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}.

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