Talbot 2018

The goal 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.

A detailed syllabus with descriptions of each talk can be found here. A preview of many of these topics can be found in the lecture notes ∞-category theory from scratch. The mentors are also writing a book entitled Elements of ∞-Category Theory (link is to their current draft).

Pre-Talbot seminars, organized by departments in the run-up to Talbot, discuss prerequisites and other material related to the workshop. For 2018, there was a pre-Talbot course at Johns Hopkins University (taught by Emily Riehl).

 The workshop discussions will have an expository character and most of the talks will be given by participants. The afternoon schedule will be kept clear for informal discussions and collaborations. The workshop will take place in a communal setting, with participants sharing living space and cooking and cleaning responsibilities.

We cover all local expenses, including lodging and food. We also have limited funding available for participant travel costs.

Talbot is meant to encourage collaboration among young researchers, particularly graduate students. To this end, the workshop aims to gather participants with a diverse array of knowledge and interests, so applicants need not be an expert in the field. In particular, students at all levels of graduate education are encouraged to apply. Our decisions are based not on applicants' credentials but on our assessment of how much they would benefit from the workshop. As we are committed to promoting diversity in mathematics, we especially encourage women and minorities to apply.

In accordance with the Statement of Inclusiveness, this workshop will be open to everybody, regardless of race, sex, religion, national origin, sexual orientation, gender identity, disability, age, pregnancy, immigration status, or any other aspect of identity. We are committed to ensuring that the Talbot Workshop is a supportive, inclusive, and safe environment for all participants, and that all participants are treated with dignity and respect.