStructure( E3, E3 ) ) = 1;
true
gap> IsZero( HomStructure( E3, E2 ) ) and
>   IsZero( HomStructure( E2, E1 ) ) and
>     IsZero( HomStructure( E3, E1 ) );
true
gap> IsZero( HomStructure( T, Shift( T, -2 ) ) ) and
>   IsZero( HomStructure( T, Shift( T, -1 ) ) ) and
>     IsZero( HomStructure( T, Shift( T, 1 ) ) ) and
>       IsZero( HomStructure( T, Shift( T, 2 ) ) );
true
gap> RankOfObject( HomStructure( T, T ) );
12
gap> A_E := AbstractionAlgebroid( seq );
Q-algebroid( {E1,E2,E3}[m1_2_1:E1-≻E2,m1_2_2:E1-≻E2,m1_2_3:E1-≻E2,m2_3_1:E2-≻E3,m2_3_2:E2-≻E3,m2_3_3:E2-≻E3] ) defined by 3 objects and 6 generating morphisms
gap> q_E := UnderlyingQuiver( A_E );
FinQuiver( "q(E1,E2,E3)[m1_2_1:E1-≻E2,m1_2_2:E1-≻E2,m1_2_3:E1-≻E2,m2_3_1:E2-≻E3,m2_3_2:E2-≻E3,m2_3_3:E2-≻E3]" )
gap> Assert( 0, Dimension( A_E ) = 12 );
gap> a := IsomorphismIntoAbstractionAlgebroid( seq );
Isomorphism: strong exceptional sequence ⟶ abstraction algebroid
gap> r := IsomorphismFromAbstractionAlgebroid( seq );
Isomorphism: abstraction algebroid ⟶ strong exceptional sequence
gap> m := A_E.("m1_2_1");
<1*m1_2_1:(E1) -≻ (E2)>
gap> Assert( 0, m = ApplyFunctor( a, ApplyFunctor( r, m ) ) );
gap> T_E := TriangulatedSubcategory( seq );
TriangulatedSubcategory( A strong exceptional sequence in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) )
gap> O0 := CreateComplex( KA_Oadd, A_O.("O0") / A_Oadd, 0 );
<An object in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ 0 ]>
gap> O1 := CreateComplex( KA_Oadd, A_O.("O1") / A_Oadd, 0 );
<An object in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ 0 ]>
gap> O2 := CreateComplex( KA_Oadd, A_O.("O2") / A_Oadd, 0 );
<An object in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ 0 ]>
gap> IsWellDefined( AsSubcategoryCell( T_E, O0 ) ) and
>     IsWellDefined( AsSubcategoryCell( T_E, O1 ) ) and
>         IsWellDefined( AsSubcategoryCell( T_E, O2 ) );
true
gap> G := ReplacementFunctorIntoHomotopyCategoryOfAdditiveClosureOfAbstractionAlgebroid( seq );
Replacement functor
gap> F := ConvolutionFunctorFromHomotopyCategoryOfAdditiveClosureOfAbstractionAlgebroid( seq );
Convolution functor
gap> G_O0 := ApplyFunctor( G, O0 );
<An object in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1,E2,E3}[m1_2_1:E1-≻E2,m1_2_2:E1-≻E2,m1_2_3:E1-≻E2,m2_3_1:E2-≻E3,m2_3_2:E2-≻E3,m2_3_3:E2-≻E3] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ 0 ]>
gap> G_O1 := ApplyFunctor( G, O1 );
<An object in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1,E2,E3}[m1_2_1:E1-≻E2,m1_2_2:E1-≻E2,m1_2_3:E1-≻E2,m2_3_1:E2-≻E3,m2_3_2:E2-≻E3,m2_3_3:E2-≻E3] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ -1 .. 0 ]>
gap> G_O2 := ApplyFunctor( G, O2 );
<An object in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {E1,E2,E3}[m1_2_1:E1-≻E2,m1_2_2:E1-≻E2,m1_2_3:E1-≻E2,m2_3_1:E2-≻E3,m2_3_2:E2-≻E3,m2_3_3:E2-≻E3] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ -2 .. 0 ]>
gap> epsilon := CounitOfConvolutionReplacementAdjunction( seq );
Counit ϵ : F∘G ⟹ Id of the adjunction F ⊣ G
gap> epsilon_O0 := ApplyNaturalTransformation( epsilon, O0 );
<A morphism in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ 0 ]>
gap> epsilon_O1 := ApplyNaturalTransformation( epsilon, O1 );
<A morphism in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ -1 .. 0 ]>
gap> epsilon_O2 := ApplyNaturalTransformation( epsilon, O2 );
<A morphism in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ -2 .. 0 ]>
gap> ForAll( [ epsilon_O0, epsilon_O1, epsilon_O2 ], IsIsomorphism );
true
gap> i := InverseForMorphisms( DirectSumFunctorial( [ epsilon_O0, epsilon_O1, epsilon_O2 ] ) );
<A morphism in Homotopy category by cochains( AdditiveClosure( Q-algebroid( {O0,O1,O2}[x0:O0-≻O1,x1:O0-≻O1,x2:O0-≻O1,y0:O1-≻O2,y1:O1-≻O2,y2:O1-≻O2] ) defined by 3 objects and 6 generating morphisms ) ) supported on the interval [ -2 .. 0 ]>
gap> IsWellDefined( i ) and IsIsomorphism( i );
true