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36
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 A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor.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 24 (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A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor.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
From loop groups to 2groups
 HHA
"... We describe an interesting relation between Lie 2algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2gr ..."
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Cited by 23 (11 self)
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We describe an interesting relation between Lie 2algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2group is a categorified version of a Lie group. If G is a simplyconnected compact simple Lie group, there is a 1parameter family of Lie 2algebras gk each having g as its Lie algebra of objects, but with a Jacobiator built from the canonical 3form on G. There appears to be no Lie 2group having gk as its Lie 2algebra, except when k = 0. Here, however, we construct for integral k an infinitedimensional Lie 2group PkG whose Lie 2algebra is equivalent to gk. The objects of PkG are based paths in G, while the automorphisms of any object form the levelk Kac– Moody central extension of the loop group ΩG. This 2group is closely related to the kth power of the canonical gerbe over G. Its nerve gives a topological group PkG  that is an extension of G by K(Z, 2). When k = ±1, PkG  can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), PkG  is none other than String(n). 1 1
Representation theory of 2groups on Kapranov and Voevodsky’s 2vector spaces
 Adv. Math
"... In this paper the 2category Rep 2MatC (G) of (weak) representations of an arbitrary (weak) 2group G on (some version of) Kapranov and Voevodsky’s 2category of (complex) 2vector spaces is studied. In particular, the set of equivalence classes of representations is computed in terms of the invaria ..."
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Cited by 10 (1 self)
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In this paper the 2category Rep 2MatC (G) of (weak) representations of an arbitrary (weak) 2group G on (some version of) Kapranov and Voevodsky’s 2category of (complex) 2vector spaces is studied. In particular, the set of equivalence classes of representations is computed in terms of the invariants π0(G), π1(G) and [α]∈H 3 (π0(G), π1(G)) classifying G. Also the categories of morphisms (up to equivalence) and the composition functors are determined explicitly. As a consequence, we obtain that the monoidal category
Categorified symplectic geometry and the classical string
, 2008
"... A Lie 2algebra is a ‘categorified ’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poi ..."
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Cited by 10 (5 self)
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A Lie 2algebra is a ‘categorified ’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an ndimensional field theory using a phase space that is an ‘nplectic manifold’: a finitedimensional manifold equipped with a closed nondegenerate (n + 1)form. Here we consider the case n = 2. For any 2plectic manifold, we construct a Lie 2algebra of observables. We then explain how this Lie 2algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2plectic structure for the string.
Twisted differential String and Fivebrane structures
, 2009
"... Abelian differential generalized cohomology as developed by Hopkins and Singer has been shown by Freed to formalize the global description of anomaly cancellation problems in String theory, such as notably the GreenSchwarz mechanism. On the other hand, this mechanism, as well as the FreedWitten an ..."
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Cited by 7 (7 self)
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Abelian differential generalized cohomology as developed by Hopkins and Singer has been shown by Freed to formalize the global description of anomaly cancellation problems in String theory, such as notably the GreenSchwarz mechanism. On the other hand, this mechanism, as well as the FreedWitten anomaly cancellation, are fundamentally governed by the cohomology classes represented by the relevant nonabelian O(n) and U(n)principal bundles underlying the tangent and the gauge bundle on target space. In this article we unify the picture by describing nonabelian differential cohomology and twisted nonabelian differential cohomology and apply it to these situations. We demonstrate that the FreedWitten mechanism for the Bfield, the GreenSchwarz mechanism for the H3field, as well as its magnetic dual version for the H7field define cocycles in twisted nonabelian differential cohomology that may be addressed, respectively, as twisted Spin(n), twisted String(n) and twisted Fivebrane(n)structures on target space, where the twist in each case is provided by the obstruction to lifting the gauge bundle through a higher connected cover of U(n). We work out the (nonabelian) L∞algebra valued connection data provided by the differential refinements of these twisted cocycles and demonstrate that this reproduces locally the differential form data with the twisted Bianchi identities as known from the
On twodimensional holonomy
 Trans. Amer. Math. Soc
"... We define the thin fundamental categorical group P2(M, ∗) of a based smooth manifold (M, ∗) as the categorical group whose objects are rank1 homotopy classes of based loops on M, and whose morphisms are rank2 homotopy classes of homotopies between based loops on M. Here two maps are rankn homotop ..."
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Cited by 5 (4 self)
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We define the thin fundamental categorical group P2(M, ∗) of a based smooth manifold (M, ∗) as the categorical group whose objects are rank1 homotopy classes of based loops on M, and whose morphisms are rank2 homotopy classes of homotopies between based loops on M. Here two maps are rankn homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie categorical group coming from a Lie crossed module G = (∂: E → G, ⊲). We construct categorical holonomies, defined to be smooth morphisms P2(M, ∗) → C(G), by using a notion of categorical connections, being a pair (ω, m), where ω is a connection 1form on P, a principal G bundle over M, and m is a 2form on P with values in the Lie algebra of E, with the pair (ω, m) satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context. Key words and phrases nonabelian gerbe; 2bundle, twodimensional holonomy; crossed module; categorical group; Wilson sphere
CATEGORIFIED SYMPLECTIC GEOMETRY AND THE STRING LIE 2ALGEBRA
"... Abstract. Multisymplectic geometry is a generalization of symplectic geometry suitable for ndimensional field theories, in which the nondegenerate 2form of symplectic geometry is replaced by a nondegenerate (n + 1)form. The case n = 2 is relevant to string theory: we call this ‘2plectic geometry ..."
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Cited by 5 (4 self)
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Abstract. Multisymplectic geometry is a generalization of symplectic geometry suitable for ndimensional field theories, in which the nondegenerate 2form of symplectic geometry is replaced by a nondegenerate (n + 1)form. The case n = 2 is relevant to string theory: we call this ‘2plectic geometry’. Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, the observables associated to a 2plectic manifold form a ‘Lie 2algebra’, which is a categorified version of a Lie algebra. Any compact simple Lie group G has a canonical 2plectic structure, so it is natural to wonder what Lie 2algebra this example yields. This Lie 2algebra is infinitedimensional, but we show here that the subLie2algebra of leftinvariant observables is finitedimensional, and isomorphic to the already known ‘string Lie 2algebra ’ associated to G. So, categorified symplectic geometry gives a geometric construction of the string Lie 2algebra. 1.
Representation theory of 2groups on finite dimensional 2vector spaces, in preparation
, 2004
"... In this paper we unfold the 2category structure of the representations of a (strict) 2group on (a suitable version of) Kapranov and Voevodsky’s 2category of finite dimensional 2vector spaces and we discuss the relationship with classical representation theory of groups on finite dimensional vect ..."
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Cited by 4 (1 self)
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In this paper we unfold the 2category structure of the representations of a (strict) 2group on (a suitable version of) Kapranov and Voevodsky’s 2category of finite dimensional 2vector spaces and we discuss the relationship with classical representation theory of groups on finite dimensional vector spaces. In particular, we prove that the monoidal category of representations of any group G appears as a full subcategory of the category of endomorphisms of a particular object in the 2category of representations of G when G is thought of as a 2group with only identity arrows. As an easy consequence of the unfolding process, we also see that every 2group with a compact Lie group as base group has a rank one representation faithful with respect to the base group, contrary to a claim by Barrett and Mackaay (unpublished work). 1
Lie 2algebras
, 2004
"... I would like to express my sincere gratitude to my advisor, John Baez, for his patience, guidance, enthusiasm, encouragement, inspiration, and humor. Additionally, I am indebted to Vyjayanthi Chari, James Dolan, and XiaoSong Lin for their willingness to share their knowledge with me numerous times ..."
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Cited by 4 (1 self)
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I would like to express my sincere gratitude to my advisor, John Baez, for his patience, guidance, enthusiasm, encouragement, inspiration, and humor. Additionally, I am indebted to Vyjayanthi Chari, James Dolan, and XiaoSong Lin for their willingness to share their knowledge with me numerous times during my graduate studies. I also thank my ‘mathematical brothers’: Miguel CarriónÁlvarez, Toby Bartels, Jeffrey Morton, and Derek Wise for their friendship and engaging, educational conversations. I am grateful for the assistance of Aaron Lauda in drawing various braid diagrams, and thank Ronnie Brown, Andrée Ehresmann, Thomas Larsson, James Stasheff, J. Scott Carter, and Masahico Saito for helpful discussions and correspondence. Finally, I am extremely appreciative of the love and support of my family, friends, and former professors during my time as a graduate student. I certainly could not have accomplished all that I have without them. iii ABSTRACT OF THE DISSERTATION