A short course on $\infty$-categories

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.

major revision, environment `perspective' added to include comments on larger picture, many references added, submitted, 77 pages

Cited by in corpus (10)

• Kato-Nakayama spaces, infinite root stacks, and the profinite homotopy type of log schemes
• Gluing restricted nerves of $\infty$-categories
• Tannaka duality and stable infinity-categories
• t-structures are normal torsion theories
• Quasi-Categories vs. Segal Spaces: Cartesian Edition
• Betti realization of varieties defined by formal Laurent series
• On autoequivalences of the (\infty, 1)-category of \infty-operads
• Mixed motives and quotient stacks: Abelian varieties
• Introduction to Abstract Homotopy Theory
• Higher Lie and Leibniz algebras