## From Finite Sets to Feynman Diagrams (2001)

Venue: | Mathematics Unlimited - 2001 And Beyond |

Citations: | 50 - 6 self |

### BibTeX

@INPROCEEDINGS{Baez01fromfinite,

author = {John C. Baez and James Dolan and Björn Engquist and Wilfried Schmid},

title = {From Finite Sets to Feynman Diagrams},

booktitle = {Mathematics Unlimited - 2001 And Beyond},

year = {2001},

pages = {29--50},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

‘Categorification ’ is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin with elementary arithmetic, where the category of finite sets serves as a categorified version of the set of natural numbers, with disjoint union and Cartesian product playing the role of addition and multiplication. We sketch how categorifying the integers leads naturally to the infinite loop space Ω ∞ S ∞ , and how categorifying the positive rationals leads naturally to a notion of the ‘homotopy cardinality ’ of a tame space. Then we show how categorifying formal power series leads to Joyal’s espèces des structures, or ‘structure types’. We also describe a useful generalization of structure types called ‘stuff types’. There is an inner product of stuff types that makes the category of stuff types into a categorified version of the Hilbert space of the quantized harmonic oscillator. We conclude by sketching how this idea gives a nice explanation of the combinatorics of Feynman diagrams. 1

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Citation Context ...pursued in many directions, especially within combinatorics. As an example, we describe how formal power series rise from decategorifying Andre Joyal's `structure types', or `especes de structures' [1=-=5, 16]-=-. A structure type is any sort of structure onsnite sets that transforms naturally under permutations; counting the structures of a given type that can be put on a set with n elements is one of the ba... |

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Citation Context ...pace G(B(FinSet 0 )) = 1 S 1 where the space on the right is a kind of limit of the n-fold loop space n S n as n !1. This space is fundamental to a branch of mathematics called stable homotopy theory =-=[1, 13]-=-. It has nonvanishing homotopy groups in arbitrarily high dimensions, so we should really think of it as an `!-groupoid'. What this means is that to properly categorify subtraction, we need to categor... |

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Citation Context ... the classifying space B(FinSet 0 ). This notion allows for a fascinating interplay between Feynman diagrams and homotopy theory. Unfortunately, for the details the reader will have to turn elsewhere =-=[7-=-]. 5 Feynman Diagrams and Stu Operators From here we could go in various directions. But since we are dreaming about the future of mathematics, let us choose a rather speculative one, and discuss some... |