The suspension-loop construction is used to define a process in a symmetric monoidal category. The algebra of such processes is that of symmetric monoidal bicategories. Processes in categories with products and in categories with sums are studied in detail, and in both cases the resulting bicategories of processes are equipped with operations called feedback. Appropriate versions of traced monoidal properties are verified for feedback, and a normal form theorem for expressions of processes is proved. Connections with existing theories of circuit design and computation are established via structure preserving homomorphisms.

. We construct a category of circuits: the objects are alphabets and the morphisms are deterministic automata. The construction differs in several respects from the bicategories of circuits appearing previously in the literature: it is parameterized by a monad which allows flexibility in the emergent notion of process. We focus on the circuits which arise from a distributive category and the exception monad. These circuits are partial in that they may, based on their state, choose to abort on some inputs. Consequently, certain circuits determine languages, and safety and liveness properties with respect to these languages are captured by circuit equations. Actually, the notions of safety and liveness arise abstractly in any copy category. Extracting the category of circuits which are both safe and live corresponds to the extensive completion of a distributive copy category. Partial circuits coincide with elements of the terminal coalgebra of a specific datatype. The co--induction princ...

The purpose of this paper is to show how one may construct from a synchronous interaction category, such as SProc, a corresponding asynchronous version. Significantly, it is not a simple Kleisli construction, but rather arises due to particular properties of a monad combined with the existence of a certain type of distributive law. Following earlier work we consider those synchronous interaction categories which arise from model categories through a quotiented span construction: SProc arises in this way from labelled transition systems. The quotienting is determined by a cover system which expresses bisimulation. Asynchrony is introduced into a model category by a monad which, in the case of transition systems, adds the ability to idle. To form a process category atop this two further ingredients are required: pullbacks in the Kleisli category, and a cover system to express (weak) bisimulation. The technical results of the paper provide necessary and sufficient conditions for a Kleisli...

This is a report on a mathematician's effort to understand some concurrency theory. The starting point is a logical interpretation of Nielsen and Winskel's [30] account of the basic models of concurrency. Upon the obtained logical structures, we build a calculus of relations which yields, when cut down by bisimulations, Abramsky's interaction category of synchronous processes [2]. It seems that all interaction categories arise in this way. The obtained presentation uncovers some of their logical contents and perhaps sheds some more light on the original idea of processes as relations extended in time. The sequel of this paper will address the issues of asynchrony, preemption, noninterleaving and linear logic in the same setting. 1 Introduction Concurrency in computation is modelled in many different ways. Several attempts at unification have been made. Most recently, Abramsky [1, 2] has proposed the paradigm of relations extended in time as a foundation for theory of processes. His in...

The compact closed bicategory Span of spans of reflexive graphs is described and it is interpreted as an algebra for constructing specifications of concurrent systems. We describe a procedure for associating to any Place/Transition system\Omega an expression \Psi\Omega in the algebra Span. The value of this expression is a system whose behaviours are the same as those of the P/T system. Furthermore, along the lines of Penrose's string diagrams, a geometry is associated to the expression \Psi\Omega which is essentially the same geometry as that usually associated to the net underlying \Omega .

Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity. Bireflective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterize them in terms of split-idempotent natural transformations on id A . In the special case that A is a presheaf category, we characterize them in terms of the domain, and prove that any bireflective subcategory of A is itself a presheaf category. We define diagonal structure on a symmetric monoidal category which is still more general than asking the tensor product to be the categorical product. We then obtain a bireflective subcategory of [C op ; Set] and deduce results relating its finite product structure with the monoidal structure of [C op ; Set] determined by that of C. We also investigate the closed structure. Finally, for completeness, we give results on bireflective subcategories in Rel(A), the category of relati...