Super rewriting theory and nondegeneracy of odd categorified ${\mathfrak{sl}}_2$

Abstract

We develop the rewriting theory for monoidal supercategories and $2$-supercategories. This extends the theory of higher-dimensional rewriting established for (linear) 2-categories to the super setting, providing a suite of tools for constructing bases and normal forms for $2$-supercategories given by generators and relations. We then employ this newly developed theory to prove the non-degeneracy conjecture for the odd categorification of quantum $\mathfrak{sl}(2)$ from A. Ellis and A. Lauda [Quantum Topol. 7 (2016), 329–433] and J. Brundan and A. Ellis [Proc. Lond. Math. Soc. (3) 115 (2017), 925–973] As a corollary, this gives a classification of dg-structures on the odd $2$-category conjectured by A. Lauda and I. Egilmez [Quantum Topol. 11 (2020), 227–294].