In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity. 1

We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of the Green-Schwarz anomaly cancelation in heterotic string theory which demands the target space to have a String structure, we observe that the “magnetic dual ” version of the anomaly cancelation condition can be read as a higher analog of String structure, which we call Fivebrane structure. This involves lifts of orthogonal and unitary structures through higher connected covers which are not just 3-but even 7-connected. We discuss the topological obstructions to the existence of Fivebrane structures. The dual version of the anomaly cancelation points to a relation of String and Fivebrane structures under

For any topological bicategory 2C, the Duskin nerve N2C of 2C is a simplicial space. We introduce the classifying topos B2C of 2C as the Deligne topos of sheaves Sh(N2C) on the simplicial space N2C. It is shown that the category of topos morphisms Hom(Sh(X), BC) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal C-bundles. As a simple consequence, the geometric realization |N2C | of the nerve N2C of a locally contractible topological bicategory 2C is the classifying space of principal 2C-bundles (on CW complexes), giving a variant of the result of Baas, Bökstedt and Kro derived in the context of bicategorical K-theory [1]. We also define classifying topoi of a topological bicategory 2C using sheaves on other types of nerves of a bicategory given by Lack and Paoli [13], Simpson [17] and Tamsamani [18] by means of bisimplicial spaces, and we examine their properties. 1