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In algebraic topology, power operations are cohomology operations in multiplicative cohomology theory which are higher-degree analogs of cup product-squares symmetrized in the appropriate homotopy-theoretic sense.
At least to some extent, power operations may be understood as the higher algebra-generalization of the ordinary $p$-power map $(-)^p$ on a commutative ring, the one that appears in the definition of Fermat quotients, p-derivations and Frobenius morphisms.
See for instance Lurie, from remark 2.2.7 on for relation to the Frobenius homomorphism and see the example below. See Guillot 06, Morava-Santhanam 12 for further discussion and speculation in this direction.
For $E$ an E-∞ ring and $X$ a topological space (∞-groupoid, homotopy type), a map $a\;\colon\;X \to E$ is a cocycle in the Whitehead-generalized cohomology of $X$ with coefficients in $E$.
The $n$-th cup product power of this $a$ is the composite
where the second map is given the product operation in the ring spectrum $E$. Since this is, by assumption, commutative up to coherent higher homotopy, this map factors through the homotopy quotient by the ∞-action of the symmetric group $\Sigma_n$
The cohomology class of this $E$-cocycle on $X \times B \Sigma_n$ is the $n$-th (symmetric) power of $a$.
On ordinary cohomology over a topological space, the power operations are the Steenrod operations;
Specifically for $n = 2$ and $E = H \mathbb{Z}_2$, the second (symmetric) power of $a \in H(X,\mathbb{Z}_2)$ is an element in $H^\bullet(\mathbb{R}P^\infty \times X, \mathbb{Z}_2) \simeq H^\bullet(X,\mathbb{Z}_2)[x]$ and the coefficients of this polynomial in $x$ are the Steenrod operations on $a$.
For $p \gt 2$ there are the Steenrod power operations (e.g. Rognes 12, around theorem 3.3, quick exposition here).
On an infinite loop space the power operations are the Kudo-Araki-Dyer-Lashof operations?
In the context of complex K-theory power operations are the Adams operations.
From this MO comment by Akhil Mathew:
Let $R$ be a K(1)-local E-∞ ring under (p-adic) complex K-theory KU. Then there exists a basic power operation $\theta \colon \pi_0 R \to \pi_0 R$ (see Hopkins) such that :
$\psi(x) \stackrel{\mathrm{def}}{=} x^p + p \theta(x)$ defines a ring homomorphism from $\pi_0 R \to \pi_0 R$.
$\theta$ satisfies all the identities needed to make $\psi$ a ring-homomorphism after “division by $p$.” For instance $\psi(x+y) = \psi(x) + \psi(y)$ implies that
where the last term is an integral polynomial in $x,y$ and is interpreted as such.
(see also Rezk 09, example 1.3)
This is a “$\theta$-algebra.”/p-derivation as in remark above.
Notice that $\psi$ is, in particular, a lift of the Frobenius homomorphism. There are generalizations of $\psi, \theta$ at higher chromatic levels, too, and there is a modular interpretation of the resulting algebraic structure in (Rezk 09).
By (Strickland 98) we have that if $G$ is the formal group associated to a Morava E-theory, then Frobenius lifts (twhich corresponds to degree $p^k$ subgroups of $G$) are classified by maps into $E^0(B \Sigma_{p^r})/I_{t r}$ where $I_{t r}$ is the transfer ideal. So, for example, the map $\psi$ above corresponds to a universal map $KU^0 \to KU^0(B \Sigma_p)/I_{t r} \simeq KU^0$.
The basic idea is nicely described in
(from which some of the above text is adapted).
More technical surveys include
Charles Rezk, Lectures on power operations (dvi) (2006)
Charles Rezk, Power operations in Morava E-theory – a survey (2009) (pdf)
Charles Rezk, Isogenies, power operations, and homotopy theory, article (pdf) and talk at ICM 2014 (pdf)
Lecture notes on the Steenrod squares and power operations include
The original articles are
Charles Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006), 969-1014 (arXiv:math/0407022)
Charles Rezk, Power operations for Morava E-theory of height 2 at the prime 2 (arXiv:0812.1320)
Charles Rezk, The congruence criterion for power operations in Morava E-theory, Homology, Homotopy and Applications, 11(2), 327-379 (arXiv:0902.2499)
More discussion in the generality of E-infinity arithmetic geometry is in
Discussion for $K(1)$-local $E_\infty$-rings is in
and discussion of power operations in Morava E-theory is in
Matthew Ando, Isogenies of formal group laws and power operations in the cohomology theories $E_n$, Duke Math. J. Volume 79, Number 2 (1995), 423-485 (Euclid)
Neil Strickland, Morava E-theory of symmetric groups (arXiv:math/9801125)
Comments on the analogy between power operations in homotopy theory and Lambda ring structure in Borger's absolute geometry are in
Pierre Guillot, Adams operations in cohomotopy (arXiv:0612327)
Jack Morava, Rakha Santhanam, Power operations and Absolute geometry, 2012 (pdf)