From Finite Sets to Feynman Diagrams

By John C. Baez, J. Dolan. 2000

201 citations recorded by the source corpus.

`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 Omega^infinity S^infinity, 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 `especes 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.

Subfields: Artificial Intelligence

View Paper: https://www.semanticscholar.org/paper/af4ee59c0a3b2531a1f75b4ce89cc31eb4f7ab58

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In Artificial Intelligence

1. Joachim Kock
2. C. Berger
3. Tom Leinster
4. Martin Brandenburg

Citations in the corpus

• Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory (2015, background): “Commutative triangle in an ∞-category C means instead ‘object in the functor ∞-category Fun(∆[2],C)’: the 2-dimensional face of ∆[2] is mapped to a 2-cell in C mediating between the composite of the 01 and 12 edges and the long edge 02.”
• Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness (2015, background): “[1,10]) The homotopy cardinality of a finite ∞-groupoid B is by definition”
• The magnitude of metric spaces (2010, background): “network of theorems describes the close relationships between this invariant and established cardinality-like invariants, including the cardinality of sets and of groupoids [1], the Euler characteristic of topological spaces and of posets, and even the Euler characteristic of orbifolds.”
• Tensor categorical foundations of algebraic geometry (2014, methodology): “We refer to the wonderful papers [BD98a] and [BD01] for the general principles of categorification.”
• Higher Dimensional Algebra VII: Groupoidification (2009, unknown): “This in turn provides a new way to categorify— or more precisely, groupoidify—the positive part of the quantum group associated to the quiver.”
• The cartesian closed bicategory of generalised species of structures (2008, background): “More recently, Baez and Dolan considered further generalisations of these structures leading to the concepts of sorted symmetric set-operad [1] and of stuff types [2] (see also [37]).”
• Iterated wreath product of the simplex category and iterated loop spaces (2005, background): “A counting lemma of Dolan [17] shows that the number of (n + d)-dimensional cells of the resulting CW-complex of type K(Z/2Z, n) is a generalised Fibonacci number, the classical one’s arising for n = 2.”
• THE EULER CHARACTERISTIC OF A CATEGORY AS THE SUM OF A DIVERGENT SERIES (2007, background): “that this is the ‘right’ value for the Euler characteristic of G (or BG) from several points of view: see [W] and [BD], for instance.”