License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.TYPES.2016.11
URN: urn:nbn:de:0030-drops-98512
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9851/
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Hou (Favonia), Kuen-Bang ; Harper, Robert

Covering Spaces in Homotopy Type Theory

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LIPIcs-TYPES-2016-11.pdf (0.5 MB)

Abstract

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.

BibTeX - Entry

@InProceedings{houfavonia_et_al:LIPIcs:2018:9851,
  author =	{Kuen-Bang {Hou (Favonia)} and Robert Harper},
  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 =	{Silvia Ghilezan and Herman Geuvers and Jelena Ivetić},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9851},
  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}
}

Keywords: homotopy type theory, covering space, fundamental group, mechanized reasoning
Collection: 22nd International Conference on Types for Proofs and Programs (TYPES 2016)
Issue Date: 2018
Date of publication: 05.11.2018

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