In preparation (apologies to those who have inquired about the draft of this paper previously announced here under the title "Spectral algebraic theories I: derived categories"; dissatisfaction led to changes in understanding which led to the ongoing rewriting)
We exhibit the derived ∞-categories of certain abelian categories, the ∞-category of derived commutative rings, and certain ∞-categories of synthetic spectra as instances of a universal construction.