Category Theory 2008,
Abstract In the last couple of years it has become apparent that there is a sensible definition of the "Euler characteristic" or "cardinality" of a category. This notion extends easily to enriched categories, and in particular to metric spaces.
I will explain what the cardinality of a metric space is, and describe some relations with convex geometry and geometric measure theory. Conjecturally, the most important invariants of geometric measure theory (such as volume, Euler characteristic and Hausdorff dimension) can all be derived from cardinality.
Slides In this pdf file (420KB).
An earlier and very productive discussion of these ideas can be found at the n-Category Café.