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

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.