enveloping group functor

 algebra  free
Notation FMon,Grp:Mon→GrpF_{\Mon,\Grp}: \Mon \to \Grp Domain category of monoidsCodomain category of groupsRight adjoint forgetful functor from groups to monoids Related FGrpF_{\Grp} External nLab link
This functor maps a monoid MM to the group F(M)F(M) that is equipped with a universal homomorphism iM:M→F(M)i_M : M \to F(M). It is called the (universal) enveloping group or the group completion of MM; in the commutative case, it is known as the Grothendieck group of MM. As a possible construction of F(M)F(M), take the free group on generators m‾\underline{m} for m∈Mm \in M subject to the relations 1‾=1\underline{1} = 1 and m⋅n‾=m‾⋅n‾\underline{m \cdot n} = \underline{m} \cdot \underline{n}.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Indistinguishable functors

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