A remark on ?homotofp fibrations?

Abstract

We define a category TopF of “homotopy fibrations with fibre F” (or rather “maps with homotopy fibre F”) and show that this category is closed under certain colimits and “homotopy colimits”. It follows that the geometric realization of a semisimplicial object in TopF is again in TopF. As a corollary we show that for a homotopy everything H-space A*(i.e. a (special) Γ-space in the sense of G. Segal (s.[9],[10])) with homotopy inverse the loop space of the classifying space of A* is homotopy equivalent (not only weakly s. [9],[iO])to A1 even without assuming that all spaces involved have the homotopy type of CW-complexes (compare [8]).