forgetful functor for groups

notation: $U_{\Grp}$ : $\Grp$ → $\Set$
Source: category of groups
Target: category of sets
Left adjoint: $F_{\Grp}$
Related functors: $U_{\Ring}$ , $U_{\Vect}$ , $U_{\Ab,\Grp}$
nLab Link

This functor maps a group $G$ to its underlying set $U_{\Grp}(G)$ .

Satisfied Properties

Assigned properties

is representable
is monadic
is finitary
preserves epimorphisms
preserves reflexive coequalizers

Deduced properties

is continuous
is conservative
is faithful
is a right adjoint
is cofinitary
is left exact
preserves products
preserves finite products
preserves equalizers
preserves monomorphisms
preserves terminal objects
preserves coreflexive equalizers

Unsatisfied Properties

Assigned properties

is not essentially injective
is not essentially surjective
does not preserve initial objects
does not preserve coequalizers

Deduced properties*

is not an equivalence
is not full
is not left-invertible
does not preserve finite coproducts
is not right exact
is not exact
is not cocontinuous
does not preserve coproducts
is not a left adjoint
is not comonadic

*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 forgetful functor for groups. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.

forgetful functor for vector spaces