Cohomology theories in the moduli of ring stacks

We show that the natural map from the syntomification of a ring $R$ to the stack of $R$-algebra stacks is fully faithful, answering a question of Drinfeld, and we describe its essential image in terms of underlying monoid stacks. We also give similar statements in the characteristic 0 filtered de Rham, $\mathrm{\ell }=p$ étale, and Betti settings.