The Euler Characteristic of Finite Categories

We associate a rational number $\chi (\mathcal{A})$ to every category $\mathcal{A}$ whose object and morphism sets are finite. We show that the assignment $\chi$ is additive under disjoint union and it preserves products. Hence we consider $\chi$ as an Euler measure on a family of categories where this assignment also obeys a version of the inclusion–exclusion principle.