nLab super 2-algebra

Idea

The notion of symmetric monoidal tensor category may be thought of as a categorification of the notion of commutative ring, a 2-ring or 2-algebra.

Now the notion of super 2-algebra is accordingly supposed to be the notion categorifying super algebra.

Related concepts

• spectral super-scheme

References

The definition of super 2-algebra starts at 33:10 in:

• Mikhail Kapranov, Categorification of supersymmetry and stable homotopy groups of spheres, talk at Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky (1953-2013) (April 2013) [pdf, video:YT]

  Abstract:. The “minimal sign skeleton” necessary to formulate the Koszul sign rule is a certain Picard category, a symmetric monoidal category with all objects and morphisms invertible. It can be seen as the free Picard category generated by one object and corresponds, by Grothendieck‘s dictionary, to the truncation of the spherical spectrum $S$ in degrees 0 and 1, so that $\{\pm 1\}$ appears as the first stable homotopy group of spheres $\pi_{n+1}(S^n)$. This suggest a “higher” or categorified versions of super-mathematics which utilize deeper structure of $S$. The first concept on this path is that of a supersymmetric monoidal category which is categorified version of the concept of a supercommutative algebra.

(while the first 30 minutes are concerned with discussion of how super-grading is implied by grading (augmented infinity-groups) over the sphere spectrum, see at superalgebra - Abstract idea).

The example of the super 2-algebra of superalgebras starts at 39:30,

the example of the exterior 2-algebra of a super-linear category starts at 47:30.