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DOI: 10.4230/LIPIcs.ITP.2023.16

Formalising Yoneda Ext in Univalent Foundations

Authors: Jarl G. Taxerås Flaten

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)

Abstract

Ext groups are fundamental objects from homological algebra which underlie important computations in homotopy theory. We formalise the theory of Yoneda Ext groups [Yoneda, 1954] in homotopy type theory (HoTT) using the Coq-HoTT library. This is an approach to Ext which does not require projective or injective resolutions, though it produces large abelian groups. Using univalence, we show how these Ext groups can be naturally represented in HoTT. We give a novel proof and formalisation of the usual six-term exact sequence via a fibre sequence of 1-types (or groupoids), along with an application. In addition, we discuss our formalisation of the contravariant long exact sequence of Ext, an important computational tool. Along the way we implement and explain the Baer sum of extensions and how Ext is a bifunctor.

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Jarl G. Taxerås Flaten. Formalising Yoneda Ext in Univalent Foundations. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{flaten:LIPIcs.ITP.2023.16,
  author =	{Flaten, Jarl G. Taxer\r{a}s},
  title =	{{Formalising Yoneda Ext in Univalent Foundations}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{16:1--16:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.16},
  URN =		{urn:nbn:de:0030-drops-183911},
  doi =		{10.4230/LIPIcs.ITP.2023.16},
  annote =	{Keywords: homotopy type theory, homological algebra, Yoneda Ext, formalisation, Coq}
}

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Documents authored by Flaten, Jarl G. Taxerås

Formalising Yoneda Ext in Univalent Foundations

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