We recast dataflow in a modern categorical light using profunctors as a generalisation of relations. The well-known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits with the view of categories of models for concurrency and the general treatment of bisimulation they provide. In particular, it fits with the recent categorical formulation of feedback using traced monoidal categories. The payoffs are: (1) explicit relations to existing models and semantics, especially the usual axioms of monotone IO automata are read off from the definition of profunctors; (2) a new definition of bisimulation for dataflow, the proof of the congruence of which benefits from the preservation properties associated with open maps; and (3) a treatment of higher-order dataflow as a biproduct, essentially by following the geometry of interaction programme.

We study typed behavioural equivalences for the $\pi$-calculus in which the type system allows a form of subtyping. This enables processes to distribute different capabilities selectively on communication channels. The equivalences considered include typed versions of testing equivalences and barbed bisimulation equivalences.

We show that each of these equivalences can be characterised via standard techniques applied to a novel labelled transition system of configurations. These consist of a process term together with two related type environments; one constraining the process and the other its computing environment.

The context of this article is a program studying the bicategory of spans of graphs as an algebra of processes, with applications to concurrency theory. The objective here is to study functorial aspects of reachability, minimisation and minimal realisation. The compositionality of minimisation has application to model-checking.

The asynchronous $\pi$-calculus has been considered as the basis of experimental programming languages (or proposals for programming languages) like Pict, Join and TyCO. However, on closer inspection, these languages are based on an even simpler calculus, called Localised$\pi$ (L$\pi$), where: (a) only the output capability of names may be transmitted; (b) there is no matching or similar constructs for testing equality between names.

We study the basic operational and algebraic theory of L$\pi$. We focus on bisimulation-based behavioural equivalences, more precisely, on barbed congruence. We prove two coinductive characterisations of barbed congruence in L$\pi$, and some basic algebraic laws. We then show applications of this theory, including: the derivability of the delayed input; the correctness of an optimisation of the encoding of call-by-name $\lambda$-calculus; the validity of some laws for Join; the soundness of Thielecke's axiomatic semantics of the Continuation Passing Style calculus.