Homotopy Type Theory
univalence axiom

The univalence axiom for a universe UU states that for all A,B:UA,B:U, the map

(A= UB)β†’(A≃B) (A=_U B) \to (A\simeq B)

defined by path induction, is an equivalence. So we have

(A= UB)≃(A≃B). (A=_U B) \simeq (A \simeq B).

A univalent universe inside a non-univalent universe

In a post to the Homotopy Type Theory Google Group, Peter LeFanu Lumsdaine wrote:

Let (x 0:X)(x_0:X) be any pointed type, and (𝒰,El)(\mathcal{U}, El) be a universe (with rules as I set out a couple of emails ago). Then X×𝒰X \times \mathcal{U} is again a universe, admitting all the same constructors as 𝒰\mathcal{U}: take

El(x,A)=El(A)El(x,A) = El(A),
(x,A)+ 𝒰(y,B)=(x 0,A+ 𝒰B)(x,A) +_\mathcal{U} (y,B) = (x_0, A +_\mathcal{U} B),

and so on; that is, constructor operations on (X×𝒰)(X \times \mathcal{U}) are constantly x 0x_0 on the first component, and mirror those of 𝒰\mathcal{U} on the second component.

Now if 𝒰\mathcal{U} is univalent, and XX has non-trivial Ο€ 0\pi_0 (e.g. X=S 1X=S^1), then 𝒰→(X×𝒰\mathcal{U} \rightarrow (X \times \mathcal{U}) gives a univalent universe sitting inside a non-univalent one (again, with the rules as I set out earlier).

Slightly more generally, given any cumulative pair of universes 𝒰 0→𝒰 1\mathcal{U}_0 \rightarrow \mathcal{U}_1, we can consider 𝒰 0β†’A×𝒰 1\mathcal{U}_0 \rightarrow A \times \mathcal{U}_1; this shows we can additionally have the smaller universe represented by an element of the larger one.

Mike Shulman added:

[N]ot only is X×𝒰X \times \mathcal{U} not univalent, it’s not even β€œunivalent on the image of 𝒰\mathcal{U}”, as was the case for the example in the groupoid model that I mentioned.

Revised on May 3, 2014 12:28:55 by Mike Shulman (108.247.151.163)