An \(\infty\)-categorical approach to \(R\)-line bundles, \(R\)-module Thom spectra, and twisted \(R\)-homology (Q2921101)

This paper is the second of two by these authors concerning Thom spectra and orientations. In the first paper, the authors give what they call here an ``algebraic'' definition of a Thom spectrum functor. Here, the authors use Joyal and Lurie's theory of quasi-categories (referred to here as \(\infty\)-categories) to give a ``geometric'' definition and then to give a comparison between the two.NEWLINENEWLINEA general theory of Thom spectra should build on the following ideas. From an \(A_\infty\)-ring spectrum, one should produce a space \(BGL_1R\), and any map of spaces \(f : X \rightarrow BGL_1R\) should give rise to an \(R\)-module spectrum \(Mf\) where orientations \(Mf \rightarrow R\) should correspond exactly to nullhomotopies of the map \(f\). In analogy with classical bundle theory, Thom spectra should arise from bundles of \(R\)-modules, but making this idea precise has been difficult. While previous work has shown that \(BGL_1S\), where \(S\) is the sphere spectrum, classifies spherical fibrations, there were technical difficulties in making a more general statement.NEWLINENEWLINEThe authors' use of quasi-categories allows sufficient flexibility to overcome these difficulties. They give a new approach to parametrized spectra as homotopy local systems of spectra, and then give a model for \(BGL_1R\) which classifies the homotopy local systems of free rank 1 \(R\)-modules. In particular, they give a definition of bundle of \(R\)-modules over a space \(X\) which can be further specified to that of a bundle of \(R\)-lines over \(X\). These definitions can be used to give a Thom \(R\)-module spectrum construction as a left adjoint functor, which has a corresponding space of orientations. Given the Thom spectrum \(Mf\) associated to a map from \(X\) to the space of \(R\)-lines, an orientation \(Mf \rightarrow R\) corresponds to a lift of \(f\) to the space of trivialized \(R\)-lines. When such a map \(f\) admits an orientation, the authors give explicit formulas for the twisted \(R\)-homology and \(R\)-cohomology spectra of \(X\).NEWLINENEWLINEWith these constructions in place, the authors give several comparison results. They prove that this ``geometric'' definition agrees with the ``algebraic'' one in their previous paper, and give a characterization of Thom spectrum functors via Morita theory. They also prove that they recover the ``neoclassical'' Thom spectrum functor of Lewis and May in the case where \(R\) is the sphere spectrum. The final section of the paper gives a sketch of an alternative method of comparisons via colimits.NEWLINENEWLINEOverall, the paper gives an elegant approach to the theory of Thom spectra and unifies the subject substantially. The authors do, however, make heavy use of Lurie's results regarding quasi-categories, and at least a basic understanding of this work is necessary to understand their constructions and arguments.