  The Lefschetz fixed point theorem follows easily from the identification of
the Lefschetz number with the fixed point index. This identification is a
consequence of the functoriality of the trace in symmetric monoidal categories.
There are refinements of the Lefschetz number and the fixed point index that
give a converse to the Lefschetz fixed point theorem. An important part of this
theorem is the identification of these different invariants. We define a
generalization of the trace in symmetric monoidal categories to a trace in
bicategories with shadows. We show the invariants used in the converse of the
Lefschetz fixed point theorem are examples of this trace and that the
functoriality of the trace provides some of the necessary identifications. The
methods used here do not use simplicial techniques and so generalize readily to
other contexts.
