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A Cocycle Model for Topological and Lie Group Cohomology
, 2011
"... We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on equal footings. In particular, we show that the cohomology of “locally continuous ” cochains (respectively “locally smooth ” in the case of Lie groups) fits ..."
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Cited by 7 (5 self)
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We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on equal footings. In particular, we show that the cohomology of “locally continuous ” cochains (respectively “locally smooth ” in the case of Lie groups) fits into this framework, which provides an easily accessible cocycle model for topological and Lie group cohomology. We illustrate the use of this unified framework and the relation between the different models in various applications. This includes the construction of cohomology classes characterizing the string group and a direct connection to Lie algebra cohomology.
QUANTUM GAUGE FIELD THEORY IN COHESIVE HOMOTOPY TYPE THEORY
"... Abstract. We implement in the formal language of homotopy type theory a new set of axioms called cohesion. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in quantum gauge field theory. This is a brief survey of work by th ..."
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Abstract. We implement in the formal language of homotopy type theory a new set of axioms called cohesion. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in quantum gauge field theory. This is a brief survey of work by the authors developed in detail elsewhere [48, 45]. Contents
Principal ∞bundles – General theory
, 2012
"... The theory of principal bundles makes sense in any ∞topos, such as the ∞topos of topological, of smooth, or of otherwise geometric ∞groupoids/∞stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fib ..."
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Cited by 4 (3 self)
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The theory of principal bundles makes sense in any ∞topos, such as the ∞topos of topological, of smooth, or of otherwise geometric ∞groupoids/∞stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure ∞group G these Gprincipal ∞bundles reproduce the theories of ordinary principal bundles, of bundle gerbes/principal 2bundles and of bundle 2gerbes and generalize these to their further higher and equivariant analogs. The induced associated ∞bundles subsume the notion of Giraud’s gerbes, Breen’s 2gerbes, Lurie’s ngerbes, and generalize these to the notion of nonabelian ∞gerbes; which are the universal local coefficient bundles for nonabelian twisted cohomology. We discuss here this general abstract theory of principal ∞bundles, observing that it is intimately related to the axioms of Giraud, ToënVezzosi, Rezk and Lurie that characterize ∞toposes. A central result is a natural equivalence between principal ∞bundles and intrinsic nonabelian cocycles, implying the classification of principal
Multiple M5branes, String 2connections, and 7d nonabelian ChernSimons theory
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A Construction of String 2Group Models using a TransgressionRegression Technique
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Principal ∞bundles – Presentations
 University of Notre Dame
, 2006
"... We discuss two aspects of the presentation of the theory of principal ∞bundles in an ∞topos, introduced in [NSSa], in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site C and for G a presheaf of simplicial groups which is Cacyclic, Gprincipal ∞bundles over a ..."
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We discuss two aspects of the presentation of the theory of principal ∞bundles in an ∞topos, introduced in [NSSa], in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site C and for G a presheaf of simplicial groups which is Cacyclic, Gprincipal ∞bundles over any object in the ∞topos over C are classified by hyperČechcohomology with coefficients in G. Then we show that over a site C with enough points, principal ∞bundles in the ∞topos are presented by ordinary simplicial bundles in the sheaf topos that satisfy principality by stalkwise weak equivalences. Finally we discuss explicit details of these presentations for the discrete site (in discrete ∞groupoids) and the smooth site (in smooth ∞groupoids, generalizing Lie groupoids and differentiable stacks). In the companion article [NSSc] we use these presentations for constructing classes of examples of (twisted) principal ∞bundles and for the discussion of various applications.
∞sheaves = ∞stacks
"... Two formalized methods of quantization: algebra algebraic quantization (deformation quantization) Isbell duality geometry geometric quantization best discussed in the context of: higher algebra ∞cosheaves considered elsewhere ..."
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Two formalized methods of quantization: algebra algebraic quantization (deformation quantization) Isbell duality geometry geometric quantization best discussed in the context of: higher algebra ∞cosheaves considered elsewhere