On bialgebras, comodules, descent data and Thom spectra in $\infty$-categories

Abstract

This paper establishes several results for coalgebraic structure in $\infty$-categories, specifically with connections to the spectral noncommutative geometry of cobordism theories. We prove that the categories of comodules and modules over a bialgebra always admit suitably structured monoidal structures in which the tensor product is taken in the ambient category (as opposed to a relative (co)tensor product over the underlying algebra or coalgebra of the bialgebra). We give two examples of higher coalgebraic structure: first, following Hess we show that for a map of $\mathbb{E}_n$-ring spectra $\varphi : A \to B$, the associated $\infty$-category of descent data is equivalent to the $\infty$-category of comodules over $B \otimes_A B$, the so-called descent coring; secondly, we show that Thom spectra are canonically equipped with a highly structured comodule structure which is equivalent to the $\infty$-categorical Thom diagonal of Ando, Blumberg, Gepner, Hopkins and Rezk (which we describe explicitly) and that this highly structured diagonal decomposes the Thom isomorphism for an oriented Thom spectrum in the expected way indicating that Thom spectra are good examples of spectral noncommutative torsors.