Functions out of higher truncations

In homotopy type theory, the truncation operator ||-||_n (for a number n ≥ -1) is often useful if one does not care about the higher structure of a type and wants to avoid coherence problems. However, its elimination principle only allows to eliminate into n-types, which makes it hard to construct functions ||A||_n → B if B is not an n-type. This makes it desirable to derive more powerful elimination theorems. We show a first general result: If B is an (n + 1)-type, then functions ||A||_n → B correspond exactly to functions A → B which are constant on all (n+1)-st loop spaces. We give one "elementary" proof and one proof that uses a higher inductive type, both of which require some effort. As a sample application of our result, we show that we can construct "set-based" representations of 1-types, as long as they have "braided" loop spaces. The main result with one of its proofs and the application have been formalised in Agda.