A short course on ∞-categories
Moritz Groth
Abstract
Abstract. These are notes on the theory of ∞-categories building on a series of talks given by the author in Warsaw in January, 2010. The aim is to give a non-technical introduction to some of the main ideas of the theory in order to facilitate the digestion of the far more voluminous tomes due to Andre Joyal [Joy08b] and Jacob Lurie [Lur09e] where the theory is developed in full detail. Besides the basic ∞-categorical notions, we mention the Joyal and Bergner model structures which are two approaches to the theory of (∞, 1)-categories. We then treat the theory of (symmetric) monoidal ∞-categories as developed in [Lur09b, Lur09c] and introduce the notion of (commutative) algebra objects. We finish with a summary of Lurie’s treatment of spectra and the smash product from the perspective of ∞-categories [Lur09a] which allows us to give the