Department of Mathematics

Select Page Style

(sets a browser cookie)

Welcome to Niles's research page

My research is focused on Morita theory in bicategorical contexts. Below you will find links to some introductory slides, and an arXiv preprint. Please contact me with questions or comments if you read any of these, and especially if they are relevant to your own work! (e-mail: my first name at math.uchicago.edu).

Six-sentence Introduction

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.

Preprint Article

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.

Niles Johnson
Mathematics
Graduate Student
University of Chicago

Office: MS 301
(directions)