A short course on $\infty$-categories

TL;DR

This paper provides a non-technical overview of $$-categories, introducing foundational concepts and key structures like monoidal, stable, and algebraic $$-categories, to facilitate understanding of advanced topics in higher category theory.

Contribution

It offers an accessible survey of $$-categories, connecting foundational ideas with applications in spectra and derived algebraic geometry, based on the work of Joyal and Lurie.

Findings

Introduces basic $$-categorical notions

Discusses model structures for $(,1)$-categories

Explores applications in spectra and derived geometry

Abstract

In this short survey we give a non-technical introduction to some main ideas of the theory of $\infty$-categories, hopefully facilitating the digestion of the foundational work of Joyal and Lurie. Besides the basic $\infty$-categorical notions leading to presentable $\infty$-categories, we mention the Joyal and Bergner model structures organizing two approaches to a theory of $(\infty,1)$-categories. We also discuss monoidal $\infty$-categories and algebra objects, as well as stable $\infty$-categories. These notions come together in Lurie's treatment of the smash product on spectra, yielding a convenient framework for the study of $\mathbb{A}_\infty$-ring spectra, $\mathbb{E}_\infty$-ring spectra, and Derived Algebraic Geometry.