  By the Lefschetz fixed point theorem, if an endomorphism of a topological
space is fixed-point-free, then its Lefschetz number vanishes. This necessary
condition is not usually sufficient, however; for that we need a refinement of
the Lefschetz number called the Reidemeister trace. Abstractly, the Lefschetz
number is a trace in a symmetric monoidal category, while the Reidemeister
trace is a trace in a bicategory; in this paper we relate these contexts using
indexed symmetric monoidal categories.
  In particular, we will show that for any symmetric monoidal category with an
associated indexed symmetric monoidal category, there is an associated
bicategory which produces refinements of trace analogous to the Reidemeister
trace. This bicategory also produces a new notion of trace for parametrized
spaces with dualizable fibers, which refines the obvious "fiberwise" traces by
incorporating the action of the fundamental group of the base space. We also
advance the basic theory of indexed monoidal categories, including introducing
a string diagram calculus which makes calculations much more tractable. This
abstract framework lays the foundation for generalizations of these ideas to
other contexts.
