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49
Higher dimensional algebra V: 2groups
 Theory Appl. Categ
"... A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to tw ..."
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Cited by 25 (2 self)
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A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2groups. A weak 2group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2group is a weak 2group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2categories of weak and coherent 2groups and an ‘improvement ’ 2functor that turns weak 2groups into coherent ones, and prove that this 2functor is a 2equivalence of 2categories. We internalize the concept of coherent 2group, which gives a quick way to define Lie 2groups. We give a tour of examples, including the ‘fundamental 2group ’ of a space and various Lie 2groups. We also explain how coherent 2groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simplyconnected compact simple Lie group G a family of 2groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2groups are built using Chern–Simons theory, and are closely related to the Lie 2algebras g � ( � ∈ R) described in a companion paper. 1 1
Combinatorics of nonabelian gerbes with connection and curvature
, 203
"... Abstract: We give a functorial definition of Ggerbes over a simplicial complex when the local symmetry group G is nonAbelian. These combinatorial gerbes are naturally endowed with a connective structure and a curving. This allows us to define a fibered category equipped with a functorial connectio ..."
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Abstract: We give a functorial definition of Ggerbes over a simplicial complex when the local symmetry group G is nonAbelian. These combinatorial gerbes are naturally endowed with a connective structure and a curving. This allows us to define a fibered category equipped with a functorial connection over the space of edgepaths. By computing the curvature of the latter on the faces of an infinitesimal 4simplex, we recover the cocycle identities satisfied by the curvature of this gerbe. The link with BFtheories suggests that gerbes provide a framework adapted to the geometric formulation of strongly coupled gauge theories.
On the passage from local to global in number theory
 Bull. Amer. Math. Soc. (N.S
, 1993
"... Would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase “in Number Theory”? The distinction “local ” versus “global”, with various connotations, has found a home in almost every part of mathematics, local problems being ..."
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Cited by 21 (0 self)
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Would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase “in Number Theory”? The distinction “local ” versus “global”, with various connotations, has found a home in almost every part of mathematics, local problems being often a steppingstone to
Higher YangMills theory
"... Electromagnetism can be generalized to Yang–Mills theory by replacing the group U(1) by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2form electromagnetism to a kind of ‘higherdimensional Yang–Mills theory’. It turns out that to do this, one should repla ..."
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Cited by 20 (1 self)
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Electromagnetism can be generalized to Yang–Mills theory by replacing the group U(1) by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2form electromagnetism to a kind of ‘higherdimensional Yang–Mills theory’. It turns out that to do this, one should replace the Lie group by a ‘Lie 2group’, which is a category C where the set of objects and the set of morphisms are Lie groups, and the source, target, identity and composition maps are homomorphisms. We show that this is the same as a ‘Lie crossed module’: a pair of Lie groups G, H with a homomorphism t: H → G and an action of G on H satisfying two compatibility conditions. Following Breen and Messing’s ideas on the geometry of nonabelian gerbes, one can define ‘principal 2bundles ’ for any Lie 2group C and do gauge theory in this new context. Here we only consider trivial 2bundles, where a connection consists of a gvalued 1form together with an hvalued 2form, and its curvature consists of a gvalued 2form together with a hvalued 3form. We generalize the Yang–Mills action for this sort of connection, and use this to derive ‘higher Yang– Mills equations’. Finally, we show that in certain cases these equations admit selfdual solutions in five dimensions. 1
Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory
, 2003
"... Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge field ..."
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Cited by 18 (3 self)
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Bundle gerbes are a higher version of line bundles, we present nonabelian bundle gerbes as a higher version of principal bundles. Connection, curving, curvature and gauge transformations are studied both in a global coordinate independent formalism and in local coordinates. These are the gauge fields needed for the construction of YangMills theories with 2form gauge potential.
Moduli of metaplectic bundles on curves and Thetasheaves
, 2004
"... We give a geometric interpretation of the Weil representation of the metaplectic group, placing it in the framework of the geometric Langlands program. For a smooth projective curve X we introduce an algebraic stack ˜ BunG of metaplectic bundles on X. It also has a local version ˜ GrG, which is a ge ..."
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Cited by 12 (7 self)
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We give a geometric interpretation of the Weil representation of the metaplectic group, placing it in the framework of the geometric Langlands program. For a smooth projective curve X we introduce an algebraic stack ˜ BunG of metaplectic bundles on X. It also has a local version ˜ GrG, which is a gerbe over the affine grassmanian of G. We define a categorical version of the (nonramified) Hecke algebra of the metaplectic group. This is a category Sph ( ˜ GrG) of certain perverse sheaves on ˜ GrG, which act on ˜ BunG by Hecke operators. A version of the Satake equivalence is proved describing Sph ( ˜ GrG) as a tensor category. Further, we construct a perverse sheaf on ˜ BunG corresponding to the Weil representation and show that it is a Hecke eigensheaf with respect to Sph ( ˜ GrG).
Interpretations of Yetter's notion of Gcoloring: simplicial fibre bundles and nonabelian cohomology
, 1995
"... this article, but beware of misprints. A more thorough treatment is given in May, [18]. We will use (standard) notation from [18] wherever possible. The way found initially around the restriction that K had to be reduced in the above loop construction was to take a maximal tree in K and to contract ..."
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Cited by 11 (2 self)
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this article, but beware of misprints. A more thorough treatment is given in May, [18]. We will use (standard) notation from [18] wherever possible. The way found initially around the restriction that K had to be reduced in the above loop construction was to take a maximal tree in K and to contract it to a point. In 1984, a groupoid version of the loop group construction was given by Dwyer and Kan, [12]. (Unfortunately the published paper has many misprints and the cleanedup version that we will use was prepared by my student Phil Ehlers as part of his master's dissertation, [13]. Alternatives have been proposed by Joyal and Tierney, and by Moerdijk and Svensson. They end up with simplicial objects in the category of groupoids, whilst the Dwyer  Kan version gives a simplicially enriched groupoid, i.e. a groupoid all of whose Homobjects are simplicial sets. A simplicially enriched groupoid is also a simplicial groupoid (simplicial object in the category of groupoids), but is one whose object of objects is a constant simplicial set.) Let SS denote the category of simplicial sets and SGpds that of simplicially enriched groupoids or as we will often call them, simply, simplicial groupoids. The loop groupoid functor is a functor
Higher gauge theory
"... I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2groups, and bundles with a suitable notion of 2bundle. To link this with previous work, I show that certain 2categories of principal 2bundles are equivalent to ce ..."
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I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2groups, and bundles with a suitable notion of 2bundle. To link this with previous work, I show that certain 2categories of principal 2bundles are equivalent to certain 2categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2category of 2bundles over a given 2space under a given 2group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2space is the 2space corresponding to a given space and the 2group is the automorphism 2group of a given group, then this 2category is equivalent to the 2category of gerbes over that space under that group (being described by the same cohomological data).