and involves an $\infty$-categorical account of parametrized spectra. In order to compare these approaches to one another and to the classical theory, we characterize the Thom spectrum functor from the perspective of Morita theory.'
address:
- |
    Department of Mathematics\
    The University of Illinois at Urbana-Champaign\
    Urbana IL 61801\
    USA
- |
    Department of Mathematics\
    Stanford University\
    Stanford CA 94305
- |
    Department of Mathematics\
    The University of Illinois at Chicago\
    Chicago IL 60607\
    USA
- |
    Department of Mathematics\
    Harvard University\
    Cambridge MA 02138
- |
    The University of Illinois at Urbana-Champaign\
    Urbana IL 61801\
    USA
author:
- Matthew Ando
- 'Andrew J. Blumberg'
- David Gepner
- 'Michael J. Hopkins'
- Charles Rezk
date: 'Version 3.56 October 2009'
title: Units of ring spectra and Thom spectra
---

[^1]

Introduction {#sec:prol-thom-isom}
============

The study of orientations of vector bundles and spherical fibrations is a classical topic in algebraic topology, deeply intertwined with the modern development of the subject. In his 1970 MIT notes [@MR2162361], Sullivan observed that to describe the obstructions to orientability and to classify orientations, one should take seriously the idea of the “units” of a multiplicative cohomology theory. In order to make such algebraic notions about 