The Euler Characteristic of Finite Categories

Abstract

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

In this version we used finite dimensional matrices instead of linear operators. To appear in Homology, Homotopy and Applications