Functions out of Higher Truncations
Paper i proceeding, 2015

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.

constancy on loop spaces

homotopy type theory

truncation elimination


Paolo Capriotti

University of Nottingham

Nicolai Kraus

University of Nottingham

Andrea Vezzosi

Chalmers, Data- och informationsteknik, Datavetenskap

24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

1868-8969 (ISSN)

Vol. 41 359-373
978-3-939897-90-3 (ISBN)


Grundläggande vetenskaper


Datavetenskap (datalogi)





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