For a fibration (with a connected base space) the Euler characteristic of the total space is the product of the Euler characteristics of the base and the fiber.  The Euler characteristic is also additive on subcomplexes.  The generalizations of the Euler characteristic to fixed point invariants, primarily the Lefschetz number and Reidemeister trace, are similarly additive and multiplicative.   Classically these results were proven using a variety of techniques.

Recently, Mike Shulman and I have shown that all of these results are consequences of a simple formal observation and some specific topological input.  We think of the Euler characteristic as an endomorphism rather an integer.  With this change in perspective, the product of integers becomes a composite of functions and the topological results follow from a more general theorem about composites of traces.