Topology Seminar: Jonathan Beardsley

Any cobordism spectrum M associated to a classifying space BG is naturally equipped with a so-called Thom diagonal M->M ^ BG+ and Thom isomorphism M ^ M -> M ^ BG+ . In the homotopy category, these maps give M the structure of a cotorsor for the coalgebra BG+.  I'll describe how this structure can be lifted to the derived setting by proving a more general theorem about derived or "stacky" quotients of En-rings in ∞-categories. This theorem states that, given an action by an n-fold loop space G on an En-ring R, the stacky or derived quotient R//G  is naturally a BG+-cotorsor (the fact that this specializes to a result for cobordism spectra follows from the work of Ando, Blumberg, Gepner, Hopkins and Rezk on Thom spectra and orientations). This result can be reinterpreted in the setting of noncommutative spectral algebraic geometry as saying that Spec(R//G) is a principal Spec(BG+)-bundle (or torsor) over Spec(R) in the category of affine En-varieties. If there is time, I'll speculate wildly about applications to geometry and noncommutative algebra.