  In this paper we describe two ways on which cofibred categories give rise to
bisimplicial sets. The "fibred nerve" is a natural extension of Segal's
classical nerve of a category, and it constitutes an alternative simplicial
description of the homotopy type of the total category. If the fibration is
splitting, then one can construct the "cleaved nerve", a smaller variant which
emerges from a distinguished closed cleavage. We interpret some classical
theorems by Thomason and Quillen in terms of our constructions, and use the
fibred and cleaved nerve to establish new results on homotopy and homology of
small categories.
