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Differential cohomology in a cohesive ∞topos
"... We formulate differential cohomology and ChernWeil theory the theory of connections on fiber bundles and of gauge fields abstractly in the context of a certain class of higher toposes that we call cohesive. Cocycles in this differential cohomology classify higher principal bundles equipped with c ..."
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Cited by 9 (6 self)
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We formulate differential cohomology and ChernWeil theory the theory of connections on fiber bundles and of gauge fields abstractly in the context of a certain class of higher toposes that we call cohesive. Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, synthetic differential, supergeometric, etc.) and equipped with connections. We discuss various models of the axioms and wealth of applications revolving around fundamental notions and constructions in prequantum field theory and string theory. In particular we show that the cohesive and differential refinement of universal characteristic cocycles constitutes a higher ChernWeil homomorphism refined from secondary caracteristic classes to morphisms of higher moduli stacks of higher gauge fields, and at the same time constitutes extended geometric prequantization – in the sense of extended/multitiered quantum field theory – of higher dimensional ChernSimonstype field theories and WessZuminoWittentype field theories. This document, and accompanying material, is kept online at ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos 1 We formulate differential cohomology and ChernWeil theory the theory of connections on fiber bundles
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
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.