An $\infty$-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 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 motivated 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 $BG{L}_{1}R$; we show that $BG{L}_{1}R$ 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 $BG{L}_{1}R$, 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.

arXiv admin note: substantial text overlap with arXiv:0810.4535

Cited by in corpus (33)

• Nilpotence and descent in equivariant stable homotopy theory
• Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory
• Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map
• A simple universal property of Thom ring spectra
• Gauge enhancement of super M-branes via parametrized stable homotopy theory
• Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation
• Brane structures in microlocal sheaf theory
• Twisted Spin cobordism and positive scalar curvature
• Relative Thom Spectra Via Operadic Kan Extensions
• Spectra of units for periodic ring spectra and group completion of graded E-infinity spaces
• Bordism for the 2-group symmetries of the heterotic and CHL strings
• Topological Quantum Gates in Homotopy Type Theory
• Twisted iterated algebraic K-theory and topological T-duality for sphere bundles
• Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces
• The factorization theory of Thom spectra and twisted non-abelian Poincaré duality
• Commutative S-algebras of prime characteristics and applications to unoriented bordism
• A noncommutative model for higher twisted K-Theory
• Homotopical and operator algebraic twisted K-theory
• Exotic Multiplications on Periodic Complex Bordism
• A Theorem on Multiplicative Cell Attachments with an Application to Ravenel's X(n) Spectra
• Generalized Thom spectra and their topological Hochschild homology
• The Smith Fiber Sequence and Invertible Field Theories
• Exponential Functors, R-Matrices and Twists
• The Quantum Monadology
• Abelian duality in topological field theory
• Higher chromatic Thom spectra via unstable homotopy theory
• Thom spectra, higher $THH$ and tensors in $\infty$-categories
• Equivariant higher twisted K-theory of SU(n) for exponential functor twists
• Toward a Galois theory of the integers over the sphere spectrum
• Bundles of spectra and algebraic K-theory
• A Higher Spin-Statistics Theorem for Invertible Quantum Field Theories
• Line Bundle Twists for Unitary Bordism are Ghosts
• Quantum and Reality