2-track algebras and the Adams spectral sequence

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 \(E_r\). 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.