Contents Online

# Homology, Homotopy and Applications

## Volume 23 (2021)

### Number 1

### Unstable algebras over an operad

Pages: 119 – 141

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n1.a8

#### Author

#### Abstract

The aim of this article is to define and study a notion of unstable algebra over an operad that generalises the classical notion of unstable algebra over the Steenrod algebra. For this study we focus on the case of characteristic $2$. We define $\star$-unstable $\mathcal{P}$-algebras, where $\mathcal{P}$ is an operad and $\star$ is a commutative binary operation in $\mathcal{P}$. We then build a functor that takes an unstable module $M$ to the free $\star$-unstable $\mathcal{P}$-algebra generated by $M$. Under some hypotheses on $\star$ and on $M$, we identify this unstable algebra as a free $\mathcal{P}$-algebra. Finally, we give some examples of this result, and we show how to use our main theorem to obtain a new construction of the unstable modules studied by Carlsson, Brown–Gitler, and Campbell–Selick, that takes into account their internal product.

#### Keywords

Steenrod algebra, operad, unstable module, unstable algebra

#### 2010 Mathematics Subject Classification

17A30, 18D50, 55S10

Received 7 February 2020

Received revised 7 April 2020

Accepted 11 April 2020

Published 26 August 2020