Monoidal supercategories

TL;DR

This paper clarifies various definitions of super monoidal categories, explores the odd Temperley-Lieb supercategory, and sets the stage for future work on super Kac-Moody 2-categories.

Contribution

It systematically compares super monoidal category notions and develops a formalism for super analogs of important algebraic structures.

Findings

Clarified the relationships between different super monoidal category definitions.

Detailed the structure of the odd Temperley-Lieb supercategory.

Laid groundwork for super Kac-Moody 2-categories.

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.