Cellular $E_k$-algebras

TL;DR

This paper develops foundational tools for cellular $E_k$-algebras, facilitating their application to homological stability through filtrations, homology theories, CW approximations, and spectral sequences.

Contribution

It introduces a comprehensive framework for cellular $E_k$-algebras, including new computational tools and theoretical results tailored for homological stability applications.

Findings

Established a homology theory with a Hurewicz theorem for $E_k$-algebras.

Developed CW approximations and spectral sequences for cellular $E_k$-algebras.

Provided foundational tools for future applications in homological stability.

Abstract

We give a set of foundations for cellular $E_k$-algebras which are especially convenient for applications to homological stability. We provide conceptual and computational tools in this setting, such as filtrations, a homology theory for $E_k$-algebras with a Hurewicz theorem, CW approximations, and many spectral sequences, which shall be used for such applications in future papers.