Quadratic algebra of square groups

Abstract

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

Smash Products for Secondary Homotopy Groups

“…As proved in [7] the tensor product of square groups is a symmetric monoidal structure on the category of square groups with unit Z nil in Definition 2.1.1. The associativity isomorphism…”

“…This tensor product, first defined in [7], originates from properties of the exterior cup-products in the next section and in Section 4.…”

“…We refer the reader to [7] where the quadratic algebra of square groups is developed. We need square groups for the definition of quadratic pair modules as follows.…”

“…We first recall the notion of square group, see [13] and [7]. where X e is a group with an additively written group law, X ee is an abelian group, P is a homomorphism, H is a quadratic map, i.e.…”

“…Here one needs the symmetric monoidal structure of the category of quadratic pair modules qpm, which is based on the symmetric monoidal structure on the category of square groups constructed in [7]. The smash product morphism (2) is compatible with the associativity isomorphisms, but it is not directly compatible with the commutativity isomorphisms.…”