Towards a 2-dimensional notion of holonomy

Ronald Brown; İlhan İçen

doi:10.3934/jgm.2022011

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2022, Volume 14, Issue 2: 349-375. Doi: 10.3934/jgm.2022011

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Towards a 2-dimensional notion of holonomy

Ronald Brown^1, and
İlhan İçen^2,

1.
Mathematics Division, School of Informatics, University of Wales, Bangor, Gwynedd LL57 1UT, UK
2.
Department of Mathematics, Faculty of Science and Art, University of İnönü, Malatya, Turkey

Received: May 2022

Published: June 2022

This article was published in Advances in Mathematics, 178, Ronald Brown and İlhan İçen "Towards a 2-dimensional notion of holonomy", 141–175, Copyright Elsevier (2003)

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Previous work (Pradines, C.R. Acad. Sci. Paris 263 (1966) 907, Aof and Brown, Topology Appl. 47 (1992) 97) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involves the $ 2 $-dimensional information.

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Mathematics Subject Classification: 18D05; 58H05; 22A22; 22E99.

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