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.
@InProceedings{capriotti_et_al:LIPIcs.CSL.2015.359, author = {Capriotti, Paolo and Kraus, Nicolai and Vezzosi, Andrea}, title = {{Functions out of Higher Truncations}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, pages = {359--373}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.359}, URN = {urn:nbn:de:0030-drops-54257}, doi = {10.4230/LIPIcs.CSL.2015.359}, annote = {Keywords: homotopy type theory, truncation elimination, constancy on loop spaces} }
Feedback for Dagstuhl Publishing