New perspectives on Euler characteristic

Euler characteristic is typically thought of as an alternating sum of ranks of homology groups, but it is often better to regard it as a generalization of cardinality or measure, as suggested by the formula $\chi(X \cup Y) = \chi(X) + \chi(Y) - \chi(X \cap Y).$ This point of view leads to useful notions of Euler characteristic for structures such as groupoids, posets, categories, and classes of space not amenable to the usual methods of algebraic topology, such as certain spaces important in complex dynamics. Some of these theories of Euler characteristic are well-established, while others are currently being developed. I will give an overview.