In an action-packed three pages of Lurie's DAG-XIII: Rational and p-adic Homotopy Theory, section 2.2: Power Operations on $\mathbb{E}_{\infty}$-algebras, one finds a construction of the power operation $P^0$ following a few observations on the $p$-power Tate construction in the category of $k$-module spectra: $\hat{T}_p: X \mapsto ( X^{\otimes p})^{tC_p}$ and its best colimit-preserving approximation, $T_p.$ For any $\mathbb{E}_\infty$ $k$-algebra $X,$ one obtains a map $T_p(X)[-1] \to X.$ $T_p(X)$ is given by tensoring with a $k$-bimodule which is equivalent on one side to $k^{tC_p}$ and this allows us to obtain operations (not $k$-linear) from elements of Tate cohomology, $ \pi_* k^{tC_p} \simeq \hat{H}^{-*}(C_p, k).$

The precise statement is in construction 2.2.6, which applies the observation that for $k$ a discrete ring of characteristic $p$, $1 \in k$ determines a canonical element of $\hat{H}^{-1}(C_p, k),$ precisely because that group is given as the kernel of the norm. This defines a map $k \to k^{tC_p}[-1]$ which upon composition with the map in the previous paragraph gives a map $X \to X.$ This map is supposed to be the derived witness to $P^0.$

Here is remark 2.2.9

Construction 2.2.6 can be generalized: given any class $x \in \hat{H}^{n-1}(\mathbb{Z} / p\mathbb{Z}; k)$, we obtain an associated map $P(x) : A \to A[n]$, which induces group homomorphisms $\pi_m(A) \to \pi_{m-n}(A)$. These operations depend functorially on $A$ and generate an algebra (the extended Steenrod algebra) of “power operations” which act on the homotopy groups of every $\mathbb{E}_\infty$-algebra over $k.$

My questions: is there any reference where this construction of the extended powers is fully elaborated? How much of the elementary structure of the Steenrod algebra (e.g. Adem relations, structure of the dual Steenrod algebra, etc.) can be translated to this point of view?

Just to fill in a few details, Lurie constructs these power operations by first rotating the fiber sequence defining the Tate construction to yield

$\hat{T}_p(X)[-1] \to (X^{\otimes p})_{hC_p} \to (X^{\otimes p})^{hC_p}$

If $X$ is an $\mathbb{E}_\infty$ $k$-algebra, one then computes the composition

$T_p(X)[-1] \to \hat{T}_p(X)[-1] \to (X^{\otimes p})_{hC_p} \to X^{\otimes p}_{h\Sigma_p} \to X$

where the maps are given by approximation, the first map in the rotated fiber sequence, a tautological map between colimits, and the $\mathbb{E}_\infty$ multiplication respectively.

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    $\begingroup$ in case anyone stumbles across the question, it looks like this got a very nice treatment at p=2 by @DylanWilson in arXiv:1905.00054 "Mod 2 power operations revisited" $\endgroup$ – pupshaw May 6 '19 at 21:38

What you are looking for is probably Lecture 24 of Lurie's lecture notes on the Sullivan Conjecture. However, these kinds of results (namely, the relation between $\Sigma_2$ and operations, or the relation between $\Sigma_4$ and relations) have a much more extensive historical background, which Dylan Wilson discusses near the beginning of Section 3 of his paper Power operations for $H\underline{\Bbb{F}}_2$ and a cellular construction of $BP{\Bbb R}$.

There is probably more to say but you should feel free to contact me by email if you want further elaboration.

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