Thom spectra, higher $THH$ and tensors in $\infty$-categories

Abstract

Let $f:G\to \mathrm{Pic}(R)$ 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 $X{\otimes }_{R}Mf$ 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 Day's 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 $g:A\to B$ of $E_{\infty }$-ring spectra satisfies $X$-base change if $X\otimes B$ is the pushout of $A\to X\otimes A$ along $g$. Building on a result of Mathew, we prove that if $g$ is étale 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.

43 pages. Final version, to appear in Algebraic & Geometric Topology

References in corpus (2)

• An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology
• Exotic Multiplications on Periodic Complex Bordism