An Axiomatic Definition of Holonomy

Caetano, Ana Paula; Picken, Roger

doi:10.1142/s0129167x94000425

Cited by 51 publications

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“…In the present paper this terminological twist should not lead to any confusion, since the paper is intended to be self-contained. Barrett's main result, also obtained in the setup of [10], shows that it is possible to reconstruct the bundle and the connection, up to equivalence, from the holonomy. This result is a nice strengthening of the well-known Ambrose-Singer theorem [1].…”

Section: Introductionmentioning

confidence: 91%

Holonomy and Parallel Transport for Abelian Gerbes

Mackaay¹

2002

Advances in Mathematics

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In this paper we establish a one-to-one correspondence between U (1)-gerbes with connections, on the one hand, and their holonomies, for simply-connected manifolds, or their parallel transports, in the general case, on the other hand. This result is a higher-order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds. The holonomy of a gerbe with * presently working as a postdoc at the University of Nottingham, UK 1 group U (1) on a simply-connected manifold M is a group morphism from the thin second homotopy group to U (1), satisfying a smoothness condition, where a homotopy between maps from [0, 1] 2 to M is thin when its derivative is of rank ≤ 2. For the non-simply connected case, holonomy is replaced by a parallel transport functor between two special Lie groupoids, which we call Lie 2-groups. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply-connected case.

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Section: Introductionmentioning

confidence: 91%

Holonomy and Parallel Transport for Abelian Gerbes

Mackaay¹

2002

Advances in Mathematics

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“…In [SW09] we have instead equipped P 1 X with a diffeology, a structure that generalizes a smooth manifold structure [Che77,Sou81]. This diffeology on P 1 X has been introduced in [CP94]. For the convenience of the reader let us recall the basic definitions (see also Appendix A.2 of [SW09]).…”

Section: Diffeological Spacesmentioning

confidence: 99%

Smooth functors vs. differential forms

Schreiber

Waldorf

2011

Homology, Homotopy and Applications

153

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We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in several fields, namely as connections on (non-abelian) gerbes, as derivatives of smooth functors and as critical points in BF theory. We demonstrate further that our dictionary provides a powerful tool to discuss the transgression of geometric objects to loop spaces.

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“…where we use Barrett's lemma [5] (see also [11,16]) stating that the trivial loop is a critical point for any holonomy. Note that Z(., j) restricted to loops based at y contained in U j satisfies the conditions to be a holonomy in the sense of Barrett's lemma, because of the thin invariance and smoothness of Z (Theorem 3.7).…”

Section: Remark 36mentioning

confidence: 99%

“…Interest in gerbes has been revived recently following a concrete approach due to Hitchin and Chatterjee [15]. Gerbes can be understood both in terms of local geometry, local functions and forms, and in terms of non-local geometry, holonomies and parallel transports, and these two viewpoints are equivalent, in a sense made precise by Mackaay and the author in [16], following on from work by Barrett [5] and Caetano and the author [11]. In [16] holonomies around spheres and parallel transports along cylinders were considered.…”

Section: Introductionmentioning

confidence: 99%

TQFT’s and gerbes

Picken¹

2004

Algebr. Geom. Topol.

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We generalize the notion of parallel transport along paths for abelian bundles to parallel transport along surfaces for abelian gerbes using an embedded Topological Quantum Field Theory (TQFT) approach. We show both for bundles and gerbes with connection that there is a one-to-one correspondence between their local description in terms of locally-defined functions and forms and their non-local description in terms of a suitable class of embedded TQFT's.

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An Axiomatic Definition of Holonomy

Cited by 51 publications

References 0 publications

Holonomy and Parallel Transport for Abelian Gerbes

Holonomy and Parallel Transport for Abelian Gerbes

Smooth functors vs. differential forms

TQFT’s and gerbes

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