  We study monoidal categories that enjoy a certain weakening of the rigidity
property, namely, the existence of a dualizing object in the sense of
Grothendieck and Verdier. We call them Grothendieck-Verdier categories. Notable
examples include the derived category of constructible sheaves on a scheme
(with respect to tensor product) as well as the derived and equivariant derived
categories of constructible sheaves on an algebraic group (with respect to
convolution).
  We show that the notions of pivotal category and ribbon category, which are
well known in the setting of rigid monoidal categories, as well as certain
standard results associated with these notions, have natural analogues in the
world of Grothendieck-Verdier categories.
