An ∞-categorical approach to R -line bundles, R -module Thom spectra, and twisted R -homology

Abstract

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.

A simple universal property of Thom ring spectra

“…This generalizes earlier structural results about Thom spectra proven by Lewis . We then show that any ‐fold loop map with Thom spectrum is canonically ‐orientable, thereby establishing a structured version of the Thom isomorphism.…”

“…This construction of Thom spectrum appears in where it is shown to agree with the definitions in . We will often apply the above definition in the special case that the map factors through .…”

“…As in , we associate two ‐groupoids to . Let be the subcategory of invertible ‐modules and all equivalences between them, that is, the core of the subcategory of on the invertible objects. Let denote the subcategory of modules equivalent to and all equivalences between them.…”

“…The Thom spectrum for a topological group arises classically in the theory of orientations of vector bundles, representing the universal cohomology theory that orients manifolds with structure group . This point of view admits a generalization to ‐ring spectra, as we briefly summarize now; see for a comparison of the various notions of orientations in the and cases, as well as .…”

“…In , the authors describe a convenient model of Thom spectra as homotopy colimits of diagrams whose shape is given by the base space. Their approach is closely related and essentially equivalent to the earlier models cited above, but is expressed in the language of ‐categories.…”