%0 Journal Article
%T A short course on $\infty$-categories
%A Moritz Groth
%J Mathematics
%D 2010
%I arXiv
%X 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.
%U http://arxiv.org/abs/1007.2925v2