A simple universal property of Thom ring spectra

Understand

We give a simple universal property of the multiplicative structure on the Thom spectrum of an $n$-fold loop map, obtained as a special case of a characterization of the algebra structure on the colimit of a lax $\mathcal{O}$-monoidal functor.

• This allows us to relate Thom spectra to $\mathbb{E}_n$-algebras of a given characteristic in the sense of Szymik.
• As applications, we recover the Hopkins--Mahowald theorem realizing $H\mathbb{F}_p$ and $H\mathbb{Z}$ as Thom spectra, and compute the topological Hochschild homology and the cotangent complex of various Thom spectra.