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#! @Chapter Examples and Tests
#! @Section Homology object
LoadPackage( "LinearAlgebraForCAP" );
#! @Example
field := HomalgFieldOfRationals( );;
vec := MatrixCategory( field );;
A := MatrixCategoryObject( vec, 1 );;
B := MatrixCategoryObject( vec, 2 );;
C := MatrixCategoryObject( vec, 3 );;
alpha := VectorSpaceMorphism( A, HomalgMatrix( [ [ 1, 0, 0 ] ], 1, 3, field ), C );;
beta := VectorSpaceMorphism( C, HomalgMatrix( [ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ] ], 3, 2, field ), B );;
IsZeroForMorphisms( PreCompose( alpha, beta ) );
#! false
IsCongruentForMorphisms(
IdentityMorphism( HomologyObject( alpha, beta ) ),
HomologyObjectFunctorial( alpha, beta, IdentityMorphism( C ), alpha, beta )
);
#! true
kernel_beta := KernelEmbedding( beta );;
K := Source( kernel_beta );;
IsIsomorphism(
HomologyObjectFunctorial(
MorphismFromZeroObject( K ),
MorphismIntoZeroObject( K ),
kernel_beta,
MorphismFromZeroObject( Source( beta ) ),
beta
)
);
#! true
cokernel_alpha := CokernelProjection( alpha );;
Co := Range( cokernel_alpha );;
IsIsomorphism(
HomologyObjectFunctorial(
alpha,
MorphismIntoZeroObject( Range( alpha ) ),
cokernel_alpha,
MorphismFromZeroObject( Co ),
MorphismIntoZeroObject( Co )
)
);
#! true
op := Opposite( vec );;
alpha_op := Opposite( op, alpha );;
beta_op := Opposite( op, beta );;
IsCongruentForMorphisms(
IdentityMorphism( HomologyObject( beta_op, alpha_op ) ),
HomologyObjectFunctorial( beta_op, alpha_op, IdentityMorphism( Opposite( C ) ), beta_op, alpha_op )
);
#! true
kernel_beta := KernelEmbedding( beta_op );;
K := Source( kernel_beta );;
IsIsomorphism(
HomologyObjectFunctorial(
MorphismFromZeroObject( K ),
MorphismIntoZeroObject( K ),
kernel_beta,
MorphismFromZeroObject( Source( beta_op ) ),
beta_op
)
);
#! true
cokernel_alpha := CokernelProjection( alpha_op );;
Co := Range( cokernel_alpha );;
IsIsomorphism(
HomologyObjectFunctorial(
alpha_op,
MorphismIntoZeroObject( Range( alpha_op ) ),
cokernel_alpha,
MorphismFromZeroObject( Co ),
MorphismIntoZeroObject( Co )
)
);
#! true
#! @EndExample
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