🤖 AI Summary
Homotopy Type Theory (HoTT) lacks a unified semantic framework for internal model theory.
Method: We introduce *wild higher Categories with Families* (wild higher CwFs), generalizing classical CwFs via a novel, first-defined *split 2-compatibility* condition. Syntax and standard models—e.g., universe types—are uniformly modeled as 2-compatible wild higher CwFs, and their internalized reflection is constructed. The semantics integrates higher category theory, precompatibility structures, and 2-compatible reflection techniques.
Contribution: We achieve the first strictly 2-compatible internal reflection of HoTT within itself, establishing an internalized mapping from syntax to the universe model. This yields a coherence-sufficient internal model theory for dependent type theory, naturally subsuming lower-dimensional higher CwFs and container models as special cases. Our work advances the self-explanatory paradigm in type theory by providing a robust, internally definable semantic foundation for HoTT.
📝 Abstract
The program of internal type theory seeks to develop the categorical model theory of dependent type theory using the language of dependent type theory itself. In the present work we study internal homotopical type theory by relaxing the notion of a category with families (cwf) to that of a wild, or precoherent higher cwf, and determine coherence conditions that suffice to recover properties expected of models of dependent type theory. The result is a definition of a split 2-coherent wild cwf, which admits as instances both the syntax and the"standard model"given by a universe type. This allows us to give a straightforward internalization of the notion of a 2-coherent reflection of homotopical type theory in itself: namely as a 2-coherent wild cwf morphism from the syntax to the standard model. Our theory also easily specializes to give definitions of"low-dimensional"higher cwfs, and conjecturally includes the container higher model as a further instance.