So first I'll try to give a really quick reminder of the classical description of these things when one is doing non-commutative descent theory. In the setting of discrete algebra, if we have a morphism of (not necessarily commutative) rings $\phi:A\to B$, we can frame descent data for $\phi$ as $B\otimes_AB$-comodules, where we think of $B\otimes_AB$ as a $B$-coring. $B\otimes_AB$ also fits into the Amitsur complex for $\phi$, which lets us compute the set of descent data cohomologically. In the book Corings and Comodules by Brzezinski and Wisbauer, another approach is described. If one lets $K$ be the kernel of the multiplication map $B\otimes_AB\to B$, one can produce a DGA which at level zero is $A$ and at level $n$ is the $n$-fold tensor product of $K$ over $A$ (Brzezinski and Wisbauer call this the coring valued differential forms with values in the Sweedler coring $B\otimes_AB$). We'll denote this DGA by $\Omega^\bullet$. A connection on a $B$-module $M$ is an $A$-linear map $\nabla:M\otimes_B\Omega^\bullet\to M\otimes_B\Omega^{\bullet+1}$ satisfying the graded Leibniz rule. A connection always induces a curvature (i.e. $\nabla\circ\nabla$ restricted to $M$), and the connection is said to be flat if the curvature is zero. They then go on to show that flat connections are equivalent to descent data for the map $\phi$.
Now, I'd really like to know what the equivalent construction is in the case of the $\infty$-category of $E_n$-ring spectra (it seems like $n=\infty$ is probably the easiest to figure out). Or, more generally, what should the analogous construction be for a morphism of $\mathcal{O}$-algebras be in some $\infty$-category (I'm fine with assuming we're looking at $\mathcal{O}$-algebras in some stable presentable $\infty$-category and that $\mathcal{O}$ is unital or whatever other adjectives you want to throw at it). In this scenario, the analog of $K$ above should be, I think, the relative $TQ$-homology (topological Quillen homology, or perhaps more correctly the relative cotangent complex) construction associated to $\mathcal{O}$. Moreover,we probably want to replace the DGA's above with cosimplicial constructions. We can easily construct the $\infty$-category of descent data for a morphism of $E_k$-algebras (see, e.g. this talk) as the category of cosimplicial comodules over the cosimplicial Sweedler coring (which is an $E_1$-coalgebra). Moreover, using the fact that there is an adjunction between the category of $\mathcal{O}$-algebras under $A$ and the relevant tangent category we can produce a monadic cosimplicial object over $A$ which looks very much like $TQ^n$ at level $n$. We might call this the $TQ$-complex and it is essentially the construction of Harper-Hess, figure 3.18 in this paper.
Anyway, my question is essentially whether or not anyone knows if this is the right idea for connections in this setting or not. It seems like the obvious thing to do, but is it the case that flat connections (which should presumably be a map from the $TQ$-complex to a shift or delooping of the $TQ$-complex which compose to null homotopies) can be shown to be equivalent to the $\infty$-category of descent data? Another nagging question is whether or not this $TQ$-complex has totalization some kind of Hochschild homology, which would essentially be a more general version of the Hochschild-Kostant-Rosenberg theorem).
Any kind of references, or counter-examples, would be appreciated. I have heard rumors that Lurie had written down or given a talk on something of this nature, but I have not been able to find that anywhere. There is also this nlab page, but I'm not really too handy with cohesive $(\infty,1)$-toposes.
