2-track algebras and the Adams spectral sequence

TL;DR

This paper introduces 2-track algebras and tertiary chain complexes to describe the $E_4$-term of the Adams spectral sequence as a tertiary Ext group, extending previous secondary and higher descriptions for better computational approaches.

Contribution

It develops the concept of 2-track algebras and tertiary chain complexes, providing a new algebraic framework to compute the $E_4$-term of the Adams spectral sequence.

Findings

Defined 2-track algebras and tertiary chain complexes.

Showed $E_4$-term as a tertiary Ext group.

Extended previous secondary and higher descriptions.

Abstract

In previous work of the first author and Jibladze, the $E_3$-term of the Adams spectral sequence was described as a secondary derived functor, defined via secondary chain complexes in a groupoid-enriched category. This led to computations of the $E_3$-term using the algebra of secondary cohomology operations. In work with Blanc, an analogous description was provided for all higher terms $E_m$. In this paper, we introduce $2$-track algebras and tertiary chain complexes, and we show that the $E_4$-term of the Adams spectral sequence is a tertiary Ext group in this sense. This extends the work with Jibladze, while specializing the work with Blanc in a way that should be more amenable to computations.