I ran across Picard categories in a totally different area of
mathematics, but maybe it helps. In short, a Picard category is
a group object in the category of groupoids. Picard categories
come up when you study Picard stacks. Roughly, a Picard stack is
a sheaf of Picard categories. The classical example is taking a
two-term (perfect??) complex of sheaves, and asssociating to
such a complex the Groupoid quotient of the one term by the
other. This is important when you want to produce a geomteric
object from such a complex. This is an important tool in
defining virtual fundamental classes as in
http://arxiv.org/abs/alg-geom/9601010. Before I tell you to many
things that are not true, here are the references I know of:
Lecture notes of Martin Olsson at
http://math.berkeley.edu/~molsson/MSRISummer07.pdf, Lecture 5.
You can even watch that on video:
http://www.msri.org/calendar/sgw/WorkshopInfo/419/show_sgw The
definitive reference is Expos´e XVIII of SGA 4. You used to be
able to get that at the Grothendieck cir