CatDat

• notation: $F_{\Mon,\Grp}: \Mon \to \Grp$
• Source: category of monoids
• Target: category of groups
• Right adjoint functor: $U_{\Grp,\Mon}$
• Related functors: $F_{\Grp}$
• nLab Link

This functor maps a monoid $M$ to the group $F(M)$ that is equipped with a universal homomorphism $i_M : M \to F(M)$. It is called the (universal) enveloping group or the group completion of $M$; in the commutative case, it is known as the Grothendieck group of $M$. As a possible construction of $F(M)$, take the free group on generators $\underline{m}$ for $m \in M$ subject to the relations $\underline{1} = 1$ and $\underline{m \cdot n} = \underline{m} \cdot \underline{n}$.