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}.


Group multiplication is a natural transformation between the appropriate functors. Similarly the defining operations of other algebraic structures are also nat. trans. between appropriate functors.

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.
(ref. Adamek-Herrlick-Strecker, Abstract and Concrete Categories, 6.2 (2).)