nLab
propositional resizing

Idea

In homotopy type theory, given a universe, $\mathcal{U}$, we may define a type of mere propositions within that universe, $\sum_{A: \mathcal{U}} isProp(A)$. For universes, $\mathcal{U}_i$ and $\mathcal{U}_{i+1}$, we have that $A: \mathcal{U}_i$ entails $A:\mathcal{U}_{i+1}$. We thus have a natural map

$$Prop_{\mathcal{U}_i} \to Prop_{\mathcal{U}_{i+1}}.$$

The axiom of proposition resizing states that this map is an equivalence. Any mere proposition in the larger universe may be resized to an equivalent mere proposition in the smaller universe.

The axiom is a form of impredicativity for mere propositions, and by avoiding its use, the type theory is said to remain predicative.

References

• Section 3.5 of The HoTT Book