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55

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

61
61

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

48
48

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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48

Sep 21, 2013
09/13

by
John C. Baez; Christopher L. Rogers

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Multisymplectic geometry is a generalization of symplectic geometry suitable for n-dimensional field theories, in which the nondegenerate 2-form of symplectic geometry is replaced by a nondegenerate (n+1)-form. The case n = 2 is relevant to string theory: we call this 2-plectic geometry. Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, the observables associated to a 2-plectic manifold form a "Lie 2-algebra", which is a categorified...

Source: http://arxiv.org/abs/0901.4721v1