nLab Grothendieck ring

Idea

Any abelian category $C$ gives rise to an abelian group $K(C)$ called its Grothendieck group (see there for more). If we apply this construction to a monoidal abelian category or generally to a rig category, $K(C)$ is a ring, called the Grothendieck ring.

If $C$ is a braided monoidal category, $K(C)$ becomes a commutative ring.

If $C$ is a symmetric monoidal category, $K(C)$ becomes a $\Lambda$-ring.

Examples

• The Grothendieck ring of the monoidal category of finite G-sets is called the Burnside ring of $G$.

References

• Peter May, Picard groups, Grothendieck rings,and Burnside rings of categories, Advances in Mathematics 163, 1–16 (2001), (pdf).