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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. [ABSTRACT FROM AUTHOR]

This paper aims to answer the following question: Given an adjunction between two categories, how is Quillen (co)homology in one category related to that in the other? We identify the induced comparison diagram, giving necessary and sufficient conditions for it to arise, and describe the various comparison maps. Examples are given. Along the way, we clarify some categorical assumptions underlying Quillen (co)homology: cocomplete categories with a set of small projective generators provide a convenient setup. [ABSTRACT FROM AUTHOR]