Van Kampen Colimits and Path Uniqueness

Fibred semantics is the foundation of the model-instance pattern of software engineering. Software models can often be formalized as objects of presheaf topoi, i.e, categories of objects that can be represented as algebras as well as coalgebras, e.g., the category of directed graphs. Multimodeling requires to construct colimits of models, decomposition is given by pullback. Compositionality requires an exact interplay of these operations, i.e., diagrams must enjoy the Van Kampen property. However, checking the validity of the Van Kampen property algorithmically based on its definition is often impossible. In this paper we state a necessary and sufficient yet efficiently checkable condition for the Van Kampen property to hold in presheaf topoi. It is based on a uniqueness property of path-like structures within the defining congruence classes that make up the colimiting cocone of the models. We thus add to the statement "Being Van Kampen is a Universal Property" by Heindel and Soboci\'{n}ski the fact that the Van Kampen property reveals a presheaf-based structural uniqueness feature

Stochastic Relational Presheaves and Dynamic Logic for Contextuality

Presheaf models provide a formulation of labelled transition systems that is useful for, among other things, modelling concurrent computation. This paper aims to extend such models further to represent stochastic dynamics such as shown in quantum systems. After reviewing what presheaf models represent and what certain operations on them mean in terms of notions such as internal and external choices, composition of systems, and so on, I will show how to extend those models and ideas by combining them with ideas from other category-theoretic approaches to relational models and to stochastic processes. It turns out that my extension yields a transitional formulation of sheaf-theoretic structures that Abramsky and Brandenburger proposed to characterize non-locality and contextuality. An alternative characterization of contextuality will then be given in terms of a dynamic modal logic of the models I put forward.Comment: In Proceedings QPL 2014, arXiv:1412.810

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

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

Transpension: The Right Adjoint to the Pi-type

Presheaf models of dependent type theory have been successfully applied to model HoTT, parametricity, and directed, guarded and nominal type theory. There has been considerable interest in internalizing aspects of these presheaf models, either to make the resulting language more expressive, or in order to carry out further reasoning internally, allowing greater abstraction and sometimes automated verification. While the constructions of presheaf models largely follow a common pattern, approaches towards internalization do not. Throughout the literature, various internal presheaf operators ($\surd$, $\Phi/\mathsf{extent}$, $\Psi/\mathsf{Gel}$, $\mathsf{Glue}$, $\mathsf{Weld}$, $\mathsf{mill}$, the strictness axiom and locally fresh names) can be found and little is known about their relative expressivenes. Moreover, some of these require that variables whose type is a shape (representable presheaf, e.g. an interval) be used affinely. We propose a novel type former, the transpension type, which is right adjoint to universal quantification over a shape. Its structure resembles a dependent version of the suspension type in HoTT. We give general typing rules and a presheaf semantics in terms of base category functors dubbed multipliers. Structural rules for shape variables and certain aspects of the transpension type depend on characteristics of the multiplier. We demonstrate how the transpension type and the strictness axiom can be combined to implement all and improve some of the aforementioned internalization operators (without formal claim in the case of locally fresh names)

On the geometric theory of local MV-algebras

We investigate the geometric theory of local MV-algebras and its quotients axiomatizing the local MV-algebras in a given proper variety of MV-algebras. We show that, whilst the theory of local MV-algebras is not of presheaf type, each of these quotients is a theory of presheaf type which is Morita-equivalent to an expansion of the theory of lattice-ordered abelian groups. Di Nola-Lettieri's equivalence is recovered from the Morita-equivalence for the quotient axiomatizing the local MV-algebras in Chang's variety, that is, the perfect MV-algebras. We establish along the way a number of results of independent interest, including a constructive treatment of the radical for MV-algebras in a fixed proper variety of MV-algebras and a representation theorem for the finitely presentable algebras in such a variety as finite products of local MV-algebras.Comment: 52 page