In this talk, I will describe a variant of the category of derived rings that we call the category of synthetic E_\infty-rings. The category of derived rings may be viewed as the derived category of the category of discrete commutative rings; analogously, the category of synthetic E_\infty-rings is a derived category of the category of E_\infty-ring spectra with homotopy groups concentrated in even degrees. The category of synthetic E_\infty-rings is closely related to the even filtration of Hahn--Raksit--Wilson, and I will explain how it can be used to formulate a new universal property of the even filtration.

As an application, I will describe how one may use this framework to define the derived versions of invariants such as the prismatic cohomology of E_\infty-ring spectra, and explain why these invariants are in fact algebraic in nature.

This is joint work with Devalapurkar, Hahn and Raksit.