Euler measure as generalized cardinality

Schanuel has pointed out that there are mathematically interesting categories whose relationship to the ring of integers is analogous to the relationship between the category of finite sets and the semi-ring of non-negative integers. Such categories are inherently geometrical or topological, in that the mapping to the ring of integers is a variant of Euler characteristic. In these notes, I sketch some ideas that might be used in further development of a theory along lines suggested by Schanuel.