An ∞‐categorical approach to R‐line bundles, R‐module Thom spectra, and twisted R‐homology
By M. Ando, Charles Rezk, Andrew Justin Blumberg, D. Gepner, M. Hopkins. 2014
101 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/51d19e126793ba71e203f6232e3389c83cf59920
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Citations in the corpus
- A simple universal property of Thom ring spectra (2014, unknown): “This construction of Thom spectrum appears in [ABG14a] where it is shown to agree with the definitions in [MS06] and [LMSM86].”
- Nilpotence and descent in equivariant stable homotopy theory (2015, background): “Using the obstruction theory of [2], it suffices to show that the composite f : BU J → BGL1S → BGL1E of based spaces is null-homotopic.”