I've already posted this question on mathoverflow a few days ago and I had no reaction, I hope it is not illegal to repost it on math.stackexchange.
I'm new in the subject of stable homotopy theory, and currently trying to understand modern research on Thom spectra, like "which spectra are Thom spectra".
I know that a systematic generalization of Thom spectra is possible with the theory of $\infty$-categories, as exposed in [ABGHR] An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology.
I was wondering if a geometric theory of generalized Thom spectra was possible, and if someone had a reference about that if it exists. By geometric theory I mean with topological spaces, constructing generalized Thom spaces by taking the cofiber of the projections and forming a spectrum, + a generalized Thom Pontrjagin theorem. At least I'm hoping that someting has been written about the case where the fibers are $p$-local spheres, as it is suggested in [MRS] The Thomified Eilenberg-Moore spectral sequence.
Thank you for your help on the reference-request question.