Abstract. 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

We show, in great detail, how the perturbative tools of quantum field theory allow one to rigorously obtain: a “categorified ” Faa di Bruno type formula for multiple composition, an explicit formula for reversion and a proof of Lagrange-Good inversion, all in the setting of multivariable power series. We took great pains to offer a self-contained presentation that, we hope, will provide any mathematician who wishes, an easy access to the wonderland of quantum field theory. Key words: Quantum field theory, Combinatorial species. I

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 variable-binding 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.

Abstract. Generalising Segal’s approach to 1-fold loop spaces, the homotopy theory of n-fold loop spaces is shown to be equivalent to the homotopy theory of reduced Θn-spaces, 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 Γ-space. In particular, each Eilenberg-MacLane 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.

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 inclusion-exclusion formula. Both rest on a generalization of Rota’s Möbius inversion from posets to categories.

The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal

Abstract. We show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pull-back or push-forward formulas for integrals over suitable groupoids.

We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for the Bernoulli and Euler numbers and polynomials.

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 normal-ordered 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 simply-laced 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 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.