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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions of ordinals in homotopy type theory, and show how they relate to each other: A notation system based on Cantor normal forms, a refined notion of Brouwer trees (inductively generated by zero, successor and countable limits), and wellfounded extensional orders. For Cantor normal forms, most properties are decidable, whereas for wellfounded extensional transitive orders, most are undecidable. Formulations for Brouwer trees are usually partially decidable. We demonstrate that all three notions have properties expected of ordinals: their order relations, although defined differently in each case, are all extensional and wellfounded, and the usual arithmetic operations can be defined in each case. We connect these notions by constructing structure preserving embeddings of Cantor normal forms into Brouwer trees, and of these in turn into wellfounded extensional orders. We have formalised most of our results in cubical Agda.

Nicolai Kraus, Fredrik Nordvall Forsberg, and Chuangjie Xu. Connecting Constructive Notions of Ordinals in Homotopy Type Theory. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 70:1-70:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kraus_et_al:LIPIcs.MFCS.2021.70, author = {Kraus, Nicolai and Nordvall Forsberg, Fredrik and Xu, Chuangjie}, title = {{Connecting Constructive Notions of Ordinals in Homotopy Type Theory}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {70:1--70:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.70}, URN = {urn:nbn:de:0030-drops-145100}, doi = {10.4230/LIPIcs.MFCS.2021.70}, annote = {Keywords: Constructive ordinals, Cantor normal forms, Brouwer trees} }

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**Published in:** LIPIcs, Volume 175, 25th International Conference on Types for Proofs and Programs (TYPES 2019)

Cube categories are used to encode higher-dimensional categorical structures. They have recently gained significant attention in the community of homotopy type theory and univalent foundations, where types carry the structure of higher groupoids. Bezem, Coquand, and Huber [Bezem et al., 2014] have presented a constructive model of univalence using a specific cube category, which we call the BCH cube category.
The higher categories encoded with the BCH cube category have the property that all morphisms are invertible, mirroring the fact that equality is symmetric. This might not always be desirable: the field of directed type theory considers a notion of equality that is not necessarily invertible.
This motivates us to suggest a category of twisted cubes which avoids built-in invertibility. Our strategy is to first develop several alternative (but equivalent) presentations of the BCH cube category using morphisms between suitably defined graphs. Starting from there, a minor modification allows us to define our category of twisted cubes. We prove several first results about this category, and our work suggests that twisted cubes combine properties of cubes with properties of globes and simplices (tetrahedra).

Gun Pinyo and Nicolai Kraus. From Cubes to Twisted Cubes via Graph Morphisms in Type Theory. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{pinyo_et_al:LIPIcs.TYPES.2019.5, author = {Pinyo, Gun and Kraus, Nicolai}, title = {{From Cubes to Twisted Cubes via Graph Morphisms in Type Theory}}, booktitle = {25th International Conference on Types for Proofs and Programs (TYPES 2019)}, pages = {5:1--5:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-158-0}, ISSN = {1868-8969}, year = {2020}, volume = {175}, editor = {Bezem, Marc and Mahboubi, Assia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.5}, URN = {urn:nbn:de:0030-drops-130694}, doi = {10.4230/LIPIcs.TYPES.2019.5}, annote = {Keywords: homotopy type theory, cubical sets, directed equality, graph morphisms} }

Document

**Published in:** LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

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.

Thorsten Altenkirch, Paolo Capriotti, and Nicolai Kraus. Extending Homotopy Type Theory with Strict Equality. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{altenkirch_et_al:LIPIcs.CSL.2016.21, author = {Altenkirch, Thorsten and Capriotti, Paolo and Kraus, Nicolai}, title = {{Extending Homotopy Type Theory with Strict Equality}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {21:1--21:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-022-4}, ISSN = {1868-8969}, year = {2016}, volume = {62}, editor = {Talbot, Jean-Marc and Regnier, Laurent}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.21}, URN = {urn:nbn:de:0030-drops-65612}, doi = {10.4230/LIPIcs.CSL.2016.21}, annote = {Keywords: homotopy type theory, coherences, strict equality, homotopy type system} }

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**Published in:** LIPIcs, Volume 39, 20th International Conference on Types for Proofs and Programs (TYPES 2014)

In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least unit, sigma, pi, and identity types), we define the type of coherently constant functions from A to B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over omega^op. Our main result is that, if the category further has propositional truncations and satisfies function extensionality, the type of coherently constant function is equivalent to the type ||A|| -> B. If B is an n-type for a given finite n, the tower of coherence conditions becomes finite and the requirement of nontrivial Reedy limits vanishes. The whole construction can then be carried out in standard syntactical homotopy type theory and generalises the universal property of the truncation. This provides a way to define functions ||A|| -> B if B is not known to be propositional, and it streamlines the common approach of finding a propositional type Q with A -> Q and Q -> B.

Nicolai Kraus. The General Universal Property of the Propositional Truncation. In 20th International Conference on Types for Proofs and Programs (TYPES 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 39, pp. 111-145, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kraus:LIPIcs.TYPES.2014.111, author = {Kraus, Nicolai}, title = {{The General Universal Property of the Propositional Truncation}}, booktitle = {20th International Conference on Types for Proofs and Programs (TYPES 2014)}, pages = {111--145}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-88-0}, ISSN = {1868-8969}, year = {2015}, volume = {39}, editor = {Herbelin, Hugo and Letouzey, Pierre and Sozeau, Matthieu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2014.111}, URN = {urn:nbn:de:0030-drops-54944}, doi = {10.4230/LIPIcs.TYPES.2014.111}, annote = {Keywords: coherence conditions, propositional truncation, Reedy limits, universal property, well-behaved constancy} }

Document

**Published in:** LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 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.

Paolo Capriotti, Nicolai Kraus, and Andrea Vezzosi. Functions out of Higher Truncations. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 359-373, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{capriotti_et_al:LIPIcs.CSL.2015.359, author = {Capriotti, Paolo and Kraus, Nicolai and Vezzosi, Andrea}, title = {{Functions out of Higher Truncations}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, pages = {359--373}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.359}, URN = {urn:nbn:de:0030-drops-54257}, doi = {10.4230/LIPIcs.CSL.2015.359}, annote = {Keywords: homotopy type theory, truncation elimination, constancy on loop spaces} }

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