Journal of Homotopy and Related Structures

, Volume 9, Issue 2, pp 465–493 | Cite as

Azumaya objects in triangulated bicategories



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 



This work developed from a portion of the author’s Ph.D. thesis. The author thanks his advisor, Peter May, for reading it many times in draft form and offering a number of suggestions both mathematical and stylistic. The author also thanks Michael Ching for his significant encouragement. The exposition has benefited from the helpful suggestions of anonymous referees, and from the audiences who have listened to talks and read drafts of this article. The terminology “Eilenberg–Watts equivalence” was suggested by Justin Noel during one of many helpful conversations.


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Copyright information

© Tbilisi Centre for Mathematical Sciences 2013

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State University NewarkNewarkUSA

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