Niles Johnson

A bicategory is a kind of (weak) 2-category, and classical Morita theory can be phrased in terms of a classical example of a bicategory. This is the bicategory whose 0-cells are rings, 1-cells are bimodules, and 2-cells are bimodule homomorphisms. The unit 1-cell over a ring is that ring considered as a bimodule over itself, and composition of 1-cells is defined by the tensor product.

A Morita equivalence of rings is simply an equivalence of 0-cells in this bicategory. That is, a 1-cell from one ring to another (a bimodule), with a 1-cell inverse, so that the composite (tensor product) of these two bimodules over one ring is isomorphic to the other, and vice-versa. Currently, I'm working to use this bicategorical perspective to give a conceptual unification of various extensions of Morita theory.

Introductory Slides

Napkin depiction of a category, an enriched category, and a bicategory. Click to enlarge (javascript).

Napkin depiction of a category, an enriched category, and a bicategory. Click to shrink (javascript).

These slides present classical Morita theory from the perspective of enriched categories. Their aim is to introduce both enriched and bicategorical concepts as they relate to Morita theory. The slides were presented at a talk during the 2008 Graduate Student Topology Conference.

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Bicategorical Perspective on Morita Theory

arXiv 0805.3673: [abs | pdf]

Abstract: We present a bicategorical perspective on derived Morita theory for rings, DG algebras, and spectra. This perspective draws a connection between Morita theory and the bicategorical Yoneda Lemma, yielding a conceptual unification of Morita theory in derived and bicategorical contexts. This is motivated by study of Rickard's theorem for derived equivalences of rings and of Morita theory for ring spectra, which we present in Sections 2 and 4. Along the way, we gain an understanding of the barriers to Morita theory for DG algebras and give a conceptual explanation for the counterexample of Dugger and Shipley.

Invertibility in Bicategories

Abstract: These slides are slightly modified from a talk presented at the 2008 Algebraic Topology Conference in Buenos Aires. We give a bicategorical perspective on invertibility beginning with Morita theory and duality, and then describing generalized Brauer groups and Azumaya objects. We develop the theory of invertibility in triangulated bicategories and give a characterization of Azumaya objects therein. One goal of this work is to develop a calculational foothold on Picard and Brauer groups in generalized contexts.

Preprint Draft: Invertibility in Bicategories. Please read with care, as this is still in a rough state. Contact Niles if you have any comments or complaints!

Morita Theory and Azumaya Objects in Bicategorical Contexts

Abstract: These slides are from a talk at the 2009 Joint Meetings Special Session on Homotopy Theory and Higher Categories. Morita theory provides a wonderful first example of bicategorical structure in classic algebra. Generalizations of the Picard group, Azumaya algebras, and the Brauer group are now a part of higher-categorical folklore. However, these are important algebraic concepts not only because they have pleasing definitions but also because they are calculationally accessible. This talk will explain how to generalize those results from algebra which make these concepts so accessible, and describe some examples of interest to topologists and algebraists. The essential tools: duality and our friend the (bicategorical) Yoneda lemma.

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For Complex Orientations Preserving Power Operations, p-typicality is Atypical

arXiv 0910.3187: [abs | pdf]

Abstract: We show, at the primes 2, 3, and 5, that no map from MU to BP defining a universal p-typical formal group law on BP preserves power operations. In particular, such a map cannot define a commutative MU-algebra structure on BP. Our results apply more generally to show that the p-typical complex orientations on a number of standard spectra are not commutative MU-algebra maps.

This builds on work of Jim McClure to determine whether Quillen's orientation on BP is an H_\infty^2 map. By direct computation, we show that the necessary condition he derives for Quillen's orientation to be H_\infty^2 fails for the first three primes. We go on to show that this implies the more general results above. There is nothing special we see about the first three primes in this work, and we conjecture that these results hold for all primes.

We also provide a reinterpretation of McClure's conditions in the language of formal group laws.

Errata: (typos)

• In line 2 of page 3, "The full set . . . were enumerated" should be "The full set . . . was enumerated"
• In Question 1.5, "are thier any" should be "are there any"
• At the end of page 14, "primes greater than 3." should be "primes greater than 5."

Computational package: For this paper, we developed a Mathematica 7 package to compute the McClure Formula. We have also developed a Mathematica notebook which serves as a guide to our calculations. If you don't have Mathematica, you can view the pdf version, or use the notebook with the free Mathematica Player.

Although we appreciate the Mathematica software, we hope to release a streamlined and more accessible version of our package using the open-source software Sage. An additional goal of this implementation will be to facilitate formal group law computations over other spectra familiar to topologists.

Here are slides showing some of the calculations (with, unfortunately, almost no explanaition).

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