Infinity category theory from scratch

Abstract

We use the terms "$\infty$-categories" and "$\infty$-functors" to mean the objects and morphisms in an "$\infty$-cosmos." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal spaces, $\theta_n$-spaces, and fibered versions of each of these are all $\infty$-categories in this sense. We show that the basic category theory of $\infty$-categories and $\infty$-functors can be developed from the axioms of an $\infty$-cosmos; indeed, most of the work is internal to a strict 2-category of $\infty$-categories, $\infty$-functors, and natural transformations. In the $\infty$-cosmos of quasi-categories, we recapture precisely the same 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 $\infty$-cosmos and introduce its "homotopy 2-category," using formal category theory to define and study equivalences and adjunctions between $\infty$-categories. In the second lecture, we study (co)limits of diagrams taking values in an $\infty$-category and the relationship between (co)limits and adjunctions. In the third lecture, we introduce comma $\infty$-categories, which are used to encode the universal properties of (co)limits and adjointness and prove "model independence" results. In the fourth lecture, we introduce (co)cartesian fibrations, describe the calculus of "modules" between $\infty$-categories, and use this framework to prove the Yoneda lemma and develop the theory of pointwise Kan extensions of $\infty$-functors.