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**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 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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**Published in:** LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

We prove a conjecture about the constructibility of conductive types - in the principled form of indexed M-types - in Homotopy Type Theory. The conjecture says that in the presence of inductive types, coinductive types are derivable. Indeed, in this work, we construct coinductive types in a subsystem of Homotopy Type Theory; this subsystem is given by Intensional Martin-Löf type theory with natural numbers and Voevodsky's Univalence Axiom. Our results are mechanized in the computer proof assistant Agda.

Benedikt Ahrens, Paolo Capriotti, and Régis Spadotti. Non-Wellfounded Trees in Homotopy Type Theory. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 17-30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{ahrens_et_al:LIPIcs.TLCA.2015.17, author = {Ahrens, Benedikt and Capriotti, Paolo and Spadotti, R\'{e}gis}, title = {{Non-Wellfounded Trees in Homotopy Type Theory}}, booktitle = {13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)}, pages = {17--30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-87-3}, ISSN = {1868-8969}, year = {2015}, volume = {38}, editor = {Altenkirch, Thorsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.17}, URN = {urn:nbn:de:0030-drops-51522}, doi = {10.4230/LIPIcs.TLCA.2015.17}, annote = {Keywords: Homotopy Type Theory, coinductive types, computer theorem proving, Agda} }