On the Hierarchy of Univalent Universes: U(n) is not n-Truncated

In recent years, it has become clear that types in intensional MartinLöf Type Theory can be seen as spaces, alternatively to the traditional view as sets or propositions. This observation motivated Voevodsky's univalence axiom and the development of a whole branch of mathematics, known as Univalent Foundations and Homotopy Type Theory (HoTT). One of the most basic consequences of univalence is that the typetheoretic universe U0 does not have unique identity proofs. We show a generalization of this result: universe Un is not n-truncated, meaning that it has a non-trivial homotopical structure above dimension n. Our solution also answers the related (and so far open) problem of the Univalent Foundations Program in Princeton (2012/2013) of constructing a type that strictly has some high truncation level without using higher inductive types. Further, we present a construction for the dual notion, connectedness. Given a type, we construct (in plain Martin-Löf Type Theory without HoTT-axioms) a type that is equivalent to the original one above a given dimension n. We show that it is trivial on all lower dimensions in the sense of a predicate we de ne, and that this predicate is equivalent to saying that the type is n-connected if the theory supports truncations. We have fully formalized and veri ed our results within the dependently typed language and proof assistant Agda.