Profunctors, Open Maps and Bisimulation

This paper studies fundamental connections between profunctors (i.e., distributors, or bimodules), open maps and bisimulation. In particular, it proves that a colimit preserving functor between presheaf categories (corresponding to a profunctor) preserves open maps and open map bisimulation. Consequently, the composition of profunctors preserves open maps as 2-cells. A guiding idea is the view that profunctors, and colimit preserving functors, are linear maps in a model of classical linear logic. But profunctors, and colimit preserving functors, as linear maps, are too restrictive for many applications. This leads to a study of a range of pseudo-comonads and how non-linear maps in their co-Kleisli bicategories preserve open maps and bisimulation. The pseudo-comonads considered are based on finite colimit completion, ``lifting'', and indexed families. The paper includes an appendix summarising the key results on coends, left Kan extensions and the preservation of colimits. One motivation for this work is that it provides a mathematical framework for extending domain theory and denotational semantics of programming languages to the more intricate models, languages and equivalences found in concurrent computation. But the results are likely to have more general applicability because of the ubiquitous nature of profunctors

Relational Semantics of Non-Deterministic Dataflow

We recast dataflow in a modern categorical light using profunctors as a generalization of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preservemuch 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 theusual 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

A Presheaf Semantics of Value-Passing Processes

This paper investigates presheaf models for process calculi withvalue passing. Denotational semantics in presheaf models are shownto correspond to operational semantics in that bisimulation obtainedfrom open maps is proved to coincide with bisimulation as definedtraditionally from the operational semantics. Both "early" and "late"semantics are considered, though the more interesting "late" semanticsis emphasised. A presheaf model and denotational semantics is proposedfor a language allowing process passing, though there remainsthe problem of relating the notion of bisimulation obtained from openmaps to a more traditional definition from the operational semantics.A tentative beginning is made of a "domain theory" supportingpresheaf models

String Diagrams for Layered Explanations

We propose a categorical framework to reason about scientific explanations: descriptions of a phenomenon meant to translate it into simpler terms, or into a context that has been already understood. Our motivating examples come from systems biology, electrical circuit theory, and concurrency. We demonstrate how three explanatory models in these seemingly diverse areas can be all understood uniformly via a graphical calculus of layered props. Layered props allow for a compact visual presentation of the same phenomenon at different levels of precision, as well as the translation between these levels. Notably, our approach allows for partial explanations, that is, for translating just one part of a diagram while keeping the rest of the diagram untouched. Furthermore, our approach paves the way for formal reasoning about counterfactual models in systems biology

Domain Theory for Concurrency

A simple domain theory for concurrency is presented. Based on a categorical model of linear logic and associated comonads, it highlights the role of linearity in concurrent computation. Two choices of comonad yield two expressive metalanguages for higher-order processes, both arising from canonical constructions in the model. Their denotational semantics are fully abstract with respect to contextual equivalence. One language derives from an exponential of linear logic; it supports a straightforward operational semantics with simple proofs of soundness and adequacy. The other choice of comonad yields a model of affine-linear logic, and a process language with a tensor operation to be understood as a parallel composition of independent processes. The domain theory can be generalised to presheaf models, providing a more refined treatment of nondeterministic branching. The article concludes with a discussion of a broader programme of research, towards a fully fledged domain theory for concurrency

A Profunctorial Scott Semantics

In this paper, we study the bicategory of profunctors with the free finite coproduct pseudo-comonad and show that it constitutes a model of linear logic that generalizes the Scott model. We formalize the connection between the two models as a change of base for enriched categories which induces a pseudo-functor that preserves all the linear logic structure. We prove that morphisms in the co-Kleisli bicategory correspond to the concept of strongly finitary functors (sifted colimits preserving functors) between presheaf categories. We further show that this model provides solutions of recursive type equations which provides 2-dimensional models of the pure lambda calculus and we also exhibit a fixed point operator on terms