Path categories and quasi-categories

TL;DR

This paper introduces the homotopy theory of quasi-categories, characterizing weak equivalences via induced equivalences on associated groupoids, which serve as higher homotopy group analogs.

Contribution

It provides a foundational framework for understanding homotopy theory in quasi-categories using groupoids to replace higher homotopy groups.

Findings

Weak equivalences characterized by induced equivalences on groupoids

Groupoids effectively replace higher homotopy groups in quasi-category theory

Establishes a foundational approach to quasi-category homotopy theory

Abstract

This paper gives an introduction to the homotopy theory of quasi-categories. Weak equivalences between quasi-categories are characterized as maps which induce equivalences on a naturally defined system of groupoids. These groupoids effectively replace higher homotopy groups in quasi-category homotopy theory.