Associative Ring Spectra and Derived Completion So, I was thinking  before that this might have some nice, simple topos theoretic explanation, but Jacob disabused me of that notion. However, I'm still very interested in the following question:
Is there a criterion for determining when a morphism $f:A\to B$ of connective $A_\infty$-ring spectra is such that $A$ can be recovered as the (homotopy) limit of the (homotopical) Amitsur complex on $f$ in the $(\infty,1)$-category of $A_\infty$-ring spectra? For what it's worth, in the context of EKMM, such a criterion is given as an (unproven) proposition in a paper of Carlsson's  (Proposition 3.3 here). There, it is stated that a morphism $f:A\to B$ of connective associative $\mathbb{S}$-algebras has the desired property if $\pi_0(f)$ is an isomorphism and $\pi_1(f)$ is onto.  I'm interested in proving this in the $\infty$-categorical context (and really proving it at all). 
 A: If by ``effective monomorphism'' you mean the categorical dual of the condition of being an effective epimorphism, then it is not (or at least not obviously) equivalent to the statement that A can be recovered as the totalization of the cosimplicial A-module given by the tensor powers of B, because tensor product is not the same as coproduct when working with associative rings.
The condition that A can be recovered in this way is often (but not always) true. It's true for example if B is faithfully flat over A, or if A admits a finite filtration by B-modules, or under the hypothesis you mention.
It doesn't seem obvious to me that you can interpret this in terms of Grothendieck topologies on associative ring spectra (again because tensor product is different from coproduct for associative rings), though there are some cases where this is possible (if you're willing to assume that things are approximately commutative and you stick to etale morphisms). In any case, I doubt that such an interpretation would be useful for actually proving that a particular map A -> B had the property you are interested in; more likely, it would just give you a way of restating the problem.
