A talk given on February 28, 2024 at the Graduate Center.

Abstract: One way to think about the language of Homotopy Type Theory [HoTT], is that it enforces that anything you can say is "up to homotopy". In particular, equality proofs are not strict, but rather carry the data of a particular [class of] deformation. In HoTT, all types have the structure of an infinity groupoid, and thus the language allows for conveniently working with certain infinitary structures synthetically. However, one of the most important and long standing open problems in the field is to analytically define infinitary structures such as semi-simplicial types [i.e. semi-simplicial sets "valued in" homotopy types]. The primary difficulty with this has been that as soon as you use the equality symbol in an attempted definition of such a structure, you fall into a pit of higher coherence issues such that infinitely many layers of higher coherences, with each depending on the proofs of all of the prior ones and growing exponentially in complexity, become required. In HoTT, therefore, one comes directly face-to-face with the core problems of homotopy coherent mathematics.

In this talk, we will construct semi-simplicial types in Displayed Type Theory [dTT], a fully semantically general homotopy type theory. Many of our main results are independent of type theory and will say something new and surprising about the homotopy theoretic notion of a classifier for semi-simplicial objects.

This talk is based on joint work with Michael Shulman. Reference: https://arxiv.org/abs/2311.18781

Slides:
http://www.sci.brooklyn.cuny.edu/~nos...