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Functions out of Higher Truncations Capriotti, Paolo Kraus, Nicolai Vezzosi, Andrea homotopy type theory truncation elimination constancy on loop spaces In homotopy type theory, the truncation operator ||-||n (for a number n greater or equal to -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 that 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Paolo Capriotti and Nicolai Kraus and Andrea Vezzosi 2015 Is Part Of LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CSL.2015.359 urn:nbn:de:0030-drops-54257 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.359 eng https://creativecommons.org/licenses/by/3.0/legalcode