Higher Structures in Homotopy Type Theory

Abstract

The intended model of the homotopy type theories used in Univalent Foundations is the ∞-category of homotopy types, also known as ∞-groupoids. The problem of higher structures is that of constructing the homotopy types needed for mathematics, especially those that aren't sets. The current repertoire of constructions, including the usual type formers and higher inductive types, suffice for many but not all of these. We discuss the problematic cases, typically those involving an infinite hierarchy of coherence data such as semi-simplicial types, as well as the problem of developing the meta-theory of homotopy type theories in Univalent Foundations. We also discuss some proposed solutions.

Two-sided cartesian fibrations of synthetic $(\infty,1)$-categories

“…generalizing Theorem 2.9, by 12 Φ b,e m : 1 for an illustration. By Q-cocartesianness of the lifts of Pvertical arrows in K one can show-analogously to the proof of Theorem 2.9-that the maps are quasi-inverse to one another.…”

“…Perspectives on directed type theories in connections with synthetic higher categories and/or fibrations are given in [12,31].…”

Synthetic fibered $(\infty,1)$-category theory

“…1.1.2 Synthetic higher category theory in simplicial homotopy type theory Outside the realm of Book HoTT, there do exist various approaches to reason type-theoretically about higher categorical structues, cf. [7] for an overview and discussion. As one solution, simplicial (homotopy) type theory (sHoTT) has been suggested by Riehl-Shulman [30].…”

“…of functions in U (viewed as type-theoretic fibrations 7 ), and families with U-small fibers, resp. Both these types naturally are fibered over U via the following maps:…”

“…Assuming universes with better structural properties-such as Segalness or directed univalence-would be fruitful for further considerations, but this is part of future work 7. The inhabitants of Fib(U) are just maps between arbitrary U-small types, but viewed as "U-small type theoretic fibrations over a U-small base".…”