Units of Ring Spectra and Thom Spectra

We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. Specifically, we show that for an E∞ ring spectrum A, the classical construction of gl1A, the spectrum of units, is the right adjoint of the functor Σ∞+ Ω ∞ : ho(connective spectra) → ho(E∞ ring spectra). To a map of spectra f : b → bgl1A, we associate an E∞ A-algebra Thom spectrum Mf , which admits an E∞ A-algebra map to R if and only if the composition b → bgl1A → bgl1R is null; the classical case developed by [MQRT77] arises when A is the sphere spectrum. We develop the analogous theory for A∞ ring spectra. If A is an A∞ ring spectrum, then to a map of spaces f : B → BGL1A we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if B → BGL1A → BGL1R is null. We note that BGL1A classifies the twists of A-theory. We take two different approaches to the A∞ theory which are of independent interest. The first involves a rigidified model of A∞ (and E∞) spaces, as developed in [Blum05, BCS08]. The second uses the theory of ∞-categories as described in [HTT]. 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.