Derived $\delta$-Rings and Relative Prismatic Cohomology

We characterize the relative prismatic cohomology of Bhatt and Scholze by a universal property by endowing it with the additional structure of a ``derived $\delta$-ring". This involves introducing an analogue of prismatic envelopes in the setting of filtered derived commutative rings and applying this construction to the Hodge filtration on infinitesimal cohomology. In particular, we recover relative prismatic cohomology from infinitesimal cohomology via a purely algebraic process.