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Register a new userAn $infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology
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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 $BGL_1 R$; we show that $BGL_1 R$ 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 $BGL_1 R$, 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.
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Let $f:Gto mathrm{Pic}(R)$ be a map of $E_infty$-groups, where $mathrm{Pic}(R)$ denotes the Picard space of an $E_infty$-ring spectrum $R$. We determine the tensor $Xotimes_R Mf$ of the Thom $E_infty$-$R$-algebra $Mf$ with a space $X$; when $X$ is the circle, the tensor with $X$ is topological Hochschild homology over $R$. We use the theory of localizations of $infty$-categories as a technical tool: we contribute to this theory an $infty$-categorical analogue of Days reflection theorem about closed symmetric monoidal structures on localizations, and we prove that for a smashing localization $L$ of the $infty$-category of presentable $infty$-categories, the free $L$-local presentable $infty$-category on a small simplicial set $K$ is given by presheaves on $K$ valued on the $L$-localization of the $infty$-category of spaces. If $X$ is a pointed space, a map $g: Ato B$ of $E_infty$-ring spectra satisfies $X$-base change if $Xotimes B$ is the pushout of $Ato Xotimes A$ along $g$. Building on a result of Mathew, we prove that if $g$ is etale then it satisfies $X$-base change provided $X$ is connected. We also prove that $g$ satisfies $X$-base change provided the multiplication map of $B$ is an equivalence. Finally, we prove that, under some hypotheses, the Thom isomorphism of Mahowald cannot be an instance of $S^0$-base change.
We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a presentable infinity category parametrized over objects of an infinity topos. We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures. Our main applications are to the study of generalized Thom spectra. We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality. In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we characterize the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.
We apply an announced result of Blumberg-Cohen-Schlichtkrull to reprove (under restricted hypotheses) a theorem of Mahowald: the connective real and complex K-theory spectra are not Thom spectra.
We explain how to reconstruct the category of Artin-Tate $mathbb{R}$-motivic spectra as a deformation of the purely topological $C_2$-equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of $C_2$-equivariant sheaves on the moduli stack of formal groups. As such, our results directly generalize the cofiber of $tau$ philosophy that has revolutionized classical stable homotopy theory. A key observation is that the Artin-Tate subcategory of $mathbb{R}$-motivic spectra is easier to understand than the previously studied cellular subcategory. In particular, the Artin-Tate category contains a variant of the $tau$ map, which is a feature conspicuously absent from the cellular category.
We compute some R-motivic stable homotopy groups. For $s - w leq 11$, we describe the motivic stable homotopy groups $pi_{s,w}$ of a completion of the R-motivic sphere spectrum. We apply the $rho$-Bockstein spectral sequence to obtain R-motivic Ext groups from the C-motivic Ext groups, which are well-understood in a large range. These Ext groups are the input to the R-motivic Adams spectral sequence. We fully analyze the Adams differentials in a range, and we also analyze hidden extensions by $rho$, 2, and $eta$. As a consequence of our computations, we recover Mahowald invariants of many low-dimensional classical stable homotopy elements.
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