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The Euler Characteristic of a Category
 DOCUMENTA MATH.
, 2008
"... The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusionexclusion formula. Both rest on a generali ..."
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Cited by 34 (4 self)
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The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusionexclusion formula. Both rest on a generalization of Rota’s Möbius inversion from posets to categories.
Iterated wreath product of the simplex category and iterated loop spaces
 Adv. Math
"... Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternat ..."
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Cited by 25 (5 self)
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Abstract. Generalising Segal’s approach to 1fold loop spaces, the homotopy theory of nfold loop spaces is shown to be equivalent to the homotopy theory of reduced Θnspaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternative description of the Segal spectrum associated to a Γspace. In particular, each EilenbergMacLane space has a canonical reduced Θnset model. The number of (n + d)dimensional cells of the resulting CWcomplex of type K(Z/2Z, n) is a generalised Fibonacci number.
Higher dimensional algebra VII: Groupoidification
, 2010
"... 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 space ..."
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Cited by 23 (5 self)
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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 from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normalordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang–Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field Fq. The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of Fq representations of a simplylaced quiver can be seen as an example of degroupoidification. 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
, 2006
"... Abstract. The concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart to the composition of generalised ..."
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Cited by 23 (3 self)
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Abstract. The concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart to the composition of generalised analytic functors, is also put forward. These definitions encompass most notions of combinatorial species considered in the literature—including of course Joyal’s original notion—together with their associated substitution operation. Our first main result exhibits the substitution calculus of generalised species as arising from a Kleisli bicategory for a pseudocomonad on profunctors. Our second main result establishes that the bicategory of generalised species of structures is cartesian closed. 1.
THE EULER CHARACTERISTIC OF A CATEGORY AS THE SUM OF A DIVERGENT SERIES
"... The Euler characteristic of a cell complex is often thought of as the alternating sum of the number of cells of each dimension. When the complex is infinite, the sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this is possible when the complex is the nerve of a finite categ ..."
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Cited by 22 (4 self)
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The Euler characteristic of a cell complex is often thought of as the alternating sum of the number of cells of each dimension. When the complex is infinite, the sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this is possible when the complex is the nerve of a finite category. This provides an alternative definition of the Euler characteristic of a category, which is in many cases equivalent to the original one. 1.
Categorified algebra and quantum mechanics
, 2006
"... Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial model for some mathematical entity is a par ..."
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Cited by 21 (3 self)
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Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial model for some mathematical entity is a particular instance of the process called “categorification”. Examples include the interpretation of as the Burnside rig of the category of finite sets with product and coproduct, and the interpretation of [x] as the category of combinatorial species. This has interesting applications to quantum mechanics, and in particular the quantum harmonic oscillator, via Joyal’s “species”, a new generalization called “stuff types”, and operators between these, which can be represented as rudimentary Feynman diagrams for the oscillator. In quantum mechanics, we want to represent states in an algebra over the complex numbers, and also want our Feynman diagrams to carry more structure than these “stuff operators ” can do, and these turn out to be closely related. We will show how to construct a combinatorial model for the quantum harmonic
COMBINATORIAL MODELS OF CREATION–ANNIHILATION
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 65 (2011), ARTICLE B65C
, 2011
"... Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB − BA = 1. This study surveys the relationships between classical combinatorial structures and the reduction to normal form of operator ..."
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Cited by 18 (3 self)
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Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB − BA = 1. This study surveys the relationships between classical combinatorial structures and the reduction to normal form of operator polynomials in such an algebra. The connection is achieved through suitable labelled graphs, or “diagrams”, that are composed of elementary “gates”. In this way, many normal form evaluations can be systematically obtained, thanks to models that involve set partitions, permutations, increasing trees, as well as weighted lattice paths. Extensions to qanalogues, multivariate frameworks, and urn models are also briefly discussed.
Mathematical models of computational and combinatorial structures. Invited address for Foundations
 of Software Science and Computation Structures (FOSSACS 2005
, 2005
"... Abstract. The general aim of this talk is to advocate a combinatorial perspective, together with its methods, in the investigation and study of models of computation structures. This, of course, should be taken in conjunction with the wellestablished views and methods stemming from algebra, category ..."
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Cited by 17 (8 self)
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Abstract. The general aim of this talk is to advocate a combinatorial perspective, together with its methods, in the investigation and study of models of computation structures. This, of course, should be taken in conjunction with the wellestablished views and methods stemming from algebra, category theory, domain theory, logic, type theory, etc. In support of this proposal I will show how such an approach leads to interesting connections between various areas of computer science and mathematics; concentrating on one such example in some detail. Specifically, I will consider the line of my research involving denotational models of the pi calculus and algebraic theories with variablebinding operators, indicating how the abstract mathematical structure underlying these models fits with that of Joyal’s combinatorial species of structures. This analysis suggests both the unification and generalisation of models, and in the latter vein I will introduce generalised species of structures and their calculus. These generalised species encompass and generalise various of the notions of species used in combinatorics. Furthermore, they have a rich mathematical structure (akin to models of Girard’s linear logic) that can be described purely within Lawvere’s generalised logic. Indeed, I will present and treat the cartesian closed structure, the linear structure, the differential structure, etc. of generalised species axiomatically in this mathematical framework. As an upshot, I will observe that the setting allows for interpretations of computational calculi (like the lambda calculus, both typed and untyped; the recently introduced differential lambda calculus of Ehrhard and Regnier; etc.) that can be directly seen as translations into a more basic elementary calculus of interacting agents that compute by communicating and operating upon structured data.
Feynman diagrams in algebraic combinatorics, Séminaire Lotharingien de Combinatoire 49
 2002–04), Article B49c, 45 pp. SPECIES AND FEYNMAN DIAGRAMS 37
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FINITENESS OBSTRUCTIONS AND EULER CHARACTERISTICS OF CATEGORIES
, 2009
"... We introduce notions of finiteness obstruction, Euler characteristic, L²Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type (FP) is a class in the projective class group K0(RΓ); the Euler characteristic and L²Euler character ..."
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Cited by 14 (2 self)
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We introduce notions of finiteness obstruction, Euler characteristic, L²Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type (FP) is a class in the projective class group K0(RΓ); the Euler characteristic and L²Euler characteristic are respectively its RΓrank and L²rank. We also extend the second author’s Ktheoretic Möbius inversion from finite categories to quasifinite categories. Our main