Super rewriting theory and nondegeneracy of odd categorified sl(2)

TL;DR

This paper extends rewriting theory to supercategories, enabling the construction of bases and normal forms, and applies it to prove the non-degeneracy of the odd categorification of quantum sl(2), leading to classification results.

Contribution

It develops the rewriting theory for monoidal supercategories and applies it to prove a key conjecture in odd categorification of quantum sl(2).

Findings

Proved the non-degeneracy conjecture for odd categorified sl(2).

Provided a classification of dg-structures on the odd 2-category.

Extended higher-dimensional rewriting theory to supercategories.

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 sl(2) from arXiv:1307.7816 and arXiv:1701.04133. As a corollary, this gives a classification of dg-structures on the odd 2-category conjectured in arXiv:1808.04924.