An 8-categorical approach to R -line bundles, R -module Thom spectra, and twisted R -homology
An 8-categorical approach to R -line bundles, R -module Thom spectra, and twisted R -homology
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R 线束、R 模 Thom 谱和扭曲 R 同源性的 8 分类方法
DOI：
10.1112/jtopol/jtt035
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发表时间：
2014
影响因子：
1.1
通讯作者：
Ando M
中科院分区：
文献类型：
--
作者：
Ando M
我们使用∞-范畴的语言发展了Thom谱理论的一个推广。这种处理暴露的托姆谱函子的概念基础：我们使用一个新的模型参数化的光谱，我们的定义是出于托姆谱的几何定义的May-Sigurdsson。对于A ∞-环谱R，我们将Thom谱与从空间X的∞-广群到自由秩1 R-模的∞-范畴的∞-范畴映射联系起来，证明了它是BGL 1 R的一个模型;我们证明了BGL 1 R对秩1 R-模的同伦层进行了分类，我们称之为R-线丛.利用R-模的Thom谱定义了映射X → BGL 1 R上R-线丛的扭R-同调和上同调，并在此基础上恢复了广义定向理论.为了比较这种方法的经典理论，我们刻画了托姆谱函子公理，从森田理论的角度。
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 anA∞-ring spectrumR, we associate a Thom spectrum to a map of ∞-categories from the ∞-groupoid of a spaceXto the ∞-category of free rank oneR-modules, which we show is a model forBGL1R; we show thatBGL1Rclassifies homotopy sheaves of rank oneR-modules, which we callR-line bundles. We use ourR-module Thom spectrum to define the twistedR-homology and cohomology ofR-line bundles over a space classified by a mapX→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.