There’s quite a bit on the $n$-lab about effective epimorphisms in $\infty$-toposes, so maybe I’m just not putting it together correctly, but I’m wondering if anyone here knows if there’s a characterization (in print somewhere, preferably) of effective epimorphisms for $A_\infty$-ring spectra. In particular, I suspect there should be some kind of statement like: For $f:A\to B$, a morphism of connected $A_\infty$-ring spectra, we can recover $A$ from the Amitsur complex (dually, the Cech nerve) of $f$ as long as $\pi_0(f)$ is an isomorphism and $\pi_1(f)$ is a surjection.

This sort of statement is proven by Gunnar Carlsson in the context of what he calls “derived completion” but he works with the $S$-algebra framework of Elmendorf, Kriz, May and Mandell, and it seems like it should be a much more general topos theoretic statement for an $\infty$-topos. Incidentally, has anyone written down anything about the topos of $A_\infty$-ring spectra at all? At first glance it seems like most of Lurie’s work on the subject is for $E_\infty$-rings.

Thanks! Jon

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