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Extending Homotopy Type Theory with Strict Equality

Altenkirch, Thorsten; Capriotti, Paolo; Kraus, Nicolai


Paolo Capriotti


In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is difficult and often impossible to handle towers of coherences. To address this, we propose a 2-level theory which features both strict and weak equality. This can essentially be represented as two type theories: an ``outer'' one, containing a strict equality type former, and an ``inner'' one, which is some version of HoTT. Our type theory is inspired by Voevodsky's suggestion of a homotopy type system (HTS) which currently refers to a range of ideas. A core insight of our proposal is that we do not need any form of equality reflection in order to achieve what HTS was suggested for. Instead, having unique identity proofs in the outer type theory is sufficient, and it also has the meta-theoretical advantage of not breaking decidability of type checking. The inner theory can be an easily justifiable extensions of HoTT, allowing the construction of ``infinite structures'' which are considered impossible in plain HoTT. Alternatively, we can set the inner theory to be exactly the current standard formulation of HoTT, in which case our system can be thought of as a type-theoretic framework for working with ``schematic'' definitions in HoTT. As demonstrations, we define semi-simplicial types and formalise constructions of Reedy fibrant diagrams.


Altenkirch, T., Capriotti, P., & Kraus, N. (2016). Extending Homotopy Type Theory with Strict Equality. In 25th EACSL Annual Conference on Computer Science Logic. , (21:1-21:17).

Conference Name 25th EACSL Annual Conference on Computer Science Logic, CSL 2016.
Conference Location Marseille, France
Start Date Aug 29, 2016
End Date Sep 1, 2016
Acceptance Date Jun 11, 2016
Online Publication Date Aug 29, 2016
Publication Date Aug 29, 2016
Deposit Date Aug 9, 2016
Publicly Available Date Aug 29, 2016
Publisher Schloss Dagstuhl - Leibniz-Zentrum für Informatik
Peer Reviewed Peer Reviewed
Volume 2016-Aug
Pages 21:1-21:17
Series Title Leibniz International Proceedings in Informatics (LIPIcs)
Series Number 62
Series ISSN 1868-8969
Book Title 25th EACSL Annual Conference on Computer Science Logic
ISBN 9783959770224
Keywords homotopy type theory, coherences, strict equality, homotopy type system
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