# nLab Grothendieck ring

Any abelian category $C$ gives rise to an abelian group $K\left(C\right)$ called its Grothendieck group. If we apply this construction to a monoidal abelian category, $K\left(C\right)$ is a ring, called the Grothendieck ring.

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

If $C$ is a symmetric monoidal category, $K\left(C\right)$ becomes a $\Lambda$-ring — even better.

If $C$ is just braided monoidal, is $K\left(C\right)$ just a commutative ring?

Revised on July 30, 2009 18:56:12 by Toby Bartels (71.104.230.172)