  Traces in symmetric monoidal categories are well-known and have many
applications; for instance, their functoriality directly implies the Lefschetz
fixed point theorem. However, for some applications, such as generalizations of
the Lefschetz theorem, one needs "noncommutative" traces, such as the
Hattori-Stallings trace for modules over noncommutative rings. In this paper we
study a generalization of the symmetric monoidal trace which applies to
noncommutative situations; its context is a bicategory equipped with an extra
structure called a "shadow." In particular, we prove its functoriality and
2-functoriality, which are essential to its applications in fixed-point theory.
Throughout we make use of an appropriate "cylindrical" type of string diagram,
which we justify formally in an appendix.
