group of units functor
- notation:
- Source: category of monoids
- Target: category of groups
- Left adjoint functor:
- nLab Link
This functor maps a monoid to its group of units , consisting of pairs satisfying . Equivalently, it takes the submonoid of invertible elements of , equipped with the inverse operation.
Satisfied Properties
Assigned properties
- is a coreflector
- is finitary
- preserves coproducts
Deduced properties
- is a right adjoint
- is right-invertible
- preserves finite coproducts
- is continuous
- is essentially surjective
- preserves binary coproducts
- preserves initial objects
- is cofinitary
- is left exact
- preserves products
- is dominant
- preserves terminal objects
- preserves finite products
- preserves equalizers
- preserves monomorphisms
- preserves binary products
- preserves coreflexive equalizers
- preserves regular monomorphisms
Unsatisfied Properties
Assigned properties
- is not essentially injective
- is not faithful
- is not full
- does not preserve regular epimorphisms
Deduced properties*
- is not regular
- is not fully faithful
- is not conservative
- is not left-invertible
- is not full on isomorphisms
- is not pseudomonic
- is not monadic
- does not preserve coequalizers
- does not preserve epimorphisms
- does not preserve reflexive coequalizers
- is not comonadic
- is not an equivalence
- is not right exact
- is not an isomorphism
- is not exact
- is not cocontinuous
- is not coregular
- is not a left adjoint
- is not a reflector
- is not representable
*This also uses the deduced satisfied properties.
Unknown properties
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