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15
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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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
R.Picken, A Cubical Set Approach to 2Bundles with Connection and Wilson Surfaces
"... In the context of nonabelian gerbes we define a cubical version of categorical group 2bundles with connection over a smooth manifold and consider their twodimensional parallel transport with the aim of defining nonabelian Wilson surface functionals. Key words and phrases: cubical set; nonabelia ..."
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In the context of nonabelian gerbes we define a cubical version of categorical group 2bundles with connection over a smooth manifold and consider their twodimensional parallel transport with the aim of defining nonabelian Wilson surface functionals. Key words and phrases: cubical set; nonabelian gerbe; 2bundle; twodimensional holonomy; twodimensional parallel transport; crossed module; categorical group; double groupoid; Higher Gauge Theory; Wilson surface; Wilson sphere; knotted sphere
THE FUNDAMENTAL CROSSED MODULE OF THE COMPLEMENT Of A Knotted Surface
, 2009
"... We prove that if M is a CWcomplex and M 1 is its 1skeleton, then the crossed module Π2(M,M 1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D 2,S 1). From this it follows that if G is a finite crossed module and M is finite, the ..."
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We prove that if M is a CWcomplex and M 1 is its 1skeleton, then the crossed module Π2(M,M 1) depends only on the homotopy type of M as a space, up to free products, in the category of crossed modules, with Π2(D 2,S 1). From this it follows that if G is a finite crossed module and M is finite, then the number of crossed module morphisms Π2(M,M 1) →Gcan be rescaled to a homotopy invariant IG(M), depending only on the algebraic 2type of M. We describe an algorithm for calculating π2(M,M (1) ) as a crossed module over π1(M (1)), in the case when M is the complement of a knotted surface Σ in S 4 and M (1) is the handlebody of a handle decomposition of M made from its 0 and 1handles. Here, Σ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant IG yields a nontrivial invariant of knotted surfaces in S 4 with good properties with regard to explicit calculations.
NORMALISATION FOR THE FUNDAMENTAL CROSSED COMPLEX OF A SIMPLICIAL SET
, 2007
"... Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. Th ..."
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Cited by 4 (2 self)
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Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes.
Twisted differential nonabelian cohomology Twisted (n−1)brane nbundles and their ChernSimons (n+1)bundles with characteristic (n + 2)classes
, 2008
"... We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shif ..."
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We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian ngroup B n−1 U(1). Notable examples are String 2bundles [9] and Fivebrane 6bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spinstructures to Stringstructures [13] and further to Fivebranestructures [133, 52], are abelian ChernSimons 3 and 7bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [35, 36]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by ∞Lieintegrating the L∞algebraic data in [132]. As a result, even if the lift fails, we obtain twisted String 2 and twisted Fivebrane 6bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted Ktheory. We explain the GreenSchwarz mechanism in heterotic string theory in terms of twisted String 2bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6bundles. We close by transgressing differential cocycles to mapping
Invariants of welded virtual knots via crossed module invariants of knotted surfaces
 Comp. Math
"... We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non trivial by calculating explicit examples. We define welded vi ..."
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We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non trivial by calculating explicit examples. We define welded virtual graphs and consider invariants of them defined in a similar way. Also at Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologia,
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2GROUP ACTIONS
"... Abstract. Transformation groupoids associated to group actions capture the interplay between global and local symmetries of structures described in settheoretic terms. This paper examines the analogous situation for structures described in categorytheoretic terms, where symmetry is expressed as t ..."
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Abstract. Transformation groupoids associated to group actions capture the interplay between global and local symmetries of structures described in settheoretic terms. This paper examines the analogous situation for structures described in categorytheoretic terms, where symmetry is expressed as the action of a 2group G (equivalently, a categorical group) on a category C. It describes the construction of a transformation groupoid in diagrammatic terms, and considers this construction internal to Cat, the
Cubical 2Bundles with Connection and Wilson Spheres
, 2009
"... We define a cubical version of categorical group 2bundles with connection and consider their twodimensional parallel transport with the aim of defining Wilson surface functionals. Key words and phrases cubical set; nonabelian gerbe; 2bundle; twodimensional holonomy; crossed module; categorical g ..."
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We define a cubical version of categorical group 2bundles with connection and consider their twodimensional parallel transport with the aim of defining Wilson surface functionals. Key words and phrases cubical set; nonabelian gerbe; 2bundle; twodimensional holonomy; crossed module; categorical group; double groupoid; Wilson sphere; Wilson surface; Higher gauge theory 2000 Mathematics Subject Classification 53C29 (primary); 18D05, 70S15 (secondary) 1
On nonabelian differential cohomology
, 2008
"... Nonabelian differential ncocycles provide the data for an assignment of “quantities ” to ndimensional “spaces ” which is • locally controlled by a given “typical quantity”; • globally compatible with all possible gluings of volumes. For n = 1 this encompasses the notion of parallel transport in fi ..."
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Nonabelian differential ncocycles provide the data for an assignment of “quantities ” to ndimensional “spaces ” which is • locally controlled by a given “typical quantity”; • globally compatible with all possible gluings of volumes. For n = 1 this encompasses the notion of parallel transport in fiber bundles with connection. In general we think of it as parallel ntransport. For low n and/or “sufficiently abelian quantities ” this has been modeled by differential characters, (n − 1)gerbes, (n − 1)bundle gerbes and nbundles with connection. We give a general definition for all n in terms of descent data for transport nfunctors along the lines of [7, 57, 58, 59]. Concrete realizations, notably ChernSimons ncocycles, are obtained by integrating L∞algebras and their higher CartanEhresmann connections [52]. Here we assume all gluing to happen through equivalences. If one