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, then $K(C)$ inherits the further structure of a ring, thus 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.

Definition

Let $C$ be an abelian category. The Grothendieck group $K(C)$ is the abelian group generated by objects of $C$, quotiented by the relation that $[A]+[B]=[C]$ if there is a short exact sequence

$$0\to A\to C\to B\to 0.$$

In the presence of a monoidal structure $\otimes$, we can turn $C$ into a ring by defining

$$[A]\cdot [B]=[A\otimes B].$$

This endows $K(C)$ with the structure of a ring because

$$A\otimes(B\oplus C)\cong (A\otimes B)\oplus (A\otimes C),$$

and every monoidal category has a tensor unit.

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).