1,477 research outputs found
Higher Auslander algebras of type and the higher Waldhausen -constructions
These notes are an expanded version of my talk at the ICRA 2018 in Prague,
Czech Republic; they are based on joint work with Tobias Dyckerhoff and Tashi
Walde. In them we relate Iyama's higher Auslander algebras of type
to Eilenberg--Mac Lane spaces in algebraic topology and to higher-dimensional
versions of the Waldhausen -construction from algebraic
-theory.Comment: 16 pages. The author's contribution to the Proceedings of the ICRA
2018, v.2 minor edits following referee repor
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.Comment: 23 pages; final version; to appear in Journal of Homotopy and Related
Structure
On realizing diagrams of Pi-algebras
Given a diagram of Pi-algebras (graded groups equipped with an action of the
primary homotopy operations), we ask whether it can be realized as the homotopy
groups of a diagram of spaces. The answer given here is in the form of an
obstruction theory, of somewhat wider application, formulated in terms of
generalized Pi-algebras. This extends a program begun in [J. Pure Appl. Alg.
103 (1995) 167-188] and [Topology 43 (2004) 857-892] to study the realization
of a single Pi-algebra. In particular, we explicitly analyze the simple case of
a single map, and provide a detailed example, illustrating the connections to
higher homotopy operations.Comment: This is the version published by Algebraic & Geometric Topology on 21
June 200
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