Here you can see a set of related talks that John Baez, Alex Hoffnung and Chris Rogers gave in Göttingen.
Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some sort of "higher gauge theory" that describes parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, we must "categorify" concepts from topology and geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, Lie algebras by Lie 2-algebras, bundles by 2-bundles, sheaves by stacks or gerbes, and so on. This overview of higher gauge theory will emphasize its relation to homotopy theory and the cohomology of groups and Lie algebras.
Click on these to see transparencies of the talks:
In 1977 K.T. Chen introduced a notion of smooth spaces as a generalization of the category of smooth manifolds. In 1979 Souriau introduced another notion, "diffeological spaces", serving the same purposes. Both of these categories have all limits and colimits, and are cartesian closed. In fact, following ideas of Dubuc, we give a unified proof that the categories of Chen spaces, diffeological spaces, and simplicial complexes are "quasitopoi" locally cartesian closed categories with finite (and in these cases all) colimits and a weak subobject classifier.The transparencies are available here.
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Just as symplectic geometry is a natural setting for the classical mechanics of point particles, 2-plectic geometry can be used to describe classical strings. Just as a symplectic manifold is equipped with a closed non-degenerate 2-form, a "2-plectic manifold" is equipped with a closed non-degenerate 3-form.The transparencies are available here.The Poisson bracket makes the smooth functions on a symplectic manifold form a Lie algebra. Similarly, any 2-plectic manifold gives a "Lie 2- algebra": the categorified analogue of a Lie algebra, where the usual laws hold only up to isomorphism. We explain these ideas and use them to give a new construction of the "string Lie 2-algebra" associated to a simple Lie group.
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