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Pseudo algebras and pseudo double categories
 J. Homotopy Relat. Struct
"... Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, an ..."
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Cited by 16 (2 self)
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Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group.
DAMTP200327 Higher Gauge Theory and a nonAbelian generalization
, 2003
"... In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of ..."
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In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelian symmetry groups, say G = U(1), there exists a generalization, known as pform electrodynamics, in which (p − 1)dimensional charged objects can be propagated along psurfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed psurfaces. In this article, we use Lie 2groups and ideas from higher category theory in order to formulate a discrete gauge theory which generalizes these models at the level p = 2 to possibly nonAbelian symmetry groups. The main new feature is that our model involves both parallel transports along paths and generalized transports along surfaces with a nontrivial interplay of these two types of variables. We construct the precise assignment of variables to the curves and surfaces, the generalized local symmetries and gauge invariant actions and we clarify which structures can be nonAbelian and which others are always Abelian. A discrete version of connections on nonAbelian gerbes is a special case of our construction. Even though the motivation sketched so far suggests applications mainly in string theory, the model presented here is also related to spin foam models of quantum gravity and may in addition provide some insight into the role of centre monopoles and vortices in lattice QCD.
STRICT 2GROUPS ARE CROSSED MODULES SVENS. PORST
, 812
"... ABSTRACT. The 2categories of strict 2groups and crossed modules are introduced and their 2equivalence is made explicit. ..."
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ABSTRACT. The 2categories of strict 2groups and crossed modules are introduced and their 2equivalence is made explicit.
Preprint typeset in JHEP style HYPER VERSION Higher Gauge Theory II: 2Connections
"... Abstract: Connections and curvings on gerbes are beginning to play a vital role in differential geometry and theoretical physics — first abelian gerbes, and more recently nonabelian gerbes and the twisted nonabelian gerbes introduced by Aschieri and Jurčo in their study of Mtheory. These concepts c ..."
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Abstract: Connections and curvings on gerbes are beginning to play a vital role in differential geometry and theoretical physics — first abelian gerbes, and more recently nonabelian gerbes and the twisted nonabelian gerbes introduced by Aschieri and Jurčo in their study of Mtheory. These concepts can be elegantly understood using the concept of ‘2bundle’ recently introduced by Bartels. A 2bundle is a generalization of a bundle in which the fibers are categories rather than sets. Here we introduce the concept of a ‘2connection’ on a principal 2bundle. We describe principal 2bundles with connection in terms of local data, and show that under certain conditions this reduces to the cocycle data for twisted nonabelian gerbes with connection and curving subject to a certain constraint — namely, the vanishing of the ‘fake curvature’, as defined by Breen and Messing. This constraint also turns out to guarantee the existence of ‘2holonomies’: that is, parallel transport over both curves and surfaces, fitting together to define a 2functor from the ‘path 2groupoid’ of the base space to the structure 2group. We give a general theory of 2holonomies and show how they are related to ordinary parallel transport on the path space of the base
PSEUDO ALGEBRAS AND PSEUDO DOUBLE CATEGORIES
 JOURNAL OF HOMOTOPY AND RELATED STRUCTURES, VOL. 2(2), 2007, PP.119–170
, 2007
"... As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to ..."
Abstract
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As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group.