Synthetic Fibered (∞,1)-Category Theory

As an alternative to set-theoretic foundations, homotopy type theory is a logical system which allows for reasoning about homotopical structures in an invariant and more intrinsic way.

Specifically, for the case of higher categories there exists an extended framework, due to Riehl-Shulman, to develop (∞,1)-category theory synthetically. The idea is to work internally to simplicial spaces, where one can define predicates witnessing that a type is (complete) Segal. This had also independently been suggested by Joyal.

Generalizing Riehl-Shulman’s previous work on synthetic discrete fibrations, we discuss the case of synthetic cartesian fibrations in this setting, leading up to a 2-Yoneda Lemma. In developing this theory, we are led by Riehl–Verity’s model-independent higher category theory, therefore adapting results from ∞-cosmos theory to the type-theoretic setting. Time permits, we’ll briefly point out generalizations to the two-sided case.

In fact, by Shulman’s recent work on strict universes, the theory at hand has semantics in Reedy fibrant diagrams in an arbitrary (∞,1)-topos, so all type-theoretically formulated results semantically translate to statements about internal (∞,1)-categories. 

This is based on joint work with Ulrik Buchholtz (https://arxiv.org/abs/2105.01724) and the speaker's recent PhD thesis.