TITLE: Non-abelian Grothendieck group
QUESTION [21 upvotes]: By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations $\underline{1}=1$, $\underline{x  \cdot y}=\underline{x}  \cdot \underline{y}$. Notice that the elements of this group have the form $\underline{x_1} \cdot \underline{x_2}^{-1}  \cdot \underline{x_3}  \cdot \underline{x_4}^{-1}  \cdot \dotsc$.
Question 1: Does this group have a name? It is analogous to the Grothendieck group, which is the left adjoint of the forgetful functor from abelian groups to commutative monoids, so perhaps we may call it the non-abelian Grothendieck group?
Question 2: Is there any criterion when an element of this group vanishes?

REPLY [9 votes]: I'd just like to point out that in the many-object case this has been called the fundamental groupoid of a category by Paré, which would suggest that it could be called the fundamental group of a monoid, a possibility strengthened by the fact that, as Benjamin Steinberg points out in the comments, it's the fundamental group of the classifying space of the monoid. Another possibility might be classifying group, but that's probably ill-advised since it's not clear (to me) what it classifies.