Linear polygraphs applied to categorification

TL;DR

This paper applies polygraph techniques to categorification, providing methods to compute Karoubi envelopes and Grothendieck decategorifications, linking rewriting system properties to algebraic decompositions.

Contribution

It extends polygraph constructions to $n$-categories and introduces new methods for decategorification and analyzing algebraic decompositions.

Findings

Extended Karoubi envelope construction to $n$-polygraphs.

Developed Grothendieck decategorification for linear polygraphs.

Linked quasi-convergence of rewriting systems to decomposition uniqueness.

Abstract

We introduce two applications of polygraphs to categorification problems. We compute first, from a coherent presentation of an $n$-category, a coherent presentation of its Karoubi envelope. For this, we extend the construction of Karoubi envelope to $n$-polygraphs and linear $(n,n-1)$-polygraphs. The second problem treated in this paper is the construction of Grothendieck decategorifications for $(n,n-1)$-polygraphs. This construction yields a rewriting system presenting for example algebras categorified by a linear monoidal category. We finally link quasi-convergence of such rewriting systems to the uniqueness of direct sum decompositions for linear $(n-1,n-1)$-categories.