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**Published in:** LIPIcs, Volume 97, 22nd International Conference on Types for Proofs and Programs (TYPES 2016)

Broadly speaking, algebraic topology consists of associating algebraic structures to topological spaces that give information about their structure. An elementary, but fundamental, example is provided by the theory of covering spaces, which associate groups to covering spaces in such a way that the universal cover corresponds to the fundamental group of the space. One natural question to ask is whether these connections can be stated in homotopy type theory, a new area linking type theory to homotopy theory. In this paper, we give an affirmative answer with a surprisingly concise definition of covering spaces in type theory; we are able to prove various expected properties about the newly defined covering spaces, including the connections with fundamental groups. An additional merit is that our work has been fully mechanized in the proof assistant Agda.

Kuen-Bang Hou (Favonia) and Robert Harper. Covering Spaces in Homotopy Type Theory. In 22nd International Conference on Types for Proofs and Programs (TYPES 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 97, pp. 11:1-11:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hou(favonia)_et_al:LIPIcs.TYPES.2016.11, author = {Hou (Favonia), Kuen-Bang and Harper, Robert}, title = {{Covering Spaces in Homotopy Type Theory}}, booktitle = {22nd International Conference on Types for Proofs and Programs (TYPES 2016)}, pages = {11:1--11:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-065-1}, ISSN = {1868-8969}, year = {2018}, volume = {97}, editor = {Ghilezan, Silvia and Geuvers, Herman and Ivetic, Jelena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2016.11}, URN = {urn:nbn:de:0030-drops-98512}, doi = {10.4230/LIPIcs.TYPES.2016.11}, annote = {Keywords: homotopy type theory, covering space, fundamental group, mechanized reasoning} }

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**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The fibrant fragment validates Voevodsky's univalence axiom and includes a circle type, while the non-fibrant fragment includes exact (strict) equality types satisfying equality reflection. Our type theory is defined by a semantics in cubical partial equivalence relations, and is the first two-level type theory to satisfy the canonicity property: all closed terms of boolean type evaluate to either true or false.

Carlo Angiuli, Kuen-Bang Hou (Favonia), and Robert Harper. Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 6:1-6:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{angiuli_et_al:LIPIcs.CSL.2018.6, author = {Angiuli, Carlo and Hou (Favonia), Kuen-Bang and Harper, Robert}, title = {{Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {6:1--6:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.6}, URN = {urn:nbn:de:0030-drops-96734}, doi = {10.4230/LIPIcs.CSL.2018.6}, annote = {Keywords: Homotopy Type Theory, Two-Level Type Theory, Computational Type Theory, Cubical Sets} }

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

Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic theorems, yielding "synthetic homotopy theory". Here we consider the Seifert-van Kampen theorem, which characterizes the loop structure of spaces obtained by gluing. This is useful in homotopy theory because many spaces are constructed by gluing, and the loop structure helps distinguish distinct spaces. The synthetic proof showcases many new characteristics of synthetic homotopy theory, such as the "encode-decode" method, enforced homotopy-invariance, and lack of underlying sets.

Kuen-Bang Hou (Favonia) and Michael Shulman. The Seifert-van Kampen Theorem in Homotopy Type Theory. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 22:1-22:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{hou(favonia)_et_al:LIPIcs.CSL.2016.22, author = {Hou (Favonia), Kuen-Bang and Shulman, Michael}, title = {{The Seifert-van Kampen Theorem in Homotopy Type Theory}}, booktitle = {25th EACSL Annual Conference on Computer Science Logic (CSL 2016)}, pages = {22:1--22:16}, 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.2016.22}, URN = {urn:nbn:de:0030-drops-65626}, doi = {10.4230/LIPIcs.CSL.2016.22}, annote = {Keywords: homotopy type theory, fundamental group, homotopy pushout, mechanized reasoning} }

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