We develop a generalization of the theory of Thom spectra using the language of infinity categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parametrized spectra, and our definition is m
otivated by the geometric definition of Thom spectra of May-Sigurdsson. For an associative ring spectrum $R$, we associate a Thom spectrum to a map of infinity categories from the infinity groupoid of a space $X$ to the infinity category of free rank one $R$-modules, which we show is a model for $BGL_1 R$; we show that $BGL_1 R$ classifies homotopy sheaves of rank one $R$-modules, which we call $R$-line bundles. We use our $R$-module Thom spectrum to define the twisted $R$-homology and cohomology of an $R$-line bundle over a space $X$, classified by a map from $X$ to $BGL_1 R$, and we recover the generalized theory of orientations in this context. In order to compare this approach to the classical theory, we characterize the Thom spectrum functor axiomatically, from the perspective of Morita theory. An earlier version of this paper was part of arXiv:0810.4535.
We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the string orientation of tmf, the spectrum of topological modular forms. We also develop the analogous theory of Thom
spectra and orientations for associative ring spectra. Our work is based on a new model of the Thom spectrum as a derived smash product. An earlier version of this paper was part of arXiv:0810.4535.
We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $mathbb{P}^infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspensi
on spectrum of $BGL$. In particular, working over $mathbb{C}$ and passing to spaces of $mathbb{C}$-valued points, we obtain new proofs of the topologic
We explore an approach to twisted generalized cohomology from the point of view of stable homotopy theory and quasicategory theory provided by arXiv:0810.4535. We explain the relationship to the twisted K-theory provided by Fredholm bundles. We show
how our approach allows us to twist elliptic cohomology by degree four classes, and more generally by maps to the four-stage Postnikov system BO<0...4>. We also discuss Poincare duality and umkehr maps in this setting.
