Symmetric and exterior powers of categories

Supermathematics is based on the symmetric
monoidal category of Z- (or Z/2-) graded vector spaces with
the Koszul sign rule. If we isolate its "sign skeleton"
(the minimal subcategory necessary to formulate the rule), we
get a Picard category P with the set of isomorphism classes of
objects being  Z (or Z/2) and the group of automorphisms
of any object being {\pm 1}, i.e., again Z/2.

By Grothendieck, Picard categories correspond to spectra with
only two homotopy groups (in degrees 0, 1), and P, being
a free Picard category on one object, corresponds to the
[0,1]-truncation of the spherical spectrum S, whose homotopy groups
(=stable homotopy groups of spheres) are:

Z, Z/2, Z/2, Z/24, ...

This suggests that  Picard n-categories obtained by less drastic
truncations of S, should also serve as skeletons for some
"higher supersymmetry". The talk will explain a first step in  this
direction for n=2: the construction of exterior powers of
categories. In particular, we construct the
categorical analog of the Koszul complex.
Joint work with N. Ganter.