‣ SomeProjectiveObjectForKernelObject( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha\). The output is the source of EpimorphismFromSomeProjectiveObjectForKernelObject applied to \(\alpha\).
‣ 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.
‣ 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 \(\pi: P \rightarrow \mathrm{KernelObject}( \alpha )\).
‣ SomeInjectiveObjectForCokernelObject( alpha ) | ( attribute ) |
Returns: an object
The argument is a morphism \(\alpha\). The output is the range of MonomorphismToSomeInjectiveObjectForCokernelObject applied to \(\alpha\).
‣ 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.
‣ 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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