A categorification of the square root of -1

TL;DR

This paper introduces a graphical calculus for a monoidal DG category that categorifies the ring of Gaussian integers, providing a new framework to understand the algebraic structure of √-1.

Contribution

It constructs a monoidal DG category whose Grothendieck group is isomorphic to , and develops a categorical action lifting the algebraic action on .

Findings

Categorical framework for  established

Graphical calculus for the monoidal DG category developed

Categorical action of the category on  constructed

Abstract

We give a graphical calculus for a monoidal DG category $\cal{I}$ whose Grothendieck group is isomorphic to the ring $\mathbb{Z}[\sqrt{-1}]$. We construct a categorical action of $\cal{I}$ which lifts the action of $\mathbb{Z}[\sqrt{-1}]$ on $\mathbb{Z}^2$.