On two-dimensional holonomy

Joao Faria Martins, Roger Picken

Research output: Contribution to journalArticlepeer-review

33 Citations (Scopus)


We define the thin fundamental categorical group P2(M,*) of a based smooth manifold (M, *) as the categorical group whose objects are rank- 1 homotopy classes of based loops on M and whose morphisms are rank-2 homotopy classes of homotopies between based loops onM. Here twomaps are rank-n homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie categorical group coming from a Lie crossed module G = (δ : E → G, δ). We construct categorical holonomies, defined to be smooth morphisms P2(M,*) → C(G), by using a notion of categorical connections, being a pair (ω,m), where ω is a connection 1-form on P, a principal G bundle over M, and m is a 2-form on P with values in the Lie algebra of E, with the pair (ω,m) satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.

Original languageEnglish
Pages (from-to)5657-5695
Number of pages39
JournalTransactions of the American Mathematical Society
Issue number11
Publication statusPublished - Nov 2010


  • 2-bundle
  • Categorical group
  • Crossed module
  • Non-abelian gerbe
  • Two-dimensional holonomy
  • Wilson sphere


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