I am working on the construction of a group free over a monoid. Maybe you know where I can find something about the (right) adjointness of forgetful functor $U:\mathbf{Grp}\rightarrow\mathbf{Mon}$. I called it forgetful functor here, but it is clear to me that you could also look at it as an inclusion functor.
It is (right) adjoint isn't it? If not then I am working in vain. I do not exclude the possibility that this question is a duplicate.
Best Answer
Adjoint is ambiguous. Left or right adjoint? In the case of $U : \mathsf{Grp} \to \mathsf{Mon}$, we have a left and a right adjoint (this is quite rare for forgetful functors). The right adjoint maps a monoid to its group of units. The left adjoint "adjoins" inverses to a monoid in order to obtain a group. It is (at least in the commutative case) known as the Grothendieck construction. If $M$ is given as a monoid by generators $X$ and relations $R$, then the Grothendieck group $G(M)$ is given by the same generators $X$ and relations $R$, but now as a group. For example, $\mathbb{N}$ is the free monoid on one generator, hence $G(\mathbb{N})$ is the free group on one generator, i.e. $G(\mathbb{N})=\mathbb{Z}$.