Classifying Spaces of Topological Monoids and Categories

Abstract

0. Introduction. In recent years, a number of curious and seemingly paradoxical properties of the bar constuction have come to light. These results have the general form that, in certain situations, the bar construction on a topological group, monoid, or category can be largely independent of the topological structure of the underlying object. Among the most prominent of these results is that of Thurston [28] which states that if G = Homeo(M) is the topological group of self-homeomorphisms of a compact manifold and G& is the same group endowed with the discrete topology, then BG6 -+ BG is a homology equivalence (cf. also MacDuff [16]). In a similar spirit there are two rather amazing results, due to KanThurston [14] and MacDuff [15], that given any connected CW complex X there is a discrete group G and a homology equivalence BG -+ X and a discrete monoid M and a homotopy equivalence BM X. Most recently Friedlander and Milnor have conjectured that for any Lie group G, BGBG is a homology equivalence with finite coefficients. This paper analyzes in detail one particular class of these phenomena, the case when BM6 -+ BM or BC 6 -+ BC is a weak homotopy equivalence for a topological monoid M or a topological category C. The author's interest in this sort of phenomenon arose in trying to understand Waldenhausen's work on the algebraic K-theory of spaces (cf. [30]). If X is a connected space, Waldhausen considers the topological category C,,k of G-equivalences of spaces having the homotopy type of G + A V kSn, where G is the Kan loop group of X. He then defines the algebraic K-theory A(X) in various ways, among them (1) A(X) = Z X limnk(BC6 ,k) and (2) A(X) = Z X lim nk(BCn,k)+. Here Cn,k is the discrete category obtained from Cn,k by discarding the topology on the function spaces of Cn,k. Both of these definitions play an important role in Waldhausen's theory: (1) is required to compare A(X) with Hatcher's higher simple homotopy theory;