The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the semantics of nondeterministic dataflow. Profunctors are shown to play a key role in relating models for concurrency and to support an interpretation as higher-order processes (where input and output may be processes). Two recent directions of research are described. One is concerned with a language and computational interpretation for profunctors. This addresses the duality between input and output in profunctors. The other is to investigate general spans of event structures (the spans can be viewed as special profunctors) to give causal semantics to higher-order processes. For this it is useful to generalise event structures to allow events which “persist.”

‘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

Polymorphic regular types are tree-like datatypes generated by polynomial type expressions over a set of free variables and closed under least fixed point. The ‘equality types ’ of Core ML can be expressed in this form. Given such a type expression with free, this paper shows a way to represent the one-hole contexts for elements of within elements of, together with an operation which will plug an element of into the hole of such a context. One-hole contexts are given as inhabitants of a regular type, computed generically from the syntactic structure of by a mechanism better known as partial differentiation. The relevant notion of containment is shown to be appropriately characterized in terms of derivatives and plugging in. The technology is then exploited to give the one-hole contexts for sub-elements of recursive types in a manner similar to Huet’s ‘zippers’[Hue97]. 1

this paper we consider the Sn -representation on the homology of certain Cohen-Macaulay subposets of \Pi n : In Section 1, we present a general technique for manipulating these homology modules. The unique properties of the partition lattice allow further simplification of these formulas, culminating in plethystic generating functions which, by recursive computation, yield the Frobenius characteristic of the representation. We illustrate our technique by giving simple derivations of three known formulas: 1. a formula for the plethystic inverse of the sum of the cycle indicators of the symmetric groups; this is essentially equivalent to Cadogan's formula; 2. a plethystic formula which determines the characteristic of the homology representation on the lattice \Pi

Chen's theory of iterated integrals provides a remarkable model for the di erential forms on the based loop space M of a di erentiable manifold M (Chen [10]; see also Hain-Tondeur [23] and Getzler-Jones-Petrack [21]). This article began as an attempt to nd an analogous model for 2 the complex of di erentiable forms on the double loop space M, motivated in part by the hope that this might provide an algebraic framework for understanding two-dimensional topological eld theories. Our approach is to use the formalism of operads. Operads can be de ned in any symmetric monoidal category, although we will mainly be concerned with dg-operads (di erential graded operads), that is, operads in the category of chain complexes with monoidal structure de ned by the graded tensor product. An operad is a sequence of objects a(k), k 0, carrying an action of the symmetric group Sk, with products a(k) a(j1) : : : a(jk) �! a(j1 + + jk) which are equivariant and associative | we give a precise de nition in Section 1.2. An operad such that a(k) = 0 for k 6 = 1 is a monoid: in this sense, operads are a non-linear generalization of monoids. If V is a chain complex, we may de ne an operad with EV (k) = Hom(V (k) ; V); where V (k) is the k-th tensor power of V. The symmetric group Sk acts on EV (k) through its action on V (k) , and the structure maps of EV are the obvious ones. This operad plays the same role in the theory of operads that the algebra End(V) does in the theory of associative algebras. An algebra over an operad a (or a-algebra) is a chain complex A together with a morphism of operads: a �! EA. In other words, A is equipped with structure maps k: a(k)

Abstract We develop a theory of abstract syntax with variable binding. To every binding signature we associate a category of models consisting of variable sets endowed with both a (binding) algebra and a substitution structure compatible with each other. The syntax generated by the signature is the initial model. This gives a notion of initial algebra semantics encompassing the traditional one; besides compositionality, it automatically verifies the semantic substitution lemma.

(0.1) Morita theory. Let A, B be two commutative rings. If their respective categories of modules are equivalent, then A and B are isomorphic. This is not anymore true if A and/or B are not assumed to be commutative. Morita theory

Abstract. We determine the character of the action of the symmetric group on the homology of the induced subposet of the lattice of partitions of the set {1, 2,...,n}obtained by restricting block sizes to the set {1,k,k+1,...}.A plethystic formula for the generating function of the Frobenius characteristic of the representation is given. We combine techniques from the theory of nonpure shellability, recently developed by Björner and Wachs, with symmetric function techniques, developed by Sundaram, for determining representations on the homology of subposets of the partition lattice. For 2 ≤ k ≤ n, the k-equal partition lattice Π (k,1 n−k) is defined to be the join-sublattice of the partition lattice Πn generated by set partitions consisting of n−k blocks of size one and one block of size k. In other words, Π (k,1 n−k) is the joinsublattice of Πn consisting of partitions having no blocks of sizes 2, 3,...,k−1. This

Abstract. Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev [V1, V2, V3]. In this paper we study the complexes of not i-connected k-hypergraphs on n vertices. We show that the complex of not 2-connected graphs has the homotopy type of a wedge of (n − 2)! spheres of dimension 2n − 5. This answers one of the questions raised by Vassiliev [V3] in connection with knot invariants. For this case the Snaction on the homology of the complex is also determined. For complexes of not 2-connected k-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not (n − 2)-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not (n − 3)-connected graphs we provide a formula for the generating function of the Euler characteristic. 1.

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