## Stacky Lie Groups (2008)

### BibTeX

@MISC{Blohmann08stackylie,

author = {Christian Blohmann},

title = { Stacky Lie Groups},

year = {2008}

}

### OpenURL

### Abstract

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie groups are defined as group objects in this weak 2-category. Introducing a graphic notation, it is shown that for every stacky Lie monoid there is a natural morphism, called the preinverse, which is a Morita equivalence if and only if the monoid is a stacky Lie group. As an example, we describe explicitly the stacky Lie group structure of the irrational Kronecker foliation of the torus.

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Citation Context ...constructive aspects of differentiable stacks by studying their presentations by Lie groups. In this spirit we could define a “stacky Lie group” to be the presentation of a differentiable group stack =-=[8]-=-. But this definition isn’t of much value if we want to study such presentations in their own right. What we need is an axiomatic definition of a stacky Lie group in terms of a Lie groupoid and group ... |

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Citation Context ...ly differential geometric problems. For example, proper étale differentiable stacks, the differential geometric analog of Deligne-Mumford stacks, are a conceptually powerful way to describe orbifolds =-=[23]-=- [24]. Another example comes from the observation that while not every Lie algebroid can be integrated by a Lie groupoid [10], it can always be integrated by a smooth stack of Lie groupoids [30] which... |

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Citation Context ...ht H-action which commute, i.e., (g · m) · h = g · (m · h), whenever defined, is called a smooth G-H bibundle. Most of the material about bibundles of this section can be found in [15] [18] [19] [25] =-=[26]-=-. We can view a G-H bibundle as a presentation of a relation between the differentiable stacks presented by G and H [5]. Definition 2.9. Let G, H be Lie groupoids and M,N be G-H smooth bibundles. A mo... |

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Citation Context ...a right H-action which commute, i.e., (g · m) · h = g · (m · h), whenever defined, is called a smooth G-H bibundle. Most of the material about bibundles of this section can be found in [15] [18] [19] =-=[25]-=- [26]. We can view a G-H bibundle as a presentation of a relation between the differentiable stacks presented by G and H [5]. Definition 2.9. Let G, H be Lie groupoids and M,N be G-H smooth bibundles.... |

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Citation Context ...MANN To see how we get from this relation the group associator we identify as before Gk ∼ = (G1) k . The inner face maps d [k] i : (G1) k → (G1) k−1 can then be identified with (18) d [2] 1 ∼ = µ , d =-=[3]-=- 1 ∼ = µ × Id , d [3] 2 ∼ = Id ×µ , d [4] 1 ∼ = µ × Id × Id , d [4] 2 ∼ = Id ×µ × Id , d [4] 3 ∼ = Id × Id ×µ , and so forth. The outer face maps d [k] 0 and d[k] k are the projections on the first k ... |

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Citation Context ...cture descends to a group structure no the coarse moduli space of the isomorphism classes of Morita equivalences of X which has been called the Picard group of G in analogy to algebraic Morita theory =-=[8]-=- [9]. In the same vein, one might call the stack presented by Aut(X) the Picard stack of X. We make no attempt here to equip Aut(X) of a differentiable stack itself with a differentiable structure. Fi... |

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Citation Context ...ction and a right H-action which commute, i.e., (g · m) · h = g · (m · h), whenever defined, is called a smooth G-H bibundle. Most of the material about bibundles of this section can be found in [15] =-=[18]-=- [19] [25] [26]. We can view a G-H bibundle as a presentation of a relation between the differentiable stacks presented by G and H [5]. Definition 2.9. Let G, H be Lie groupoids and M,N be G-H smooth ... |

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Citation Context ...e is the existence of the preinverse, which is a morphism in the category of groupoids and bibundles. The stacky preinverse is the geometric analog of the algebraic concept of the hopfish preantipode =-=[4]-=- [29] and has already appeared in [33]. 4.6. The stacky Lie group of a reduced torus foliation. A Lie 2-group is a group object in the category of Lie groupoids and differentiable homomorphisms of gro... |

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Citation Context ...cture gives rise to a hopfish structure on the noncommutative torus algebra [4]. Another motivation for this work was to understand stacky Lie groups as the geometric counterparts of hopfish algebras =-=[29]-=-. The content of this paper is as follows: Section 2 presents review material in order to make the paper reasonably self-contained and fix the terminology. We give a brief overview of differentiable s... |

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Citation Context ...he material about bibundles of this section can be found in [15] [18] [19] [25] [26]. We can view a G-H bibundle as a presentation of a relation between the differentiable stacks presented by G and H =-=[5]-=-. Definition 2.9. Let G, H be Lie groupoids and M,N be G-H smooth bibundles. A morphism of smooth bibundles is a smooth map ϕ : M → N which is biequivariant, i.e., lN(ϕ(m)) = lM(m), rN(ϕ(m)) = rM(m) f... |

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Citation Context ...stacky Lie groups and a natural source of examples. While not every stacky Lie group is a strict Lie 2group, it can be shown that every stacky Lie group is a weak Lie 2-group up to Morita equivalence =-=[32]-=- [33]. There is a 1-to-1 correspondence between strict 2-groups and crossed modules [1] [8] [27]. An obvious example of a crossed module is given by the inclusion ϕ : A ֒→ B of a normal subgroup with ... |

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(Show Context)
Citation Context ... and a right H-action which commute, i.e., (g · m) · h = g · (m · h), whenever defined, is called a smooth G-H bibundle. Most of the material about bibundles of this section can be found in [15] [18] =-=[19]-=- [25] [26]. We can view a G-H bibundle as a presentation of a relation between the differentiable stacks presented by G and H [5]. Definition 2.9. Let G, H be Lie groupoids and M,N be G-H smooth bibun... |

1 |
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Citation Context ...e descends to a group structure no the coarse moduli space of the isomorphism classes of Morita equivalences of X which has been called the Picard group of G in analogy to algebraic Morita theory [8] =-=[9]-=-. In the same vein, one might call the stack presented by Aut(X) the Picard stack of X. We make no attempt here to equip Aut(X) of a differentiable stack itself with a differentiable structure. First,... |