2-track algebras and the Adams spectral sequence

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 3term 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.

Segal-type algebraic models ofn–types

“…Since all mapping spaces in this category are themselves simplicial F p -vector spaces, the associated n-track category is correspondingly simplified. The case n = 1 was treated in great detail in [5], and some progress on the case n = 2 has been made in [9]. However, it is clear from [7] that a better conceptual framework, such as an algebraic model for such "linear" n-track categories, will be needed before any further progress can be made for n ≥ 2.…”