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We develop a generalization of the theory of Thom spectra using the language of ∞-categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parameterized spectra, and our definition is motivated by the geometric definition of Thom spectra of May-Sigurdsson. For an A∞-ring spectrum R, we associate a Thom spectrum to a map of ∞-categories from the ∞-groupoid of a space X to the ∞-category of free rank one R-modules, which we show is a model for BGL1R; we show that BGL1R 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 R-line bundles over a space classified by a map X → BGL1R, 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.

Abstract. In [AHS01] the authors constructed a natural map, called the sigma orientation, from the Thom spectrum M U 6 to any elliptic spectrum in the sense of [Hop95]. M U 6 is an H∞ ring spectrum, and in this paper we show that if (E, C, t) is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p > 0, then the sigma orientation is a map of H∞ ring spectra.

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We introduce a general theory of parametrized objects in the setting of ∞-categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a presentable ∞-category parametrized over objects of an ∞-topos. We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures.Our main applications are to the study of generalized Thom spectra. We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality. In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we characterize the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.Pre(Orb G ), or sheaves of spaces on a Grothendieck site, such as the site associated to a topological space.Let S denote the ∞-category of spaces. Since S is freely generated under colimits by its final object (the point), for any ∞-category M with small limits, the ∞category of limit-preserving functors S op → M is equivalent (via evaluation at the point) to M itself. If C is an object of M, then we will writefor the resulting functor. Now, to say that C /(−) preserves limits is to say that it satisfies descent, and so we call such a functor a sheaf on S with values in M. For example, the ∞-category Cat ∞ of (not necessarily small) ∞-categories is complete, and so any ∞-category C uniquely determines, and is determined by, a sheaf