# Talbot 2018: Model-independent theory of $\infty$-categories

## Mentored by Emily Riehl and Dominic Verity

###
May 27-June 2, 2018

Government Camp, OR

**Topic:**
The aim of the 2018 Talbot workshop is to develop the theory of
$\infty$-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 $\infty$-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 $\infty$-cosmos, which axiomatizes the universe in which $\infty$-categories live as objects. Here the term "$\infty$-category" is used very broadly to mean any structure to which category theory generalizes in a homotopy coherent manner. Several models of $(\infty,1)$-categories are $\infty$-categories in this sense but our $\infty$-categories also include certain models of ($\infty$,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 $\infty$-categories takes place not in the full $\infty$-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 $\infty$-categories closely resemble classical ones for ordinary categories except that the universal properties that characterize, e.g. when a functor between $\infty$-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 $\infty$-categories, limits and colimits in $\infty$-categories, homotopy coherent adjunctions and monads borne by $\infty$-categories as a mechanism for universal algebra, cartesian and cocartesian fibrations and their groupoidal variants, the calculus of modules (aka profunctors or correspondences) between $\infty$-categories, Kan extensions, representable functors, the Yoneda lemma, and the Yoneda embedding.

A preview of many of these topics can be found in the lecture notes $\infty$-category theory from scratch. More detailed references will appear on the syllabus.

**Mentors: **
The 2018 Talbot workshop will be mentored by Prof. Emily Riehl of Johns Hopkins
University and Prof. Dominic Verity of Macquarie University.

**Format:** 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.

**Timeline:**
The 2018 Talbot workshop will take place from May 27 to June 2, 2018. Check back here in the next couple weeks for the application, which will be due in late January.

**Funding:** We cover all local expenses including lodging and food. We also have limited funding available for participants' travel costs.

**Who should apply:** 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 also especially encourage women and minorities to apply.

**Inclusiveness statement: **
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.

**Contact Information: **
Please email the organizers at talbotworkshop(at)gmail.com if you have any questions.