The universe U n is not an n - type Nicolai Kraus Christian Sattler

We show that, in Martin-Löf Type Theory with a hierarchy of univalent universes, the universe Un is not an n-type. At the same time, our construction leads to an n+1-type that is not an n-type, solving one of the open problems of the Univalent Foundations Program in Princeton. A formalized version of the proof is available. Background to these Notes One of the most basic consequences of the univalence axiom is that the universe of types is not a set, i.e. does not have unique identity proofs. This is proved by arguing that U0 has a more complicated structure than a suitably chosen type that lives in it (e.g., the boolean type 2). Hence, it is plausible to expect that the next universe U1 is not a groupoid, i.e. its path spaces are not in general sets, but proving this turns out to be already surprisingly di cult. At the Univalent Foundations Program in Princeton, the rst-named author presented a proof that Un is not an n-type.[1] The argument works in basic Martin-Löf Type Theory with Σ-, Πand identity types. We require a hierarchy of universes, indexed over the natural numbers, and the univalence axiom, but no other features that are sometimes used in homotopy type theory (such as truncations and higher inductive types). We want to present this proof in more detail here. We have also formalized that proof in Agda,[2] but hope that these notes might be more enlightening. Regarding notation, we try to stay very close to the standard reference.[3] For the special cases of n ≡ −2,−1, 0, instead of is-n-type(X), we will write isContr(X), isProp(X), isSet(X), respectively, to maintain readability. The problems of a naive approach The standard proof of ¬isSet(U0) goes as follows. Suppose isSet(U0). Then, by de nition of is-(·)-type, we have isProp(2 = 2), where 2 is the boolean type with elements 12 and 02. However, there are two automorphisms on 2, and by univalence, they respectively induce equality proofs refl2 and swap. In formulae: isSet(U0) =⇒ isProp(2 = 2) =⇒ refl2 = swap =⇒ 12 = 02. Let us now try to prove ¬is-1-type(U1) in a similar way, choosing two inhabitants of U1 that seem complicated enough: is-1-type(U1) =⇒ isSet(U0 = U0) (1)