Unstable algebras over an operad

TL;DR

This paper generalizes the concept of unstable algebras over the Steenrod algebra to operads in characteristic 2, introducing a new framework and constructions for unstable modules and algebras.

Contribution

It defines $ ext{ extsterling}$-unstable $ ext{P}$-algebras over operads, constructs free unstable algebras, and connects these to known unstable modules with internal products.

Findings

Identifies conditions under which unstable algebras are free $ ext{P}$-algebras.

Provides new constructions for unstable modules with internal products.

Examples illustrating the theoretical framework.

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.