Functions out of higher truncations

Paolo Capriotti, Nicolai Kraus*, Andrea Vezzosi

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Citations (Scopus)


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.

Original languageEnglish
Title of host publication24th EACSL Annual Conference on Computer Science Logic (CSL 2015)
EditorsStephan Kreutzer
PublisherSchloss Dagstuhl
ISBN (Electronic)9783939897903
Publication statusPublished - 1 Sept 2015
Event24th EACSL Annual Conference on Computer Science Logic, CSL 2015 - Berlin, Germany
Duration: 7 Sept 201510 Sept 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference24th EACSL Annual Conference on Computer Science Logic, CSL 2015


  • Constancy on loop spaces
  • Homotopy type theory
  • Truncation elimination

ASJC Scopus subject areas

  • Software

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