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Sep 22, 2013
09/13

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John C. Baez; Aaron D. Lauda

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A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call "weak" and "coherent" 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a "weak inverse":...

Source: http://arxiv.org/abs/math/0307200v3

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932

Sep 22, 2013
09/13

by
John C. Baez; Aaron Lauda

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This paper traces the growing role of categories and n-categories in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts which manifest themselves in Feynman diagrams, spin networks, string theory, loop quantum gravity, and topological quantum field theory. Our chronology ends around 2000, with just a taste of later developments such as open-closed topological string theory, the categorification of quantum groups, Khovanov homology, and...

Source: http://arxiv.org/abs/0908.2469v1

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Sep 18, 2013
09/13

by
John C. Baez; Alejandro Perez

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BF theory is a topological theory that can be seen as a natural generalization of 3-dimensional gravity to arbitrary dimensions. Here we show that the coupling to point particles that is natural in three dimensions generalizes in a direct way to BF theory in d dimensions coupled to (d-3)-branes. In the resulting model, the connection is flat except along the membrane world-sheet, where it has a conical singularity whose strength is proportional to the membrane tension. As a step towards...

Source: http://arxiv.org/abs/gr-qc/0605087v1

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96

Jul 22, 2013
07/13

by
John C. Baez; Alexander E. Hoffnung

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A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's "diffeological...

Source: http://arxiv.org/abs/0807.1704v4

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Sep 22, 2013
09/13

by
John C. Baez; Alexander E. Hoffnung; Christopher D. Walker

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Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of "degroupoidification": a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally...

Source: http://arxiv.org/abs/0908.4305v3

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Sep 22, 2013
09/13

by
John C. Baez; Alexander E. Hoffnung; Christopher D. Walker

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Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of 'degroupoidification': a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from...

Source: http://arxiv.org/abs/0812.4864v1

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Sep 21, 2013
09/13

by
John C. Baez; Alexander E. Hoffnung; Christopher L. Rogers

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A Lie 2-algebra is a "categorified" version of a Lie algebra: that is, a category equipped with structures analogous those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an "n-plectic...

Source: http://arxiv.org/abs/0808.0246v1

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Sep 22, 2013
09/13

by
John C. Baez; Alissa S. Crans

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The theory of Lie algebras can be categorified starting from a new notion of "2-vector space", which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector spaces as objects, "linear functors" as morphisms and "linear natural transformations" as 2-morphisms. We define a "semistrict Lie 2-algebra" to be a 2-vector space L equipped with a skew-symmetric bilinear functor satisfying the Jacobi identity up to a completely...

Source: http://arxiv.org/abs/math/0307263v6

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Sep 18, 2013
09/13

by
John C. Baez; Alissa S. Crans; Danny Stevenson; Urs Schreiber

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We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras g_k each having Lie(G) as its Lie algebra...

Source: http://arxiv.org/abs/math/0504123v2

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Sep 22, 2013
09/13

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John C. Baez; Aristide Baratin; Laurent Freidel; Derek K. Wise

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A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector spaces", which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional...

Source: http://arxiv.org/abs/0812.4969v2

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Sep 22, 2013
09/13

by
John C. Baez; Derek K. Wise; Alissa S. Crans

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After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a...

Source: http://arxiv.org/abs/gr-qc/0603085v2