Monoidal supercategories
Jonathan Brundan, Alexander P. Ellis
TL;DR
This work develops a cohesive framework for Z/2-graded (super) monoidal and higher categories, clarifying competing notions by introducing supercategories, $\Pi$-structures, and their envelopes. It proves key equivalences between $\Pi$-supercategories and $\Pi$-categories, and extends these ideas to $2$-categories with $\Pi$- and grading data, including $Q$-gradings, via $\Pi$-envelopes and $(Q,\Pi)$-envelopes. The paper provides universal properties, adjunctions, and diagrammatic tools that enable passing between ordinary, super, and $\Pi$-enhanced worlds, and it develops graded versions that integrate $\mathbb{Z}$-gradings with parity. As illustrative instances, it analyzes the odd Brauer/Wed categories and odd Temperley-Lieb, connecting to representation theory and potential super analogs of Kac–Moody 2-categories. The resulting framework lays a solid foundation for constructing and studying super analogs of categorified quantum groups and related 2-categorical structures.
Abstract
In the literature, one finds several competing notions for the super (i.e., Z/2-graded) analog of a monoidal category. The goal of this paper is to clarify these definitions and the connections between them. We also discuss in detail the example of the odd Temperley-Lieb supercategory. In a forthcoming article, we will exploit the formalism developed here in order to define super analogs of the Kac-Moody 2-categories of Khovanov-Lauda and Rouquier.