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.
We construct and study an algebraic theory which closely approximates the theory of power operations for Morava E-theory, extending previous work of Charles Rezk in a way that takes completions into account. These algebraic structures are made explic
it in the case of K-theory. Methodologically, we emphasize the utility of flat modules in this context, and prove a general version of Lazards flatness criterion for module spectra over associative ring spectra.
The homotopy groups of a space are endowed with homotopy operations which define the Pi-algebra of the space. An Eilenberg-MacLane space is the realization of a Pi-algebra concentrated in one degree. In this paper, we provide necessary and sufficient
conditions for the realizability of a Pi-algebra concentrated in two degrees. We then specialize to the stable case, and list infinite families of such Pi-algebras that are not realizable.
