A Quillen model for classical Morita theory and a tensor categorification of the Brauer group
type: Item
instance of: scholarly article
title: A Quillen model for classical Morita theory and a tensor categorification of the Brauer group
zbMATH Open document ID: 1297.55021
author: Ivo Dell'Ambrogio
author: Gonçalo Tabuada
published in: Journal of Pure and Applied Algebra
publication date: 2014-07-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1211.2309
review text: Let \(\mathbb{K}\) be a commutative ring and \(Cat_{\mathbb{K}}\) be the category of small \(\mathbb{K}\)-categories. In this paper, the authors show that the category \(Cat_{\mathbb{K}}\) carries a combinatorial symmetric monoidal Quillen model structure. In particular, two \(\mathbb{K}\)-algebras become isomorphic in the associated homotopy category \(Ho(Cat_{\mathbb{K}})\) if and only if they are Morita equivalent, see [\textit{K. Morita}, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 6, 83--142 (1958; Zbl 0080.25702)]. As an application, the authors prove that the symmetric monoidal category \(Ho(Cat_{\mathbb{K}})^{\otimes}\) of \(\otimes\)-inversible objects in the homotopy category \(Ho(Cat_{\mathbb{K}})\) is a \(\otimes\)-categorification of the Brauer group \(Br(\mathbb{K})\). Finally, the authors prove that the contravariant corestriction map for finite Galois extensions also lifts to this categorification.
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16K50
Mathematics Subject Classification ID: 46M99
zbMATH DE Number: 6315949
zbMATH Keywords: Quillen model
zbMATH Keywords: Morita theory
zbMATH Keywords: Brauer group
zbMATH Keywords: Galois extension
reviewed by: Saïd Zarati
MaRDI profile type: Publication
OpenAlex ID: W2065583337
arXiv ID: 1211.2309
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DOI: 10.1016/J.JPAA.2014.04.004
description: scientific article