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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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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.
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.
Higher Gauge Theory II: 2Connections
"... 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 el ..."
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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
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
unknown title
, 2003
"... Higher gauge theory — differential versus integral formulation ..."
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unknown title
, 2004
"... Higher gauge theory — differential versus integral formulation ..."
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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 ..."
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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.