Higher homotopies in a hierarchy of univalent universes

Journal article, 2015

For Martin-Löf type theory with a hierarchy U_0 : U_1 : U_2 : … of univalent universes, we show that U_n is not an n-type. Our construction also solves the problem of finding a type that strictly has some high truncation level without using higher inductive types. In particular, U_n is such a type if we restrict it to n-types.

We have fully formalized and verified our results within the dependently typed language and proof assistant Agda.