The simplicial bundle of a CW fibration

Abstract

Suppose we are given a fibration p : E → B p:E \to B over a connected base, with both base and fibre having the homotopy type of CW complexes. We construct a fibre bundle over B B fibre homotopy equivalent to the given fibration and with fibre a simplicial complex. Further, the transformations of the fibre arising from the transition functions of this bundle are simplicial maps. From this, we deduce that the weak spectral sequence constructor axioms are sufficient to determine the Serre spectral sequence of a CW fibration.