Semisimplicial spaces

Abstract

This is an exposition of homotopical results on the geometric realisation of semisimplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The topics considered include: fibrancy conditions on topological categories; the effect on classifying spaces of freely adjoining units; approximate notions of units; Quillen’s Theorems A and B for nonunital topological categories; the effect on classifying spaces of changing the topology on the space of objects; the group-completion theorem.