# Mod 2 power operations revisited

## TL;DR

This paper provides a modern, homotopical perspective on mod 2 power operations in $ ext{E}_ ext{infinity}$-ring spectra, including a new proof of Adem relations and streamlined derivations of their actions.

## Contribution

It offers a homotopical approach to power operations, including a quick proof of Adem relations using Tate-valued Frobenius and simplified derivations of their actions.

## Key findings

- Quick proof of Adem relations using Tate-valued Frobenius.
- Streamlined derivation of power operations on dual Steenrod algebra.
- Modern account of mod 2 power operations in $ ext{E}_ ext{infinity}$-ring spectra.

## Abstract

In this mostly expository note we take advantage of homotopical and algebraic advances to give a modern account of power operations on the mod 2 homology of $\mathbb{E}_{\infty}$-ring spectra. The main advance is a quick proof of the Adem relations utilizing the Tate-valued Frobenius as a homotopical incarnation of the total power operation. We also give a streamlined derivation of the action of power operations on the dual Steenrod algebra.