Adjoint functors and bar constructions

Abstract

The notions of universal bundle, classifying space and bar are closely linked but not interchangeable. The relations between them are well understood in some contexts, less so in others. Our aim here is not to elucidate these relations but, merely, as a first step in such an elucidation, to clarify the notion of a bar construction. The literature on bar constructions is too large to survey here. The name is due to Eilenberg and MacLane. Their is extensively described in [4]. Milgram [8] gave a bar for topological monoids; the elegant description by Steenrod [10] motivated this paper. MacLane [5] gives yet another elegant description of Milgram's construction. A bar for topological categories was defined by Segal [9] and an abstract bar construction of very great generality was introduced by May [7]. The cobar of Adams [1] is a dual sort of thing which ought to be considered under the same rubric. The universal bundle of Milnor, on the other hand, is not a bar construction. All these bar constructions bear to one another what is evidently more than a mere family resemblance. But the means for saying precisely in what this resemblance consists appear to be lacking. It is this deficiency which we propose to remedy. We begin by defining the notion of bar in a broad categorical setting by means of a universal property. For these we can prove a rather general existence theorem (5.8). Finally we indicate how some of the bar constructions mentioned above, and the cobar as well, are instances of the notion introduced here. Our existence theorem for bar constructions actually results from an existence theorem for coproducts of monads (cf. MacLane [4]). As a