On two-dimensional holonomy

Abstract

Abstract. We define the thin fundamental categorical group P 2 (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 on M . Here two maps 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 = (∂ : E → G, ). We construct categorical holonomies, defined to be smooth morphisms P 2 (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.

Smooth functors vs. differential forms

“…Let us presume that "a surface holonomy" is at least a 2-functorial assignment, i.e., is described by a 2-functor on the path 2-groupoid. This assumption is supported by the approach via "2-holonomies" [MP10], as well as by transport 2-functors [SWA]. Similar considerations have also been made for ordinary holonomy [CP94,SW09].…”

“…The proof that this is independent of the choice of h requires a technical computation carried out in [MP10]. Another important fact is that the two compositions • and • are compatible with each other in the sense that…”

An invitation to higher gauge theory

“…This work was deeply inspired by the ideas of Breen and Messing [28,29], who considered a special class of 2-groups, and omitted the equation t(B) = d A + A ∧ A, since their sort of connection did not assign holonomies to surfaces. One should also compare the closely related work of Mackaay, Martins, and Picken [65,67,68], and the work of Pfeiffer and Girelli [76,56].…”

“…But for actual calculations, this process is not very convenient. More practical formulas for computing holonomies over surfaces can be found in the work of Schreiber and Waldorf [88,89], Martins and Picken [67,68].…”

Higher holonomies: Comparing two constructions

“…We will briefly review the definitions and basic facts regarding Lie crossed modules, following closely Faria Martins-Picken [9,10,11]. Definition 3.1.…”