Abstract
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.
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