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Internal categorical structure in homotopical algebra
 Proceedings of the IMA workshop ?nCategories: Foundations and Applications?, June 2004, (to appear). CROSSED MODULES AND PEIFFER CONDITION 135 [Ped95] [Por87
, 1995
"... Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1. ..."
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Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1.
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
Adding inverses to diagrams II: Invertible homotopy theories are spaces, preprint available at math.AT/0710.2254
"... Abstract. In previous work, we showed that there are appropriate model ..."
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Abstract. In previous work, we showed that there are appropriate model
A PREHISTORY OF nCATEGORICAL PHYSICS
, 2008
"... We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, me ..."
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We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, membranes and spin foams.