Quadratic categories and square rings

Abstract

We consider quadratic categories which generalize the classical additive categories. An additive category A is a category for which morphism sets are abelian groups and the composition fg is bilinear, and for which sums exist in A . A quadratic category Q is slightly more general in the sense that morphism sets are groups and the composition fg is linear in g and quadratic in f. This implies that morphism sets are groups of nilpotency degree 2. We describe below many examples of quadratic categories in algebra and topology which motivate the systematic study of quadratic categories started here; it may be considered as an extension of the investigation of quadratic functors in [4]. The properties of a quadratic category and its subcategories lead to the new notion of a “square ring” which is exactly the quadratic analogue of the classical notion of a “ring”. Indeed each object X in an additive category A yields an endomorphism ring given by all morphisms X → X in A; similarly each object in a quadratic category yields the endomorphism square ring End( X) of X. Hence the connection between square rings and quadratic categories is similar to the relation of rings and additive categories studied by Mitchell [18]. The initial object in the category of rings is the ring Z of integers for which the category of modules is the category of abelian groups. We here determine the initial object Z nil in the category of square rings for which the category of modules is the category of groups of nilpotency degree 2. We compute various square rings explicitly, for example, the endomorphism square rings of the suspended projective planes ΣR P 2 and ΣC P 2. This yields as an application an algebraic description of the homotopy category of all Moore spaces M( V,2) where V is a Z/2-vector space; in fact this category is equivalent to the full category of free objects in the category of 2-restricted nil(2)-groups. There has been recently a lot of interest in operads [9]. In fact, operads O = {O n } with O n = 0 for n ≥ 3 are the same as special square rings. Therefore the theory of square rings shows naturally how the theory of operads has to be modified in order to deal with nilpotent groups.