# Categorifying reduced rings

## TL;DR

The paper constructs new monoidal stable $mbda$-categories with specified $K_0$ groups for various rings, addressing a problem posed by Khovanov.

## Contribution

It provides functorial constructions of monoidal stable $mbda$-categories with prescribed $K_0$ groups for reduced rings, solving Khovanov's problem.

## Key findings

- Constructed a rigid symmetric monoidal stable $mbda$-category with $K_0$ equal to a given domain of characteristic zero.
- Built a rigid braided monoidal stable $mbda$-category with $K_0$ matching any reduced commutative ring.
- Addressed and solved a problem posed by Khovanov regarding categorification of rings.

## Abstract

Given a domain of characteristic zero $R$, we functorially construct a rigid symmetric monoidal stable $\infty$-category whose $K_0$ is $R$, solving a problem of Khovanov. We also functorially construct for any reduced commutative ring $R$ a rigid braided monoidal stable $\infty$-category whose $K_0$ is $R$.