abelianization functor for groups

notation: $(-)^{\ab}$ : $\Grp$ → $\Ab$
Source: category of groups
Target: category of abelian groups
Right adjoint: $U_{\Ab,\Grp}$
nLab Link

This functor maps a group $G$ to its abelianization $G^{\ab} \coloneqq G/[G,G]$ .

Satisfied Properties

Assigned properties

is a left adjoint
is essentially surjective
preserves finite products

Deduced properties

preserves terminal objects
is cocontinuous
is finitary
preserves coproducts
is right exact
preserves finite coproducts
preserves coequalizers
preserves epimorphisms
preserves initial objects
preserves reflexive coequalizers

Unsatisfied Properties

Assigned properties

is not essentially injective
is not faithful
is not full
does not preserve monomorphisms
does not preserve products
is not representable

Deduced properties*

is not an equivalence
is not continuous
is not cofinitary
is not left exact
does not preserve equalizers
is not left-invertible
is not monadic
is not conservative
is not comonadic
is not a right adjoint
is not exact
does not preserve coreflexive equalizers

*This also uses the deduced satisfied properties.

Unknown properties

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Undistinguishable functors

These functors in the database currently have exactly the same properties as the abelianization functor for groups. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.

enveloping group functor