Title: Semi-simplicial spaces,
Abstract: This is an exposition of homotopical results on the geometric realization of
semi-simplicial 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 non-unital topological categories; the
effect on classifying spaces of changing the topology on the space of objects;
the Group-Completion Theorem.