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1 Basic operations
 1.1 Abelian constructions

1 Basic operations

1.1 Abelian constructions

1.1-1 SomeProjectiveObjectForKernelObject
‣ SomeProjectiveObjectForKernelObject( alpha )( attribute )

Returns: an object

The argument is a morphism \(\alpha\). The output is the source of EpimorphismFromSomeProjectiveObjectForKernelObject applied to \(\alpha\).

1.1-2 EpimorphismFromSomeProjectiveObjectForKernelObject
‣ EpimorphismFromSomeProjectiveObjectForKernelObject( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}(P,\mathrm{KernelObject}( \alpha ))\)

The argument is a morphism \(\alpha\). The output is an epimorphism \(\pi: P \rightarrow \mathrm{KernelObject}( \alpha )\) with \(P\) a projective object.

1.1-3 EpimorphismFromSomeProjectiveObjectForKernelObjectWithGivenSomeProjectiveObjectForKernelObject
‣ EpimorphismFromSomeProjectiveObjectForKernelObjectWithGivenSomeProjectiveObjectForKernelObject( alpha )( operation )

Returns: a morphism in \(\mathrm{Hom}(P,\mathrm{KernelObject}( \alpha ))\)

The arguments are a morphism \(\alpha\) and an object \(P = \mathrm{SomeProjectiveObjectForKernelObject}( \alpha )\). The output is an epimorphism \(\pi: P \rightarrow \mathrm{KernelObject}( \alpha )\).

1.1-4 SomeInjectiveObjectForCokernelObject
‣ SomeInjectiveObjectForCokernelObject( alpha )( attribute )

Returns: an object

The argument is a morphism \(\alpha\). The output is the range of MonomorphismToSomeInjectiveObjectForCokernelObject applied to \(\alpha\).

1.1-5 MonomorphismToSomeInjectiveObjectForCokernelObject
‣ MonomorphismToSomeInjectiveObjectForCokernelObject( alpha )( attribute )

Returns: a morphism in \(\mathrm{Hom}(\mathrm{CokernelObject}( \alpha ), I)\)

The argument is a morphism \(\alpha\). The output is a monomorphism \(\iota: \mathrm{CokernelObject}( \alpha ) \rightarrow I\) with \(I\) an injective object.

1.1-6 MonomorphismToSomeInjectiveObjectForCokernelObjectWithGivenSomeInjectiveObjectForCokernelObject
‣ MonomorphismToSomeInjectiveObjectForCokernelObjectWithGivenSomeInjectiveObjectForCokernelObject( alpha )( operation )

Returns: a morphism in \(\mathrm{Hom}(\mathrm{CokernelObject}( \alpha ), I)\)

The arguments are a morphism \(\alpha\) and an object \(I = \mathrm{SomeInjectiveObjectForCokernelObject}( \alpha )\). The output is a monomorphism \(\iota: \mathrm{CokernelObject}( \alpha ) \rightarrow I\).

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