Azumaya Objects in Triangulated Bicategories

Abstract: 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.

• The paper presents a comprehensive characterization of Azumaya objects using duality and separability within triangulated bicategories.
• The paper extends the classical Brauer group concept to a homotopical setting, establishing an isomorphism with the Brauer group of underlying commutative rings.
• The paper applies its theoretical framework to derived categories and structured ring spectra, effectively bridging algebraic topology and homotopy theory.

Azumaya Objects in Triangulated Bicategories: An Expert Overview

The paper "Azumaya Objects in Triangulated Bicategories" by Niles Johnson presents a mathematical exploration into the structure and characterization of Azumaya objects within homotopy-theoretic settings, specifically within triangulated bicategories. This research serves to provide a more generalized understanding of classical Brauer groups and Azumaya algebras by extending these concepts into the field of homotopical algebra, including applications to derived categories of rings and ring spectra.

Core Contributions

The paper introduces the concept of Azumaya objects in the context of triangulated bicategories, which generalizes the traditional notion of Azumaya algebras over commutative rings. This generalization encompasses broader algebraic structures that appear in various fields such as algebraic topology, category theory, and homotopy theory. The main contributions of the paper are as follows:

1. Characterization of Azumaya Objects: The paper offers an in-depth characterization theorem for Azumaya objects, utilizing concepts such as duality and invertibility within the framework of Eilenberg-Watts bicategories. The equivalence of various conditions is established to identify Azumaya objects, making use of central and separable properties, which serve as extensions from classical algebra to these more general settings.
2. Homotopical Brauer Groups: By exploring the homotopical aspects, the research extends the classical Brauer group concept to a homotopical setting, defining what is referred to as the "homotopical Brauer group." The paper outlines how these generalized groups apply to Eilenberg-Mac~Lane spectra and articulates the isomorphism between the homotopical Brauer group of an Eilenberg-Mac~Lane spectrum and the Brauer group of its underlying commutative ring.
3. Applications to Derived Categories and Spectra: The framework presented enables a direct application to derived categories of rings and structured ring spectra, demonstrating how the theoretical foundations laid by the characterization of Azumaya objects can be applied to compute Brauer groups in these contexts, effectively uniting various mathematical disciplines under a cohesive theory.
4. Relation to Existing Theories: The paper situates its contributions within the corpus of existing work by aligning the newly defined concepts with prior results in the field of derived Morita theory and stable homotopy theory. This serves to validate the framework and bridge its new abstractions to established mathematical theories.

Implications and Future Directions

The implications of this research are significant for both theoretical advancement and practical computation within algebra and topology. By generalizing classical concepts to encompass more complex structures such as bicategories, the research opens new avenues for investigation into intersections of category theory, algebraic geometry, and stable homotopy theory. The characterization and classification tools developed provide a basis for deeper exploration of algebraic structures encountered in modern mathematical frameworks.

Future directions of this research could explore the nuances and extensions of Azumaya objects beyond triangulated settings into more intricate categorical structures, potentially revealing new algebraic invariants and equivalences. Additionally, the computation of Brauer groups in varied homotopical contexts might unveil further applications in fields such as algebraic K-theory, representation theory, and beyond.

In conclusion, "Azumaya Objects in Triangulated Bicategories" presents a comprehensive examination of abstract algebraic structures, extending classical concepts to new mathematical settings and providing robust tools and frameworks applicable in diverse branches of mathematics. The paper's contributions significantly enhance our understanding of the interplay between algebraic and topological structures, offering an invaluable resource for further research in the field.