Bar and cobar constructions. II (Q579690)

The author [Isr. J. Math. 48, 331-339 (1984; Zbl 0587.55013)] generalized Quillen's Theorems A and B on homotopy properties of the classical classifying space functor [see \textit{D. Quillen}, Lect. Notes Math. 341, 85-147 (1973; Zbl 0292.18004)] to bar constructions in the category \({\mathcal T}op\) of topological spaces. Further generalizations were announced in that paper. These generalizations are carried out in the present paper. The main points are that the category \({\mathcal T}op\) is replaced by a suitable category with homotopy and that homotopy equivalences or quasi-fibrations are replaced by more general families of morphisms. The resulting main theorem (Quillen's theorem B for bar constructions) of the paper covers a great number of existing results on homotopy limits and classifying spaces due to Bousfield-Kan, May, Milgram, Stasheff et al. As a matter of independent interest a general theory of geometrical realizations begun in the first part of the paper [the author, J. Pure Appl. Algebra 33, 163-207 (1984; Zbl 0542.57036)] is continued in the present paper.