Let $S$ be an abelian semigroup. The Grothendieck group of $S$ is $K(S) = S\times S/\mathord{\sim}$ , where $\sim$ is the equivalence relation: $(s,t) \sim (u,v)$ if there exists $r \in S$ such that $s+v+r = t+u+r$ . This is indeed an abelian group with zero element $(s,s)$ (any $s \in S$ ), inverse $-(s,t) = (t,s)$ and addition given by $(s,t)+(u,v) = (s+u, t+v)$ . It is common to use the suggestive notation $t-s$ for $(t,s)$ .

The Grothendieck group construction is a functor from the category of abelian semigroups to the category of abelian groups. A morphism $f\colon S \to T$ induces a morphism $K(f)\colon K(S) \to K(T)$ which sends an element $(s^+,s^-) \in K(S)$ to $(f(s^+),f(s^-)) \in K(T)$ .

Let $C$ be a (essentially small) symmetric monoidal category. Its Grothendieck group is $K([C])$ , i.e. the Grothendieck group of the isomorphism classes of objects of $C$ .