Simplicial sets in topology, category theory, and beyond

TL;DR

This paper introduces simplicial sets, highlighting their development from algebraic topology and category theory, and emphasizes their growing importance in various fields with a guide for new users.

Contribution

It provides an accessible overview of simplicial sets, their evolution, and practical guidance for researchers adopting higher categorical methods.

Findings

Clarifies the role of simplicial sets in modern mathematics

Provides a user-friendly introduction and overview

Highlights their increasing applications beyond traditional fields

Abstract

The notion of a simplicial set originated in algebraic topology, and has also been utilized extensively in category theory, but until relatively recently was not used outside of those fields. However, with the increasing prominence of higher categorical methods in a wide range of applications, it is important for researchers in a range of fields to have a good working knowledge of them. This paper is intended as an introduction to simplicial sets, both as an overview of their development from other concepts, and as a user's guide for someone wanting to read modern literature that makes use of them.