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Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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For a copy with the handdrawn figures please email
Higher dimensional algebra III: ncategories and the algebra of opetopes
, 1997
"... We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads ..."
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We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads over O. Letting I be the initial operad with a oneelement set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘ndimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak ncategory is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ωcategories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘ncoherent Oalgebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal ncategories’, ‘stable ncategories’, ‘virtual nfunctors ’ and ‘representable nprestacks’. We also describe how ncoherent Oalgebra objects may be defined in any (n + 1)coherent Oalgebra.
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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‘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
A closed model structure for ncategories, internal Hom, nstacks and generalized SeifertVan Kampen
, 1997
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On the BreenBaezDolan stabilization hypothesis for Tamsamani’s weak ncategories
, 1998
"... In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for ncategories of the wellknown stabilization theorems in homotopy theory. To explain the statement, recall that BaezDol ..."
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In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for ncategories of the wellknown stabilization theorems in homotopy theory. To explain the statement, recall that BaezDolan introduce the notion of kuply monoidal ncategory which is an n + kcategory having only one imorphism for all i < k. This includes the notions previously defined and examined by many authors, of monoidal (resp. braided monoidal, symmetric monoidal) category (resp. 2category) and so forth, as is explained in [2] [4]. See the bibliographies of those preprints as well as that of the the recent preprint [9] for many references concerning these types of objects. In the case where the ncategory in question is an ngroupoid, this notion is—except for truncation at n—the same thing as the notion of kfold iterated loop space, or “Ekspace ” which appears in Dunn [10] (see also some anterior references from there). The fully stabilized notion of kuply monoidal ncategories for k ≫ n is what Grothendieck calls Picard ncategories in [12]. The stabilization hypothesis [2] states that for n + 2 ≤ k ≤ k ′ , the kuply monoidal
Exit paths and constructible stacks
 Compositio Math
"... Abstract. For a Whitney stratification S of a space X (or more generally a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an Sconstructible stack of categories on X. The motivating example is the stack of Sconstructible perverse sheaves. We introduce ..."
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Abstract. For a Whitney stratification S of a space X (or more generally a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an Sconstructible stack of categories on X. The motivating example is the stack of Sconstructible perverse sheaves. We introduce a 2category EP≤2(X, S), called the exitpath 2category, which is a natural stratified version of the fundamental 2groupoid. Our main result is that the 2category of Sconstructible stacks on X is equivalent to the 2category of 2functors 2Funct(EP≤2(X, S),Cat) from the exitpath 2category to the 2category of small categories. 1.
Categorification
 Contemporary Mathematics 230. American Mathematical Society
, 1997
"... Categorification is the process of finding categorytheoretic analogs of settheoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘c ..."
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Categorification is the process of finding categorytheoretic analogs of settheoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘coherence laws’. Iterating this process requires a theory of ‘ncategories’, algebraic structures having objects, morphisms between objects, 2morphisms between morphisms and so on up to nmorphisms. After a brief introduction to ncategories and their relation to homotopy theory, we discuss algebraic structures that can be seen as iterated categorifications of the natural numbers and integers. These include tangle ncategories, cobordism ncategories, and the homotopy ntypes of the loop spaces Ω k S k. We conclude by describing a definition of weak ncategories based on the theory of operads. 1
Limits in ncategories
, 1997
"... One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim ” for short. The purpose of the this paper is to begin to develop ..."
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One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim ” for short. The purpose of the this paper is to begin to develop
Secondary KodairaSpencer classes and nonabelian Dolbeault cohomology
, 1998
"... One of the nicest things about variations of Hodge structure is the “infinitesimal variation of Hodge structure ” point of view [25]. A variation of Hodge structure (V = V p,q, ∇) over a base S gives rise at any point s ∈ S to the KodairaSpencer map κs: T(S)s → Hom(V p,q s, V p−1,q+1 s). In the geo ..."
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One of the nicest things about variations of Hodge structure is the “infinitesimal variation of Hodge structure ” point of view [25]. A variation of Hodge structure (V = V p,q, ∇) over a base S gives rise at any point s ∈ S to the KodairaSpencer map κs: T(S)s → Hom(V p,q s, V p−1,q+1 s). In the geometric situation of a family X → S we have V p,q s = Hq (Xs, Ω p