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Differentiable Stacks and Gerbes
, 2008
"... We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹bundles and S¹gerbes over differentiable stacks. In particular, we establish the relationship between S¹gerbes and groupoid S¹central extensions. We define connections and curvings for groupoid S¹ ..."
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Cited by 14 (3 self)
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We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹bundles and S¹gerbes over differentiable stacks. In particular, we establish the relationship between S¹gerbes and groupoid S¹central extensions. We define connections and curvings for groupoid S¹central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S¹gerbes over manifolds. We develop a ChernWeil theory of characteristic classes in this general setting by presenting a construction of Chern classes and DixmierDouady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both S¹bundles and S¹gerbes extending the wellknown result of Weil and Kostant. In particular, we give an explicit construction of S¹central extensions with prescribed curvaturelike data.
Origins and breadth of the theory of higher homotopies
, 2007
"... Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least ..."
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Cited by 7 (3 self)
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Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940’s. Prompted by the failure of the AlexanderWhitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor’s classifying space construction to associative Hspaces, and a careful examination of this extension led Stasheff to the discovery of Anspaces and A∞spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950’s, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic
NOTES ON 1 AND 2GERBES
, 2006
"... The aim of these notes is to discuss in an informal manner the construction and some properties of 1 and 2gerbes. They are for the most part based on the author’s texts [1][4]. Our main goal is to describe the construction which associates to a gerbe or a 2gerbe the corresponding nonabelian coh ..."
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The aim of these notes is to discuss in an informal manner the construction and some properties of 1 and 2gerbes. They are for the most part based on the author’s texts [1][4]. Our main goal is to describe the construction which associates to a gerbe or a 2gerbe the corresponding nonabelian cohomology class. We begin by reviewing the wellknown theory for principal bundles and show how to extend this to biprincipal bundles (a.k.a bitorsors). After reviewing the definition of stacks and gerbes, we construct the cohomology class associated to a gerbe. While the construction presented is equivalent to that in [4], it is clarified here by making use of diagram (5.1.9), a definite improvement over the corresponding diagram [4] (2.4.7), and of (5.2.7). After a short discussion regarding the role of gerbes in algebraic topology, we pass from 1 − to 2−gerbes. The construction of the associated cohomology classes follows the same lines as for 1gerbes, but with the additional degree of complication entailed by passing from 1 to 2categories, so that it now involves diagrams reminiscent of those in [5]. Our emphasis will be on explaining how the fairly elaborate equations which define cocycles and coboundaries may be reduced to terms which can be described in the tradititional formalism of nonabelian cohomology. Since the concepts discussed here are very general, we have at times not made explicit the mathematical