  Groupoids graded by the groupoid of bijections between finite sets admit
generating functions which encode the groupoid cardinalities of their graded
components. As suggested in the work of Baez and Dolan, we use analytic
continuation of such generating functions to define a complex-valued
cardinality for groupoids whose usual groupoid cardinality diverges. The
complex nature of such a cardinality invariant is shown to reflect a recursion
of structure which we refer to as `nested equivalence'.
