The **univalence axiom** for a [[universe]] $U$ states that for all $A,B:U$, the map $$ (A=_U B) \to (A\simeq B) $$ defined by [path induction](/nlab/show/inductive+type#PathInduction), is an equivalence. So we have $$ (A=_U B) \simeq (A \simeq B). $$ ## A univalent universe inside a non-univalent universe In a post to the [Homotopy Type Theory Google Group](https://groups.google.com/forum/#!forum/homotopytypetheory), Peter LeFanu Lumsdaine [wrote](https://groups.google.com/d/msg/homotopytypetheory/3wp32xV7OX0/2HfAnBt44XoJ): > Let $(x_0:X)$ be any [[loop space|pointed type]], and $(\mathcal{U}, El)$ be a universe (with rules as I set out a couple of emails ago). Then $X \times \mathcal{U}$ is again a universe, admitting all the same constructors as $\mathcal{U}$: take > $El(x,A) = El(A)$, > $(x,A) +_\mathcal{U} (y,B) = (x_0, A +_\mathcal{U} B)$, > and so on; that is, constructor operations on $(X \times \mathcal{U})$ are constantly $x_0$ on the first component, and mirror those of $\mathcal{U}$ on the second component. > Now if $\mathcal{U}$ is univalent, and $X$ has non-trivial $\pi_0$ (e.g. $X=S^1$), then $\mathcal{U} \rightarrow (X \times \mathcal{U}$) 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 $\mathcal{U}_0 \rightarrow \mathcal{U}_1$, we can consider $\mathcal{U}_0 \rightarrow A \times \mathcal{U}_1$; this shows we can additionally have the smaller universe represented by an element of the larger one. [[Mike Shulman]] [added](https://groups.google.com/d/msg/homotopytypetheory/3wp32xV7OX0/kIapR0lo1UEJ): > [N]ot only is $X \times \mathcal{U}$ not univalent, it's not even "univalent on the image of $\mathcal{U}$", as was the case for the example in the groupoid model that I mentioned. [[!redirects univalence]] [[!redirects univalent]] [[!redirects univalent universe]] [[!redirects univalent universes]]