Abstract

The aim of the 2018 Talbot workshop is to develop the theory of ∞-categories from first principles in a "model-independent" fashion, that is, using a common axiomatic framework that is satisfied by a variety of models. By the end of the week, we will also demonstrate that even "analytic" theorems about ∞-categories — which, in contrast to the "synthetic" proofs that may be interpreted simultaneously in many models, are proven using the combinatorics of a particular model — transfer across specified "change of model" functors to establish the same results for other equivalent models. In more detail, the "synthetic" theory is developed in any ∞-cosmos, which axiomatizes the universe in which ∞-categories live as objects. Here the term "∞-category" is used very broadly to mean any structure to which category theory generalizes in a homotopy coherent manner. Several models of (∞,1)-categories are ∞-categories in this sense but our ∞-categories also include certain models of (∞,n)-categories, and sliced versions of all of the above. This usage is meant to interpolate between the classical one, which refers to any variety of weak infinite-dimensional category, and the common one, which is often taken to mean quasi-categories or complete Segal spaces. Much of the development of the theory of ∞-categories takes place not in the full ∞-cosmos but in a quotient that we call the homotopy 2-category. The homotopy 2-category is a strict 2-category — like the 2-category of categories, functors, and natural transformations — and in this way proofs for ∞-categories closely resemble classical ones for ordinary categories except that the universal properties that characterize, e.g. when a functor between ∞-categories defines a cartesian fibration, are slightly weaker than in the classical case. Over the course of the workshop, we will define and develop the notions of equivalence and adjunction between ∞-categories, limits and colimits in ∞-categories, homotopy coherent adjunctions and monads borne by ∞-categories as a mechanism for universal algebra, cartesian and cocartesian fibrations and their groupoidal variants, the calculus of modules (aka profunctors or correspondences) between ∞-categories, Kan extensions, representable functors, the Yoneda lemma, and the Yoneda embedding.