Coalgebraic Structure and Intermediate Hopf-Galois Extensions of Thom Spectra in Quasicategories

We extend Lurie's work on derived algebraic geometry to define highly structured E-n-coalgebras, bialgebras and comodules in the homotopy theorist's category of spectra. We then show that representable comonads give examples of coalgebras in categories of module spectra for E-n-rings. This immediately leads to an identification of spectral (and more generally quasicategorical) descent data with certain quasicategories of comodules. Using this new framework we then extend the work of Rognes and Hess to define Hopf-Galois extensions of E-n-ring spectra. We use this machinery to produce many new examples of intermediate Hopf-Galois extensions. Such structures, unlike the intermediate extensions of Galois covers, are not generally controlled by a Galois correspondence. We do however show that intermediate Hopf-Galois extensions are ubiquitous among Thom spectra. Of particular interest are a number of classical cobordism spectra, e.g. MU and MSpin, that can now be described as quotients of other cobordism spectra, e.g. MU is the quotient of an action of the circle on MSU, and MSpin is the quotient of action of K(Z,3) on MString. Producing such intermediate extensions is accomplished by recognizing the Thom spectrum of a morphism of Kan complexes f:X→ BGL1(R) as a quotient of R by an action ΩX→ GL1(R). As a result, given a fibration F→E→B of n-fold loop spaces, and a morphism of n-fold loop spaces f:E→BGL1(R), we can produce a sequence of Hopf-Galois extensions R→ R/ΩF→ R/ΩE. Importantly, the bialgebra associated to the former extension is F and associated to the latter is E/F=B, which is distinctly reminiscent of the classical Galois correspondence.
algebraic topology, homotopy theory, thom spectra, derived algebraic geometry, topology, category theory, Hopf-Galois, Galois, quasicategory