On bialgebras, comodules, descent data and Thom spectra in \(\infty\)-categories (Q6071793)

The paper studies \(\infty\)-categorical coalgebras and bialgebras which are not necessarily commutative nor cocommutative as well as their \(\infty\)-categories of modules and comodules. The authors establish several results for coalgebraic structure in \(\infty\)-categories, specifically with connections to the spec tral noncommutative geometry of cobordism theories. It is proved 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. The authors give two examples of higher coalgebraic structure. First, they prove that for a map of \(\mathbb{E}_n\)-ring spectra \(\phi: A \to B\), 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, they show that Thom spectra are canonically equipped with a highly structured comodule structure which is equivalent to the \(\infty\)-categorical Thom diagonal of \textit{M. Ando} et al. [J. Topol. 7, No. 3, 869--893 (2014; Zbl 1312.55011)]. They also show 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.