Interpreting type theory in a quasicategory: a Yoneda approach

Abstract: We make use of a higher version of the Yoneda embedding to construct, from a given quasicategory, a simplicially enriched category, as a subcategory of a well-behaved simplicial model category, whose simplicial nerve is equivalent to the former quasicategory. We then show that, when the quasicategory is locally cartesian closed, it is possible to further endow such a simplicial category with enough structure for it to provide a model of Martin-L\"of type theory. This correspondence restricts so that elementary higher topoi are seen to model homotopy type theory.