## Holonomy and parallel transport for abelian gerbes, Preprint math.DG/0007053

Citations: | 54 - 10 self |

### BibTeX

@MISC{Mackaay_holonomyand,

author = {Marco Mackaay and Área Departamental De Matemática and Roger Picken},

title = {Holonomy and parallel transport for abelian gerbes, Preprint math.DG/0007053},

year = {}

}

### OpenURL

### Abstract

In this paper we establish a one-to-one correspondence between S 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 S 1 on a simply connected manifold M is a group morphism from the thin second homotopy group to S 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 monoidal Lie groupoids. The reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the non-abelian case via the theory of double Lie groupoids. 1

### Citations

472 |
Foundations of Differential Geometry
- Kobayashi, Nomizu
- 1963
(Show Context)
Citation Context ...t see [11]. 3 Ordinary holonomy It is well-known that a principal G-bundle with connection over M allows one to define the notion of holonomy around any smooth closed curve on M (Kobayashi and Nomizu =-=[19]-=-). In particular, given a fixed basepoint ∗ in M, and a point in the fibre over ∗, this data induces an assignment of an element of G, to each loop in M, based at ∗. Such an assignment is called a hol... |

188 |
Lie Groupoids and Lie Algebroids in Differential Geometry
- Mackenzie
- 1987
(Show Context)
Citation Context ...the right language is that of Lie groupoids and groupoid morphisms, i.e. functors, instead of Lie groups and group homomorphisms. A thorough account of Lie groupoids and their history can be found in =-=[22]-=-. We note that a Lie groupoid with only one object is precisely a Lie group. We will show how to translate Barrett’s [3] results to this more general framework. As pointed out by Brylinski [8], a gerb... |

138 | Higher-dimensional algebra and topological quantum field theory
- Baez, Dolan
- 1995
(Show Context)
Citation Context ...een as a Lie 2-groupoid with only one object. (This is analogous to the statement that a Lie groupoid with one object is nothing but a Lie group. For this kind of general remark about ncategories see =-=[2, 4, 5]-=- for example.) Without the smoothness condition this kind of groupoid goes under a variety of names in the homotopy literature. Yetter [27] calls them categorical groups, for example. This particular ... |

57 | The geometry of four-manifolds. Oxford Mathematical Monographs - Donaldson, Kronheimer - 1990 |

45 |
Singer A theorem on holonomy
- Ambrose, I
- 1953
(Show Context)
Citation Context ...y of Nottingham, UK 1group S 1 on a simply connected manifold M is a group morphism from the thin second homotopy group to S 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 monoidal Lie groupoids. The reconstruction of th... |

38 |
Categorification, in Higher Category Theory
- Dolan, Baez
- 1998
(Show Context)
Citation Context ...One gets isomorphisms between isomorphisms, etc, up to the highest level where one requires a cocycle condition to hold. This feature is known as categorification, and we recommend the reader to read =-=[4, 5]-=- on the general concept of categorification. Finally we note that there is also a notion of n-gerbe connection, and the parallel transport of an n-gerbe connection is defined along n-dimensional paths... |

34 | Double Lie algebroids and second-order geometry
- Mackenzie
(Show Context)
Citation Context ...ject accessible to a broader group of mathematicians and physicists. Finally, our setup provides a link with the theory of double Lie groupoids which is not necessarily restricted to the Abelian case =-=[24, 25]-=-. We are only aware of one type of concrete examples of non-Abelian gerbes, which define the obstruction to lifting a principal G-bundle to a ˆ G-bundle, where ˆ G is a non-abelian extension of G. To ... |

33 |
spaces, characteristic classes and geometric quantization, volume 107
- Loop
- 1993
(Show Context)
Citation Context ...ions. Gerbes were first introduced by Giraud [15] in an attempt to understand non-Abelian cohomology. Ironically only Abelian gerbes have developed into a nice geometric theory so far. Several people =-=[8, 11, 13, 18]-=- have studied Abelian gerbes. In this paper when we say gerbe we always mean a U(1)gerbe. Just as a line-bundle on M can be defined by a set of transition functions on double intersections of open set... |

33 |
Coherence for bicategories and indexed categories
- Lane, Paré
- 1985
(Show Context)
Citation Context ...his way the groupoid C(M, ∗) is a weak monoidal groupoid, with weak inverses for the objects because loops only form a group up to thin homotopy. Instead of using the abstract strictification theorem =-=[25]-=-, which is not very practical for the concrete application to gerbe-holonomy, we “strictify” C(M, ∗) by hand by dividing out by the monoidal subgroupoid of only the thin homotopies, N(M, ∗). Dividing ... |

32 | Spherical 2-categories and 4-manifold invariants, Adv
- Mackaay
- 1999
(Show Context)
Citation Context ... above. Another motivation for our approach is that we are trying to understand the differential geometry behind the four-dimensional state-sum models defined by the first author of the present paper =-=[21, 20]-=-. For the understanding of these state-sums it would be helpful to find a categorical way of perceiving the relation between homotopy theory and differential geometry. Concretely the motivation was to... |

30 |
On the classification of 2-gerbes and 2-stacks
- Breen
- 1994
(Show Context)
Citation Context ...SU(2). Therefore, Thompson’s examples are just the coboundaries of 1cochains, which are trivial as gerbes. For an introduction to the general theory of non-Abelian gerbes and its history one can read =-=[7]-=-. However, the only concrete example in the book is the one we mentioned above. Another motivation for our approach is that we are trying to understand the differential geometry behind the four-dimens... |

27 | Geometry of Deligne cohomology
- Gajer
- 1997
(Show Context)
Citation Context ...ions. Gerbes were first introduced by Giraud [15] in an attempt to understand non-Abelian cohomology. Ironically only Abelian gerbes have developed into a nice geometric theory so far. Several people =-=[8, 11, 13, 18]-=- have studied Abelian gerbes. In this paper when we say gerbe we always mean a U(1)gerbe. Just as a line-bundle on M can be defined by a set of transition functions on double intersections of open set... |

26 |
Holonomy and path structures in general relativity and Yang-Mills theory
- Barrett
- 1991
(Show Context)
Citation Context ...l for the simply connected case. Our approach to abelian gerbes with connections holds out prospects for generalizing to the non-abelian case via the theory of double Lie groupoids. 1 Introduction In =-=[3]-=- Barrett studied the holonomy of connections in principal bundles and proved a reconstruction theorem which showed that in a very precise sense all information about the connections and the bundles is... |

26 |
R.W.: A homotopy 2–groupoid of a hausdorff space
- Hardie, Kamps, et al.
(Show Context)
Citation Context ...re it is almost immediate. As it stands we cannot prove that C2 2(M, ∗) is a true strictification of C(M, ∗), however it is clear that C2 2 (M, ∗) suits our purpose in this paper very well indeed. In =-=[16]-=- the reader can find a more restricted notion of thin homotopy for which the conjecture can be shown. However, this notion of thin homotopy is not suited to the smooth context of parallel transport. B... |

25 |
An axiomatic definition of holonomy
- Caetano, Picken
- 1994
(Show Context)
Citation Context ... assignment of an element of G, to each loop in M, based at ∗. Such an assignment is called a holonomy map, or simply a holonomy. In Barrett [3], and in a slightly different fashion in Caetano-Picken =-=[9]-=-, it was shown that suitably defined holonomy maps are in one-to-one correspondence with G-bundles plus connection plus the choice of a point in the fibre over ∗, up to isomorphism. This result should... |

24 | Spherical categories
- Barrett, Westbury
- 1999
(Show Context)
Citation Context ...ight ones for our construction of the quotient monoidal groupoid, which we explain below, to be well-defined. The only reference for the definition of a normal monoidal subgroupoid that we know of is =-=[6]-=-, but it might be that it can be found in earlier papers on monoidal categories. We suspect that this definition goes back to the time when monoidal categories were defined for the first time. Lemma 6... |

21 | An introduction to n-categories. In
- Baez
- 1997
(Show Context)
Citation Context ...n gerbe-connections in general along arbitrary homotopies between arbitrary loops. For this discussion we assume some basic knowledge about groupoids and 2-groupoids, which can be obtained by reading =-=[2, 4, 22]-=- for example. We should warn the reader that this section is different in flavour from the rest of the paper. Whereas we have followed a down-to-earth approach in the rest of the paper, here some read... |

16 | Finite groups, spherical 2-categories, and 4-manifold invariants
- Mackaay
(Show Context)
Citation Context ... above. Another motivation for our approach is that we are trying to understand the differential geometry behind the four-dimensional state-sum models defined by the first author of the present paper =-=[21, 20]-=-. For the understanding of these state-sums it would be helpful to find a categorical way of perceiving the relation between homotopy theory and differential geometry. Concretely the motivation was to... |

16 |
TQFTs from homotopy 2-types
- Yetter
- 1993
(Show Context)
Citation Context .... For this kind of general remark about ncategories see [2, 4, 5] for example.) Without the smoothness condition this kind of groupoid goes under a variety of names in the homotopy literature. Yetter =-=[27]-=- calls them categorical groups, for example. This particular Lie 2-group we denote by G(G, A, M). Let us now define the Lie 2-group of thin cylinders, denoted by C2 2(M, ∗). We define C2 2 (M, ∗) as t... |

12 |
Catégories avec multiplication
- BÉNABOU
- 1963
(Show Context)
Citation Context ...might be that it can be found in earlier papers on monoidal categories and 2-categories. We suspect that this definition goes back to the time when monoidal categories were defined for the first time =-=[7]-=-, but we have been unable to find a precise written reference in the older literature. Lemma 6.8 The monoidal subgroupoid N(M, ∗) is normal in C(M, ∗). Proof The first condition is obviously satisfied... |

10 |
Higher holonomies, geometric loop groups and smooth Deligne cohomology
- Gajer
- 1998
(Show Context)
Citation Context ...o a notion of n-gerbe connection, and the parallel transport of an n-gerbe connection is defined along n-dimensional paths. Some details of these notions remain to be worked out completely, but Gajer =-=[13, 14]-=- has worked out a considerable part already. The truly categorical nature of gerbe-connections comes to light when we study the general setting of their parallel transport. To understand the need for ... |

10 |
Cohomologie non-abelienne, volume 179 of Grundl., Springer-Verlag
- Giraud
- 1971
(Show Context)
Citation Context ... answer is that, for a 1-connected manifold M, group homomorphisms π2 2(M) → U(1) correspond bijectively to equivalence classes of U(1)-gerbes with connections. Gerbes were first introduced by Giraud =-=[15]-=- in an attempt to understand non-Abelian cohomology. Ironically only Abelian gerbes have developed into a nice geometric theory so far. Several people [8, 11, 13, 18] have studied Abelian gerbes. In t... |

10 |
R.W.: A homotopy bigroupoid of a topological space
- Hardie, Kamps, et al.
(Show Context)
Citation Context ...rove that the second condition holds. Denote the group of homotopies γ → γ in C(M, ∗) by C(M, ∗)(γ). This group is isomorphic to C(M, ∗)(γ⋆µ), for any µ ∈ Ω(M, ∗) (this is well known, see for example =-=[17]-=-). The isomorphism, which clearly preserves thinness, is given by φ: G ↦→ G ⋆ 1µ. 36Under φ the homotopy GHG −1 : γ → γ is mapped to (G ⋆ 1µ)(H ⋆ 1µ)(G −1 ⋆ 1µ): γ⋆µ → γ⋆µ. Clearly this is thin homot... |

8 | On a family of topological invariants similar to homotopy groups - Caetano, Picken - 1998 |

8 |
Lectures on special Langrangian submanifolds, inWinter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds
- Hitchin
- 1999
(Show Context)
Citation Context ...ions. Gerbes were first introduced by Giraud [15] in an attempt to understand non-Abelian cohomology. Ironically only Abelian gerbes have developed into a nice geometric theory so far. Several people =-=[8, 11, 13, 18]-=- have studied Abelian gerbes. In this paper when we say gerbe we always mean a U(1)gerbe. Just as a line-bundle on M can be defined by a set of transition functions on double intersections of open set... |

1 |
Introducing quaternionic gerbes. preprint available as math.DG/0009201
- Thompson
(Show Context)
Citation Context ... help to understand how the differential geometry of Abelian gerbes fits into a framework of 2-dimensional parallel transport which is not confined to the Abelian case. We should remark that Thompson =-=[26]-=- has worked out in detail the definition of quaternionic gerbes and some of their properties. Unfortunately his only examples, related to conformal 4-manifolds, are actually just Čech 1-cochains with ... |