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.

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