Secondary derived functors and the Adams spectral sequence

Abstract

Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of "additive groupoid enriched categories", in which a secondary analog of homological algebra can be performed. We introduce secondary chain complexes and secondary resolutions leading to the concept of secondary derived functors. As a main result we show that the E 3 -term of the Adams spectral sequence can be expressed as a secondary derived functor. This result can be used to compute the E 3 -term explicitly by an algorithm.

Higher order derived functors and the Adams spectral sequence

“…For n = 1 this result is proved in [BJ2,Section 7]. Theorems 15.9 and 15.11 are proved for all n ≥ 1 in the following Section.…”

“…Let C be a category enriched in groupoids with zero maps and let C be abelian. Then the algebra of left 1-cubical balls Nul 1 (C) is defined and a triple Toda bracket δ 1 , δ 2 , δ 3 in Nul 1 (C) coincides with the classical triple Toda bracket in C. Moreover, a 1-st order chain complex in Nul 1 (C) as defined in §11.4 coincides with a secondary chain complex in C as studied in [BJ2].…”

“…Remark. In view of Lemma 3.6 (a) in [BJ2], a first-order resolution in Nul 1 C is a secondary resolution in the sense of [BJ2].…”

“…The Lemma is proved in [BJ2] in the context of track categories, above we use only algebras of left 1-cubical balls. The proof that d 2 d 2 = 0 requires the product rule below.…”

“…Remark. As shown in [BJ2], at the secondary level (that is, computing the d 2 -differential), this algebra of cubical balls can be replaced by an ordinary differential graded algebra. In [BaF], the first author and Martin Frankland show that this can also be done at the tertiary level.…”