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Eliminating Reversals from Cubical Type Theories

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Title: Eliminating Reversals from Cubical Type Theories
Authors: Cavallo, Evan, Sattler, Christian
Contributors: Evan Cavallo and Christian Sattler
Publisher Information: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Publication Year: 2026
Collection: DROPS - Dagstuhl Research Online Publication Server (Schloss Dagstuhl - Leibniz Center for Informatics )
Subject Terms: Dependent type theory, univalence, cubical type theory
Description: Cubical type theories are designed around an abstract unit interval from which types of paths, used to represent equalities, are defined. Varying the operations available on this interval yields different type theories. A reversal is an involutive operator on the interval that swaps its two endpoints. We show that for cubical type theories with self-dual interval theories, such as the minimal theory of two endpoints or the theory of a bounded distributive lattice, the extension of the theory with a reversal that internalizes the duality is a conservative extension. The key tool is a "twist construction": the product of an interval and its dual is again an interval with a reversal given by swapping coordinates. Our conservativity result applies to "opaque" cubical type theories, without strict equations reducing the filling operator at concrete type formers or eliminators from higher inductive types at path constructors. Using the same twist construction, we also construct models of strict cubical type theory with reversals in categories of cubical sets without reversals. We thereby give the first model of a theory with reversals whose homotopy theory corresponds to that of topological spaces.
Document Type: article in journal/newspaper
conference object
File Description: application/pdf
Language: English
Relation: Is Part Of LIPIcs, Volume 380, 41st Annual Symposium on Logic in Computer Science (LICS 2026); https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.LICS.2026.27
DOI: 10.4230/LIPIcs.LICS.2026.27
Availability: https://doi.org/10.4230/LIPIcs.LICS.2026.27
https://nbn-resolving.org/urn:nbn:de:0030-drops-268148
https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.LICS.2026.27
Rights: https://creativecommons.org/licenses/by/4.0/legalcode
Accession Number: edsbas.2CAC4D40
Database: BASE
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  – Url: https://doi.org/10.4230/LIPIcs.LICS.2026.27#
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  Data: Eliminating Reversals from Cubical Type Theories
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  Data: <searchLink fieldCode="AR" term="%22Cavallo%2C+Evan%22">Cavallo, Evan</searchLink><br /><searchLink fieldCode="AR" term="%22Sattler%2C+Christian%22">Sattler, Christian</searchLink>
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  Data: Evan Cavallo and Christian Sattler
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  Data: Cubical type theories are designed around an abstract unit interval from which types of paths, used to represent equalities, are defined. Varying the operations available on this interval yields different type theories. A reversal is an involutive operator on the interval that swaps its two endpoints. We show that for cubical type theories with self-dual interval theories, such as the minimal theory of two endpoints or the theory of a bounded distributive lattice, the extension of the theory with a reversal that internalizes the duality is a conservative extension. The key tool is a "twist construction": the product of an interval and its dual is again an interval with a reversal given by swapping coordinates. Our conservativity result applies to "opaque" cubical type theories, without strict equations reducing the filling operator at concrete type formers or eliminators from higher inductive types at path constructors. Using the same twist construction, we also construct models of strict cubical type theory with reversals in categories of cubical sets without reversals. We thereby give the first model of a theory with reversals whose homotopy theory corresponds to that of topological spaces.
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