# Why the Grothendieck group/ring?

[alg] [cat.theory]
Given an additive category $\mathcal{C}$ we can form the Grothendieck group $\mathcal{K}(\mathcal{C})$. If further $\mathcal{C}$ is a tensor category then $\mathcal{K}(\mathcal{C})$ is also a ring, the Grothendieck ring.

If $\mathcal{C}$ is semisimple then any equality in $\mathcal{K}(\mathcal{C})$ corresponds to an isomorphism in $\mathcal{C}$.

e.g. Let $S, U : \mathbf{Grp} \rightarrow \mathbf{Set}$ where $S$ sends $G \stackrel{f}{\longrightarrow} H$ to $G^2 \stackrel{f^2}{\longrightarrow} H^2$ (the “squaring” functor), and $U$ is the forgetful functor. The set maps $\tau_G : G^2 \rightarrow G$ defined by the multiplication of $G$ are components of a natural transformation from $S$ to $U$.