A Quillen model for classical Morita theory and a tensor categorification of the Brauer group

TL;DR

This paper develops a symmetric monoidal Quillen model structure on small K-categories to enhance Morita theory and provides a tensor categorification of the Brauer group, elucidating its functorial properties.

Contribution

It introduces a novel Quillen model structure on small K-categories and applies it to categorify the Brauer group with functoriality.

Findings

Constructed a symmetric monoidal Quillen model structure on small K-categories.

Achieved a tensor categorification of the Brauer group.

Demonstrated functoriality of the categorified Brauer group.

Abstract

Let K be a commutative ring. In this article we construct a symmetric monoidal Quillen model structure on the category of small K-categories which enhances classical Morita theory. We then use it in order to obtain a natural tensor categorification of the Brauer group and of its functoriality.