Infinity category theory from scratch

These lecture notes were written to accompany a mini course given at the 2015 Young Topologists' Meeting at Ecole Polytechnique Federale de Lausanne, videos of which can be found here. We use the terms ∞-categories and ∞-functors to mean the objects and morphisms in an ∞-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of "fibrant objects." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal spaces, Θn-spaces, and fibered versions of each of these are all ∞-categories in this sense. We show that the basic category theory of ∞-categories and ∞-functors can be developed from the axioms of an ∞-cosmos; indeed, most of the work is internal to a strict 2-category of ∞-categories, ∞-functors, and natural transformations. In the ∞-cosmos of quasi-categories, we recapture precisely the same category theory developed by Joyal and Lurie, although in most cases our definitions, which are 2-categorical rather than combinatorial in nature, present a new incarnation of the standard concepts.

In the first lecture, we define an ∞-cosmos and introduce its homotopy 2-category, the strict 2-category mentioned above. We illustrate the use of formal category theory to develop the basic theory of equivalences of and adjunctions between ∞-categories. In the second lecture, we study limits and colimits of diagrams taking values in an ∞-category and relate these concepts to adjunctions between ∞-categories. In the third lecture, we define comma ∞-categories, which satisfy a particular weak 2-dimensional universal property in the homotopy 2-category. We illustrate the use of comma ∞-categories to encode the universal properties of (co)limits and adjointness. Because comma ∞-categories are preserved by all functors of ∞-cosmoi and created by certain weak equivalences of ∞-cosmoi, these characterizations form the foundations for "model independence'' results. In the fourth lecture, we introduce (co)cartesian fibrations, a certain class of ∞-functors, and also consider the special case with groupoidal fibers. We then describe the calculus of modules between ∞-categories --- comma ∞-categories being the prototypical example --- and use this framework to state and prove the Yoneda lemma and develop the theory of pointwise Kan extensions of ∞-functors.