  We provide a necessary and sufficient condition for a simple object in a
pivotal k-category to be ambidextrous. In turn, these objects imply the
existence of nontrivial trace functions in the category. These functions play
an important role in low-dimensional topology as well as in studying the
category itself. In particular, we prove they exist for factorizable ribbon
Hopf algebras, modular representations of finite groups and their quantum
doubles, complex and modular Lie (super)algebras, the $(1,p)$ minimal model in
conformal field theory, and quantum groups at a root of unity.
