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24
Higher gauge theory
"... Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1dimensional objects (e.g. strings) using 2connections on 2bundles. A 2bundle is a categorified version of a bundle: that is, one where t ..."
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Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1dimensional objects (e.g. strings) using 2connections on 2bundles. A 2bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge theory uses Lie groups and Lie algebras, higher gauge theory uses their categorified analogues: Lie 2groups and Lie 2algebras. We describe a theory of 2connections on principal 2bundles and explain how this is related to Breen and Messing’s theory of connections on nonabelian gerbes. The distinctive feature of our theory is that a 2connection allows parallel transport along paths and surfaces in a parametrizationindependent way. In terms of Breen and Messing’s framework, this requires that the ‘fake curvature ’ must vanish. In this paper we summarize the main results of our theory without proofs. 1
Convenient Categories of Smooth Spaces
, 2008
"... A ‘Chen space ’ is a set X equipped with a collection of ‘plots ’ — maps from convex sets to X — satisfying three simple axioms. While an individual ..."
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Cited by 38 (0 self)
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A ‘Chen space ’ is a set X equipped with a collection of ‘plots ’ — maps from convex sets to X — satisfying three simple axioms. While an individual
An Invitation to Higher Gauge Theory
, 2010
"... In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2connections on 2bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2group’. We focus on 6 examples. First, every abelian Lie gr ..."
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Cited by 20 (5 self)
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In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2connections on 2bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2group’, which serves as a gauge 2group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2group’. We also touch upon higher structures such as the ‘gravity 3group’, and the Lie 3superalgebra that governs 11dimensional supergravity. 1
A convenient differential category
, 2011
"... We show that the category of convenient vector spaces in the sense of Frölicher ..."
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Cited by 20 (2 self)
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We show that the category of convenient vector spaces in the sense of Frölicher
The Fundamental Groupoid as a Topological Groupoid
 Proc. Edinburgh Math. Soc
, 1975
"... Let X be a topological space. Then we may define the fundamental groupoid nX and also the quotient groupoid (nX)/N for N any wide, totally disconnected, normal subgroupoid N of nX (1). The purpose of this note is to show that if X is locally pathconnected and semilocally 1connected, then the topo ..."
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Cited by 14 (5 self)
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Let X be a topological space. Then we may define the fundamental groupoid nX and also the quotient groupoid (nX)/N for N any wide, totally disconnected, normal subgroupoid N of nX (1). The purpose of this note is to show that if X is locally pathconnected and semilocally 1connected, then the topology of X
Internal categories, anafunctors and localisations
"... In this article we review the theory of anafunctors introduced by Makkai and Bartels, and show that given a subcanonical site S, one can form a bicategorical localisation of various 2categories of internal categories or groupoids at weak equivalences using anafunctors as 1arrows. This unifies a nu ..."
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Cited by 7 (1 self)
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In this article we review the theory of anafunctors introduced by Makkai and Bartels, and show that given a subcanonical site S, one can form a bicategorical localisation of various 2categories of internal categories or groupoids at weak equivalences using anafunctors as 1arrows. This unifies a number of proofs throughout the literature, using the fewest assumptions possible on S. 1.
Geometric structures encoded in the Lie structure of an Atiyah algebroid
, 2009
"... We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic, then the corresponding base manifolds are necessarily diffe ..."
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We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic, then the corresponding base manifolds are necessarily diffeomorphic. Further, we give two characterizations of the isomorphisms of the Lie algebras of sections for Atiyah algebroids associated to principle bundles with semisimple structure groups. For instance we prove that in the semisimple case the Lie algebras of sections are isomorphic if and only if the corresponding Lie algebroids are, or, as well, if and only if the integrating principal bundles are locally diffeomorphic. Finally, we apply these results to describe the isomorphisms of sections in the case of reductive structure groups – surprisingly enough they are no longer determined by vector bundle isomorphisms and involve divergences on the base manifolds.