Exploring the ∞-categorical semantics of homotopy type theory

This talk is meant to give an overview of the semantics of HoTT in (∞,1)-categories, tackling the key challenge of rigidification: the process of bridging homotopy-coherent notions phrased in models of higher categories such as quasicategories, and their type-theoretical counterparts that must account for the inherent strictness of type theory.
After quickly recalling the usual categorical semantics of extensional Martin-Löf type theory within locally cartesian closed category, the plan is to present some useful ideas to carry over some of the argument to the higher setting. Specifically, I will discuss the process of turning an elementary higher topos into a model of HoTT, and sketch important ideas used to prove the internal language conjecture for locally cartesian closed (∞,1)-categories.