Localizing infinity-categories with hypercovers

TL;DR

This paper provides a new method to describe the mapping spaces in localizations of infinity-categories using spans and constructs a Segal space model for these localizations, enhancing understanding of their structure.

Contribution

It introduces a novel description of localization mapping spaces via spans and constructs a Segal space model for the localization of infinity-categories.

Findings

Mapping spaces are described as group completions of span categories.

Span categories serve as the mapping objects in an $( abla, 2)$-category.

A Segal space model for the localization is established after Kan fibrant replacement.

Abstract

Given an $\infty$-category with a set of weak equivalences which is stable under pullback, we show that the mapping spaces of the corresponding localization can be described as group completions of $\infty$-categories of spans. Furthermore, we show how these $\infty$-categories of spans are the mapping objects of an $(\infty, 2)$-category, which yields a Segal space model for the localization after a Kan fibrant replacement.