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.