Journal of Homotopy and Related Structures

, Volume 9, Issue 2, pp 465–493

Azumaya objects in triangulated bicategories


DOI: 10.1007/s40062-013-0035-6

Cite this article as:
Johnson, N. J. Homotopy Relat. Struct. (2014) 9: 465. doi:10.1007/s40062-013-0035-6


We introduce the notion of Azumaya object in general homotopy-theoretic settings. We give a self-contained account of Azumaya objects and Brauer groups in bicategorical contexts, generalizing the Brauer group of a commutative ring. We go on to describe triangulated bicategories and prove a characterization theorem for Azumaya objects therein. This theory applies to give a homotopical Brauer group for derived categories of rings and ring spectra. We show that the homotopical Brauer group of an Eilenberg–Mac Lane spectrum is isomorphic to the homotopical Brauer group of its underlying commutative ring. We also discuss tilting theory as an application of invertibility in triangulated bicategories.


Brauer group Ring spectrum Homotopical algebra 

Mathematics Subject Classification (2000)

55U99 18D35 16K50 14F22 

Copyright information

© Tbilisi Centre for Mathematical Sciences 2013

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State University NewarkNewarkUSA

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