Grothendieck fibrations and classifying spaces

Grothendieck fibrations have played an important role in homotopy theory. Among others, theywereused byThomason to describehomotopy colimits of small categories and byQuillen to derive long exact sequences of higher K-theory groups. We construct simplicial objects, namely the fibred and the cleaved nerve, to characterize the homotopy type of a Grothendieck fibration by using the additional structure. From these, we derive long exact sequences of homotopy groups and spectral sequences for homology groups, establishing new results and placing those of Thomason and Quillen into our framework.