A short course on $\mathrm{\infty }$-categories

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