No 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.

No abstract

Basic examples of monoidal categories are the following: Example 1.1. Let C be any category with nite products. Then these products may be used to give it a monoidal structure C = (C; ; 1; a; l; r), where is the binary product, 1 is the terminal object (which exists as the empty product), and a, l, r are uniquely determined by the universal property of the products. A monoid in this monoidal category is what is usually called an internal monoid in a category with products. Also in this \cartesian" situation one may de ne what it means for a monoid G = (G; : G G ! G; : 1 ! G) to be an internal group object: there must exist an endomorphism : G ! G satisfying (G)d = p = (G)d where d : X ! X X and p : X ! 1 are the canonical morphisms (which are only available in the cartesian case). In particular, taking C to be the category Ens of sets and functions, one obtains just monoids and groups in the ordinary sense; or, taking the categories of spaces, simplicial sets, etc., one obtains topological or simplicial monoids and groups. Example 1.2. The category R-mod of modules over a commutative ring R may be given a monoidal structure using the tensor product over R. We shall denote this monoidal category

Square groups are quadratic analogues of abelian groups. Many properties of abelian groups are shown to hold for square groups. In particular, there is a symmetric monoidal tensor product of square groups generalizing the classical tensor product.There is a long-standing problem of algebra to extend the symmetric monoidal structure of abelian groups, given by the tensor product, to a non-abelian setting, see for example [13]. In this paper we show the somewhat surprising fact that such an extension is possible. Moreover our non-abelian tensor product remains even right exact and balanced. We describe the new nonabelian tensor product in the context of quadratic algebra which extends linear algebra."Linear algebra" is the algebra of rings and modules. A ring is a monoid in the symmetric monoidal category of abelian groups (Ab, ⊗, Z). ✩ The second and third authors are grateful to the