1, Infinity category theory from scratch, Emily Riehl's lecture 1

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, iterated complete Segal spaces, and fibered versions of each of these are all all ∞-categories in this sense. In joint work with Dominic Verity, we show that the basic category theory of ∞-categories and ∞-functors can be developed only in reference to 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 classical concept.

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 reflected 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 their groupoidal variants. 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 along ∞-functors.