A Homotopy 2-Groupoid of a Hausdorff Space

Hardie, K. A.; Kamps, Klaus; Kieboom, R. W.

doi:10.1007/978-94-017-2529-3_11

Cited by 25 publications

(22 citation statements)

References 20 publications

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“…As it stands we cannot prove that C 2 2 ðM; * Þ is a true strictification of CðM; * Þ; but it is clear that C 2 2 ðM; * Þ suits our purpose in this paper very well indeed. In [17] the reader can find a more restricted notion of thin homotopy for which the conjecture can be shown. However, this notion of thin homotopy does not seem to be suited to the smooth context of parallel transport.…”

Section: Parallel Transport In Gerbesmentioning

confidence: 97%

Holonomy and Parallel Transport for Abelian Gerbes

Mackaay¹,

Picken²

2002

Advances in Mathematics

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In this paper, we establish a one-to-one correspondence between Uð1Þ-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 group Uð1Þ on a simply connected manifold M is a group morphism from the thin second homotopy group to Uð1Þ; satisfying a smoothness condition, where a homotopy between maps from ½0; 1 2 to M is thin when its derivative is of rank 42: 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. # 2002 Elsevier Science (USA)

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Section: Parallel Transport In Gerbesmentioning

confidence: 97%

Holonomy and Parallel Transport for Abelian Gerbes

Mackaay¹,

Picken²

2002

Advances in Mathematics

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“…As it stands we cannot prove that C 2 2 (M, * ) is a true strictification of C(M, * ), but it is clear that C 2 2 (M, * ) suits our purpose in this paper very well indeed. In [17] the reader can find a more restricted notion of thin homotopy for which the conjecture can be shown. However, this notion of thin homotopy does not seem to be suited to the smooth context of parallel transport.…”

Section: Definition 64mentioning

confidence: 99%

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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“…For the definition of a 2-groupoid and of a sesquigroupoid see [HKK00,Mac98,Str96]. Let X = (∂ : E → G, ⊲) be a group crossed module.…”

Section: The Group Crossed Module Casementioning

confidence: 99%

Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy

Martins

2016

Journal of Geometry and Physics

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Article:Faria Martins, J orcid.org/0000-0001- 8113-3646 (2016) ReuseUnless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Abstract After a thorough treatment of all algebraic structures involved, we address two dimensional holonomy operators with values in crossed modules of Hopf algebras and in crossed modules of associative algebras (called here crossed modules of bare algebras.) In particular, we will consider two general formulations of the two-dimensional holonomy of a (fully primitive) Hopf 2-connection (exact and blur), the first being multiplicative the second being additive, proving that they coincide in a certain natural quotient (defining what we called the fuzzy holonomy of a fully primitive Hopf 2-connection).

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A Homotopy 2-Groupoid of a Hausdorff Space

Cited by 25 publications

References 20 publications

Holonomy and Parallel Transport for Abelian Gerbes

Holonomy and Parallel Transport for Abelian Gerbes

Holonomy and Parallel Transport for Abelian Gerbes

Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy

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