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From Finite Sets to Feynman Diagrams
 Mathematics Unlimited  2001 And Beyond
, 2001
"... ‘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 ..."
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Cited by 62 (6 self)
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‘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
Representable Multicategories
 Advances in Mathematics
, 2000
"... We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe ..."
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We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows . We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2equivalence between the 2category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a se...
Higherdimensional algebra IV: 2Tangles
"... Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we p ..."
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Cited by 34 (10 self)
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Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain 2morphisms in the 2category of ‘2tangles in 4 dimensions’. Using the work of Carter, Rieger and Saito, we prove that this 2category is the ‘free semistrict braided monoidal 2category with duals on one unframed selfdual object’. By this universal property, any unframed selfdual object in a braided monoidal 2category with duals determines an invariant of 2tangles in 4 dimensions. 1
Operads In HigherDimensional Category Theory
, 2004
"... The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n < ..."
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The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higherdimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to ncategories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
Generalized Centers of Braided and Sylleptic Monoidal 2Categories
, 1997
"... Recent developments in higherdimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic and symmetric monoidal 2categories, and a center construction for monoidal 2categories which gives a braided monoidal 2category. I give ge ..."
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Cited by 26 (3 self)
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Recent developments in higherdimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic and symmetric monoidal 2categories, and a center construction for monoidal 2categories which gives a braided monoidal 2category. I give generalized center constructions for braided and sylleptic monoidal 2categories which give sylleptic and symmetric monoidal 2categories respectively, and I correct some errors in the original center construction for monoidal 2categories. 1 Introduction The initial motivation for the study of braided monoidal categories was twofold: from homotopy theory, where braided monoidal categories of a particular kind arise as algebraic 3types of arcconnected, simply connected spaces, and from higherdimensional category theory, where braided monoidal categories arise as one object monoidal bicategories [16]. These motivations have subsequently been brought together by the definition of tricategori...
A closed model structure for ncategories, internal Hom, nstacks and generalized SeifertVan Kampen
, 1997
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An introduction to ncategories
 Proc. 7th Conf. Category Theory and Computer Science
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Descente pour les nchamps
"... We develop the theory of nstacks (or more generally Segal nstacks which are ∞stacks such that the morphisms are invertible above degree n). This is done by systematically using the theory of closed model categories (cmc). Our main results are: a definition of nstacks in terms of limits, which sh ..."
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We develop the theory of nstacks (or more generally Segal nstacks which are ∞stacks such that the morphisms are invertible above degree n). This is done by systematically using the theory of closed model categories (cmc). Our main results are: a definition of nstacks in terms of limits, which should be perfectly general for stacks of any type of objects; several other characterizations of nstacks in terms of “effectivity of descent data”; construction of the stack associated to an nprestack; a strictification result saying that any “weak ” nstack is equivalent to a (strict) nstack; and a descent result saying that the (n + 1)prestack of nstacks (on a site) is an (n + 1)stack. As for other examples, we start from a “left Quillen presheaf ” of cmc’s and introduce the associated Segal 1prestack. For this situation, we prove a general descent result, giving sufficient conditions for this prestack to be a stack. This applies to the case of complexes, saying how complexes of sheaves of Omodules can be glued together via quasiisomorphisms. This was the problem that originally motivated us. Résumé