Sigma-closure of the universe of types admitting N-elimination

Determine whether the internal universe U in the sheaf topos of primitive recursive functions—whose elements are U0-small types X equipped with an elimination principle from the natural numbers object yN—is closed under dependent sum (Σ) types; likewise, determine whether the parametrized variant U that allows an arbitrary context Γ is closed under Σ-types. Establishing Σ-closure would permit setting the small universe U0 of the theory equal to U and would simplify the interpretation of Primitive Recursive Dependent Type Theory.

Background

To interpret Primitive Recursive Dependent Type Theory in the sheaf topos built over the category of arities and primitive recursive functions, the paper introduces internal universes intended to collect exactly those types into which the natural numbers object yN admits an induction (elimination) principle.

The authors define two candidates: U, consisting of U0-small types X possessing a non-contextual elimination from yN, and a parametrized variant U that allows an arbitrary context Γ. Ideally, the small universe U0 of the syntax would be interpreted as U, but this requires closure of U (and similarly of the parametrized variant) under Σ-types. Unable to establish this, they instead use a different universe of retracts of yN. Showing Σ-closure (or providing a counterexample) remains unresolved.