Define ∞-categories and ∞-categorical models in homotopical MLTT

Develop a type-theoretic definition of ∞-categories and, consequently, ∞-categorical models entirely within homotopical Martin-Löf type theory (MLTT), without departing from the features available in homotopical MLTT itself.

Background

The paper aims to study internal models of homotopical MLTT without assuming uniqueness of identity proofs (UIP). Existing approaches that internalize ∞-categorical structures typically rely on stronger ambient theories (e.g., two-level type theory or simplicial type theory with modalities). In contrast, the authors intentionally work within plain homotopical MLTT and adopt wild categories as an interim notion because a native definition of ∞-categories in this setting is not yet available.

This open problem concerns providing a definition of ∞-categories and ∞-categorical models fully internal to homotopical MLTT, thereby eliminating the need for additional strict equality or modal structure and enabling a unified internal model theory for homotopy type theory.

References

Since defining a type of \infty-categories and, a fortiori, \infty-categorical models in this theory remains an open problem, we take the notion of a wild, or precoherent higher, category with families as our starting point for "internal model of homotopical MLTT" (\Cref{sec:wild-cwfs}).

— 2-Coherent Internal Models of Homotopical Type Theory  (2503.05790 - Chen, 28 Feb 2025) in Section 1.3 (Contributions)

To avoid an infinite tower of coherence conditions (an unsolved problem in HoTT), we only consider diagrams over graphs, which are type-theoretic versions of free categories [HHF25, Section 3.2].

— On Left Adjoints Preserving Colimits in Homotopy Type Theory  (2608.28473 - Hart, 28 Aug 2026) in Section 4, p. 9