Univalent Foundations and Higher Inductive Types within Quasi-Categorical Topoi

Mathematical Applications of Science Fiction We establish a hyper-canonical framework for the in-terpretation of Martin-Löf Type Theory (MLTT) aug-mented with the Univalence Axiom and Higher In-ductive Types (HITs) within the semantic context ofGrothendieck (∞, 1)-topoi. By leveraging the Joyalmodel structure on simplicial sets and the theory oflocally Cartesian closed quasi-categories, we prove thatevery object classifier in an ∞-topos yields a univa-lent universe. Furthermore, we demonstrate that HITscorrespond to transfinite homotopy colimits preservedby the geometric realization functor. We rigorouslyverify the coherence of these constructions up to infi-nite homotopy levels, providing a complete soundnessproof for Homotopy Type Theory (HoTT) in the ∞-categorical setting.