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I n t r o d u c t i o nThe starting point of the investigation whose results are to be presented in a series of three papers, of which this is the first, was an a t t e m p t to give precise meaning to and to answer the question how group structures arise in homotopy theory and why these group structures satisfy the familiar conditions of naturality. This problem, set in the restricted context of homotopy theory, was discussed in [2], but it was already clear that, by restricting the category of study in this way, we were disguising the generality of the approach and of the results. A more general treatment was indicated in Chapter 14 of [6] but not taken very far.A group is a set A with a multiplication m : A × A -> A satisfying certain axioms. The basic idea of this paper is to consider categories ~ rich enough in objects and maps to enable us to formulate a set of axioms which, in the case where ~ is the category ~ of (based) sets, are equivalent to the group axioms. Of course these axioms are formulated entirely in 'terms of the maps of the category ~. I t is then a basic observation that if (A, m) is a "group" in ~ or, as we shall prefer to say, a G-object, and if H(X, A) is the set of maps from X to A, then H(X, A) acquires a group-structure, in the familiar sense, from the structure m a p m; and the group-structure is commutative if m is "commutative" in a sense applicable to the category ~. Moreover the group-structure is natural with respect to maps X -> Y in ~ in the sense that, for such a m a p ], the