group of units functor

notation: $(-)^{\times}$ : $\Mon$ → $\Grp$
Source: category of monoids
Target: category of groups
Left adjoint: $U_{\Grp,\Mon}$
nLab Link

This functor maps a monoid $M$ to its group of units $M^{\times}$ , consisting of pairs $(a,b) \in M^2$ satisfying $ab=ba=1$ . Equivalently, it takes the submonoid of invertible elements of $M$ , equipped with the inverse operation.

Satisfied Properties

Assigned properties

is essentially surjective
is a right adjoint
is finitary
preserves coproducts

Deduced properties

is continuous
preserves finite coproducts
is cofinitary
is left exact
preserves products
preserves initial objects
preserves finite products
preserves equalizers
preserves monomorphisms
preserves terminal objects
preserves coreflexive equalizers

Unsatisfied Properties

Assigned properties

is not essentially injective
is not faithful
is not full
does not preserve epimorphisms
is not representable

Deduced properties*

is not an equivalence
is not conservative
is not left-invertible
is not monadic
is not right exact
is not comonadic
is not exact
is not cocontinuous
does not preserve coequalizers
is not a left adjoint
does not preserve reflexive coequalizers

*This also uses the deduced satisfied properties.

Unknown properties

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