Path spaces of higher inductive types in homotopy type theory

Nicolai Kraus, Jakob von Raumer

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

5 Citations (Scopus)
284 Downloads (Pure)


The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman [1]), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman [2]). We provide a formalization of the main technical results in the proof assistant Lean.
Original languageEnglish
Title of host publication2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
PublisherIEEE Computer Society Press
Number of pages13
ISBN (Electronic)9781728136080, 9781728136073
ISBN (Print)9781728136097
Publication statusPublished - 27 Jun 2019
EventThirty-Fourth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2019) - Vancouver, Canada
Duration: 24 Jun 201927 Jun 2019


ConferenceThirty-Fourth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2019)

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