Identifying the homotopy fiber of a map of semi-Segal spaces

TL;DR

This paper extends Quillen's theorems to semi-Segal spaces, providing a method to represent homotopy fibers via geometric realization of semi-simplicial spaces, advancing the understanding of their homotopical properties.

Contribution

It introduces a bi-semi-simplicial resolution for maps of semi-Segal spaces, enabling explicit representation of homotopy fibers.

Findings

Homotopy fiber of a map of semi-Segal spaces can be realized as a geometric realization.

Constructs a bi-semi-simplicial resolution analogous to non-unital topological categories.

Provides a new tool for analyzing homotopical properties of semi-Segal spaces.

Abstract

We prove a version of Quillen's theorems for a map of semi-Segal spaces. We construct a bi-semi-simplicial resolution similar to the one associated to a functor of non-unital topological categories. As a consequence we can represent the homotopy fiber of a map of semi-Segal spaces as the geometric realization of a certain semi-simplicial space.