Elements of ∞-Category Theory

Emily Riehl, Johns Hopkins University

Zoom link: https://princeton.zoom.us/j/96282936122

Passcode: 998749

Confusingly for the uninitiated, experts in higher category theory such as André Joyal, Jacob Lurie, Charles Rezk, and Carlos Simpson, make use of different definitions of an ∞-category, and theorems in the ∞-categorical literature are often proven "analytically", in reference to the combinatorial specifications of a particular model. In this talk, we present a new point of view on the foundations of ∞-category theory, which allows us to develop the basic theory of ∞-categories --- adjunctions, limits and colimits, co/cartesian fibrations, and pointwise Kan extensions --- "synthetically" starting from axioms that describe the infinite-dimensional category in which ∞-categories live as objects. 

We show that theorems proven in this manner specialize to the standard models of ∞-categories and demonstrate that the ∞-category theory being developed is "model-independent", i.e., invariant under change of model. Time permitting, we also describe a formal language with the feature that any statement about ∞-categories that is expressible in that language is also invariant under change of model, regardless of whether it is proven through synthetic or analytic techniques.

This is joint work with Dominic Verity.