A Whirlwind Tour of the World of $(\infty,1)$-categories

TL;DR

This paper provides an accessible overview of $( abla,1)$-categories, explaining their intuition, relevance, generalizations from ordinary categories, and applications, aimed at guiding readers through higher category theory.

Contribution

It offers a conceptual introduction to higher category theory, focusing on $( abla,1)$-categories, their motivation, and their role within the broader mathematical landscape.

Findings

Clarifies the intuition behind $( abla,1)$-categories

Explains how concepts from ordinary category theory extend to higher categories

Provides references and context for further exploration

Abstract

This introduction to higher category theory is intended to a give the reader an intuition for what $(\infty,1)$-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from ordinary category theory generalize to this new setting, and what uses people have put the theory to. It is a rough guide to a vast terrain, focuses on ideas and motivation, omits almost all proofs and technical details, and provides many references.