Monoidal Supercategories

Abstract

This work is a companion to our article “Super Kac–Moody 2-categories,” which introduces super analogs of the Kac–Moody 2-categories of Khovanov–Lauda and Rouquier. In the case of ${\mathfrak{sl}_2}$, the super Kac–Moody 2-category was constructed already in [A. Ellis and A. Lauda, “An odd categorification of ${U_q(\mathfrak{sl}_2)}$”], but we found that the formalism adopted there became too cumbersome in the general case. Instead, it is better to work with 2-supercategories (roughly, 2-categories enriched in vector superspaces). Then the Ellis–Lauda 2-category, which we call here a ${\Pi}$-2-category (roughly, a 2-category equipped with a distinguished involution in its Drinfeld center), can be recovered by taking the superadditive envelope then passing to the underlying 2-category. The main goal of this article is to develop this language and the related formal constructions in the hope that these foundations may prove useful in other contexts.