@FrankScience: there is a fiber sequence of A-bimodules: L_A-->U(A)-->A, where L_A is the E_1-cotangent complex and U(A) is the enveloping algebra A \otimes A^{op}. Tensoring over U(A) with A gives HH(A) as approximately a shift of L_A\otimes_{U(A)}A.

Also: there is no issue about simplicial/vs spectral for E_1-algebras.

(that's only for E_infty stuff)

Also, for noncommutative things, it's not true that the stabilization/abelianization of algebras augmented over A is A-modules, it's A-bimodules. (Hence the appearence of U(A) above). Notice that the derived functor of derivations involves deriving bimodule maps out of the cotangent complex, in Quillen's setup

that's why you see a tensor product over U(A) when you're doing homology vs. cohomology.

Finally, @DenisNardin, (topological) Hochschild cohomology does have to do with deformations: THC^{(n)}(A)[n], the E_n Hochschild cohomology shifted by n, turns out to have a spectral Lie algebra structure that controls deformations of the category of E_n-A-modules. (This is the point of view of John Francis's thesis, you can read about it there.) Specifically, this shifted THC is the tangent complex to an E_n-moduli problem.

In the E_1-case this is classical, that hochschild cohomology tells you about deforming the category of bimodules.

@DylanWilson Thank you that's interesting. I knew the classical case, of course, it's just that even knowing the proof I don't have a lot of intuitive "feel" for why it should be the case. Maybe I should read Francis' thesis

On the other hand, when we are working in noncommutative rings the interpretation of THH as free loop spaces does not work, so I guess maybe I'm looking for intuition where there isn't one expected

is there a reference for how exactly π_* R{x} gives all the information about power operations on an E_∞-ring R? Here R{x} is the free algebra on one generator. I'm specifically interested in the case R=F_2. Any pointers would be greatly appreciated!

@RyanKeleti Have you looked at Charles Rezk's notes?

Although I think you need π_*R{x_1,...,x_n} for all n

It's pretty much just a Yoneda argument

I'm sorry which notes?

faculty.math.illinois.edu/~rezk/power-operation-lectures.dvi

The case of $\mathbb{F}_2$ is around page 32

thanks so much!