Completions in abstract homotopy theory

Abstract

In this paper the considerations of [6] will be supplemented by a theory of completions of h-c-categories. The utility of such a theory is most strikingly demonstrated by the remarkable representation theorem for half-exact functors due to E. H. Brown [2]. It may also play a role in derivations of the spectral sequences of Adams for stable homotopy theory and of Dyer-Kahn for fibre bundles. These matters, however, are not within the scope of this paper. The primary example, which provides the intuitive basis for the notion of an h-c-category, is that of the category of finite CW-complexes. The completion in this case is the category of all CW-complexes. The peculiarities of the relevant notion of completion are quite evident in this case : the latter category is neither complete nor cocomplete in the usual senses of category theory. What is the case is that certain families, e.g., increasing families of subcomplexes, have colimits. Starting with this model Boardman [1] defined a completion of the stabilized category of finite CW-complexes which seems to be generally recognized as the appropriate category for stable homotopy theory. The concern of this paper is at once to generalize and to amplify the work of Boardman. The generalization lies in the fact that the constructions are applicable to all h-c-categories, rather than just to the category of finite h-c-complexes. The amplification is the observation that their application leads to new h-c-categories to which the formal theorems of homotopy theory are still applicable.