# Semi-simplicial spaces

## TL;DR

This paper explores homotopical properties of semi-simplicial spaces and their implications for classifying spaces of topological categories, including foundational results and theorems like Quillen's A and B.

## Contribution

It provides new foundational results on classifying spaces of possibly non-unital topological categories using semi-simplicial space techniques.

## Key findings

- Fibrancy conditions influence classifying space properties
- Adjoining units affects the homotopy type of classifying spaces
- Quillen's Theorems A and B are extended to non-unital categories

## 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.