My main goal is to understand the Grothendieck group construction and its generalization to non-commutative setting. I understand its explicit construction via an equivalence relation, but I want to fully understand as an adjoint functor. Here is what I know about this so far: There is a forgetful functor $U: \mathbf{Ab}\to\mathbf{CMon}$ from the category of abelian groups to the category of commutative monoids, which has a "free" left adjoint $Gr.$ If I understood correctly we can write a Kan extension formula for it as $\require{AMScd}$ \begin{CD} \mathbf{Ab} @>\text{Id}>> \mathbf{Ab}\\ @V U VV @|\\ \mathbf{CMon} @>>\text{Lan}_U(\text{Id})> \mathbf{Ab} \end{CD} By the pointwise formula $Gr(M)=\text{Lan}_U(\text{Id})(M)=\underset{\mathcal{D}}{colim}\,\text{Id}(G),$ where $\mathcal{D}$ is the diagram whose objects are $G\in\mathbf{Ab}$ with a commutative monoid homomorphism $U(G)\to M,$ and morphisms are commutative triangles. Until this point we did not use any structure of these categories. Now I wonder,
- how to translate this into the equivalence relation that we know? and
- what changes as we go to non-abelian groups?
Thank you in advance for your help.