Algebraic K$_0$ for unpointed Categories

Abstract

We construct a natural generalization of the Grothendieck group $\mathrm{K}_0$ to the case of possibly unpointed categories admitting pushouts by using the concept of heaps recently introduced by Brezinzki. In case of a monoidal category, the defined $\mathrm{K}_0$ is shown to be a truss. It is shown that the construction generalizes the classical $\mathrm{K}_0$ of an abelian category as the group retract along the isomorphism class of the zero object. We finish by applying this construction to construct the integers with addition and multiplication as the decategorification of finite sets and show that in this $\mathrm{K}_0\left(\underline{\mathrm{Top}}\right)$ one can identify a CW-complex with the iterated product of its cells.