This paper explores the relationship amongst the various simplicial and
pseudosimplicial objects characteristically associated to any bicategory
. It
proves the fact that the geometric realizations of all of these possible candidate “nerves
of
” are
homotopy equivalent. Any one of these realizations could therefore be taken as
the classifying
space
of the bicategory. Its other major result proves a direct extension of Thomason’s
“Homotopy Colimit Theorem” to bicategories: When the homotopy colimit
construction is carried out on a diagram of spaces obtained by applying the
classifying space functor to a diagram of bicategories, the resulting space has the
homotopy type of a certain bicategory, called the “Grothendieck construction on the
diagram”. Our results provide coherence for all reasonable extensions to
bicategories of Quillen’s definition of the “classifying space” of a category as the
geometric realization of the category’s Grothendieck nerve, and they are
applied to monoidal (tensor) categories through the elemental “delooping”
construction.