The Experts below are selected from a list of 2181 Experts worldwide ranked by ideXlab platform
Glynn Winskel - One of the best experts on this subject based on the ideXlab platform.
-
LICS - On the Expressivity of Symmetry in Event Structures
2010 25th Annual IEEE Symposium on Logic in Computer Science, 2010Co-Authors: Sam Staton, Glynn WinskelAbstract:This paper establishes a bridge between Presheaf models for concurrency and the more operationally-informative world of event structures. It concentrates on a particular Presheaf category, consisting of presheaves over finite partial orders of events; such presheaves form a model of nondeterministic processes in which the computation paths have the shape of partial orders. It is shown how with the introduction of symmetry event structures represent all presheaves over finite partial orders. This is in contrast with plain event structures which only represent certain separated presheaves. Specifically a coreflection from the category of presheaves to the category of event structures with symmetry is exhibited. It is shown how the coreflection can be cut down to an equivalence between the Presheaf category and the subcategory of graded event structures with symmetry. Event structures with strong symmetries are shown to represent precisely all the separated presheaves. The broader context and specific applications to the unfolding of higher-dimensional automata and Petri nets, and weak bisimulation on event structures are sketched.
-
Presheaf models for ccs like languages
Theoretical Computer Science, 2003Co-Authors: Gian Luca Cattani, Glynn WinskelAbstract:The aim of this paper is to harness the mathematical machinery around presheaves for the purposes of process calculi. Joyal, Nielsen and Winskel proposed a general definition of bisimulation from open maps. Here we show that open-map bisimulations within a range of Presheaf models are congruences for a general process language, in which CCS and related languages are easily encoded. The results are then transferred to traditional models for processes. By first establishing the congruence results for Presheaf models, abstract, general proofs of congruence properties can be provided and the awkwardness caused through traditional models not always possessing the cartesian liftings, used in the breakdown of process operations, are side stepped. The abstract results are applied to show that hereditary history-preserving bisimulation is a congruence for CCS-like languages to which is added a refinement operator on event structures as proposed by van Glabbeek and Goltz.
-
A Linear Metalanguage for Concurrency
BRICS Report Series, 1998Co-Authors: Glynn WinskelAbstract:A metalanguage for concurrent process languages is introduced. Within it a range of process languages can be defined, including higher-order process languages where processes are passed and received as arguments. (The process language has, however, to be linear, in the sense that a process received as an argument can be run at most once, and not include name generation as in the Pi-Calculus.) The metalanguage is provided with two interpretations both of which can be understood as categorical models of a variant of linear logic. One interpretation is in a simple category of nondeterministic domains; here a process will denote its set of traces. The other interpretation, obtained by direct analogy with the nondeterministic domains, is in a category of Presheaf categories; the nondeterministic branching behaviour of a process is captured in its denotation as a Presheaf. Every Presheaf category possesses a notion of (open-map) bisimulation, preserved by terms of the metalanguage. The conclusion summarises open problems and lines of future work.
-
An Operational Understanding of Bisimulation from Open Maps
Electronic Notes in Theoretical Computer Science, 1998Co-Authors: Glynn WinskelAbstract:AbstractModels can be given to a range of programming languages combining concurrent and functional features in which Presheaf categories are used as the semantic domains (instead of the more usual complete partial orders). Once this is done the languages inherit a notion of bisimulation from the “open” maps associated with the Presheaf categories. However, although there are methodological and mathematical arguments for favouring semantics using Presheaf categories—in particular, there is a “domain theory” based on Presheaf categories which systematises bisimulation at higher-order—it is as yet far from a routine matter to read off an “operational characterisation”; by this I mean an equivalent coinductive definition of bisimulation between terms based on the operational semantics. I hope to illustrate the issues on a little process-passing language. This is joint work with Gian Luca Cattani
-
Presheaf models for the pi calculus
Lecture Notes in Computer Science, 1997Co-Authors: Gian Luca Cattani, Ian Stark, Glynn WinskelAbstract:Recent work has shown that Presheaf categories provide a general model of concurrency, with an inbuilt notion of bisimulation based on open maps. Here it is shown how this approach can also handle systems where the language of actions may change dynamically as a process evolves. The example is the π-calculus, a calculus for ‘mobile processes’ whose communication topology varies as channels are created and discarded. A denotational semantics is described for the π-calculus within an indexed category of profunctors; the model is fully abstract for bisimilarity, in the sense that bisimulation in the model, obtained from open maps, coincides with the usual bisimulation obtained from the operational semantics of the π-calculus. While attention is concentrated on the ‘late’ semantics of the π-calculus, it is indicated how the ‘early’ and other variants can also be captured.
John F. Jardine - One of the best experts on this subject based on the ideXlab platform.
-
Fuzzy sets and presheaves
Compositionality, 2019Co-Authors: John F. JardineAbstract:This note presents a Presheaf theoretic approach to the construction of fuzzy sets, which builds on Barr's description of fuzzy sets as sheaves of monomorphisms on a locale. A Presheaf-theoretic method is used to show that the category of fuzzy sets is complete and co-complete, and to present explicit descriptions of classical fuzzy sets that arise as limits and colimits. The Boolean localization construction for sheaves and presheaves on a locale L specializes to a theory of stalks if L approximates the structure of a closed interval in the real line. The system V(X) of Vietoris-Rips complexes for a data cloud X becomes both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed in this paper, in stages.
-
Fibred sites and stack cohomology
Mathematische Zeitschrift, 2006Co-Authors: John F. JardineAbstract:The usual notion of the site associated to a stack is expanded to a definition to a site \(\mathcal{C}/A\) fibred over a Presheaf of categories A on a site \(\mathcal{C}\). If the Presheaf of categories is a Presheaf of groupoids G, then the associated homotopy theory is Quillen equivalant to the homotopy theory of simplicial presheaves over BG, and so the homotopy theory for the fibred site \(\mathcal{C}/G\) is an invariant of the homotopy type of G. Similar homotopy invariance results obtain for presheaves of spectra and presheaves of symmetric spectra on \(\mathcal{C}/G\). In particular, stack cohomology can be calculated on the fibred site for any representing Presheaf of groupoids within a fixed homotopy type.
-
Fibred sites and stack cohomology
arXiv: Algebraic Topology, 2004Co-Authors: John F. JardineAbstract:The usual notion of a site fibred over a stack is expanded to a definition of a site C/A fibred over a Presheaf of categories A. Presheaves of simplicial sets on the site fibred over a Presheaf of categories A are contravariant enriched diagrams defined on A, taking values in simplicial sets. The standard model structure for presheaves of simplicial sets induces a coarse equivariant structure for enriched contravariant A-diagrams. If the Presheaf of categories is a Presheaf of groupoids G, then the associated homotopy theory is Quillen equivalent to the homotopy theory of simplicial presheaves over BG, and so the homotopy theory for the fibred site C/G is an invariant of the homotopy type of G. Similar homotopy invariance results obtain for presheaves of spectra and presheaves of symmetric spectra on C/G. In particular, stack cohomology can be calculated on the fibred site for a representing Presheaf of groupoids.
-
Localization theories for simplicial presheaves
Canadian Journal of Mathematics, 1998Co-Authors: Paul G. Goerss, John F. JardineAbstract:This work was motivated in part by the following question of Soule: given a simplicial Presheaf X on a site C, how does one produce a map of simplicial presheaves X → LHZX in such a way that each of the maps in sections X(U) → LHZX(U), U ∈ C, is an integral homology localization map in the sense of Bousfield? Secondly, if Y is a simplicial Presheaf which is integrally homology local in a suitable sense, is it the case that the map X → LHZX induces an isomorphism
-
Generalized étale cohomology
Generalized Etale Cohomology Theories, 1997Co-Authors: John F. JardineAbstract:A generalized etale cohomology theory is a graded group \(H*(T,F)\)(T,F), which is associated to a Presheaf of spectra F on an etale site for a scheme T
Olivia Caramello - One of the best experts on this subject based on the ideXlab platform.
-
Oxford Scholarship Online - Theories of Presheaf type: general criteria
Oxford Scholarship Online, 2018Co-Authors: Olivia CaramelloAbstract:This chapter carries out a systematic investigation of the class of geometric theories of Presheaf type (i.e. classified by a Presheaf topos), by using in particular the results on flat functors established in Chapter 5. First, it establishes a number of general results on theories of Presheaf type, notably including a definability theorem and a characterization of the finitely presentable models of such a theory in terms of formulas satisfying a key property of irreducibility. Then it presents a fully constructive characterization theorem providing necessary and sufficient conditions for a theory to be of Presheaf type expressed in terms of the models of the theory in arbitrary Grothendieck toposes. This theorem is shown to admit a number of simpler corollaries which can be effectively applied in practice for testing whether a given theory is of Presheaf type as well as for generating new examples of such theories.
-
Oxford Scholarship Online - Quotients of a theory of Presheaf type
Oxford Scholarship Online, 2018Co-Authors: Olivia CaramelloAbstract:In this chapter the quotients of a given theory of Presheaf type are investigated by means of Grothendieck topologies that can be naturally attached to them, establishing a ‘semantic’ representation for the classifying topos of such a quotient as a subtopos of the classifying topos of the given theory of Presheaf type. It is also shown that the models of such a quotient can be characterized among the models of the theory of Presheaf type as those which satisfy a key property of homogeneity with respect to a Grothendieck topology associated with the quotient. A number of sufficient conditions for the quotient of a theory of Presheaf type to be again of Presheaf type are also identified: these include a finality property of the category of models of the quotient with respect to the category of models of the theory and a rigidity property of the Grothendieck topology associated with the quotient.
-
Oxford Scholarship Online - Theories, Sites, Toposes
Oxford Scholarship Online, 2018Co-Authors: Olivia CaramelloAbstract:This book is devoted to a general study of geometric theories from a topos-theoretic perspective. After recalling the necessary topos-theoretic preliminaries, it presents the main methodology it uses to extract ‘concrete’ information on theories from properties of their classifying toposes—the ‘bridge’ technique. As a first implementation of this methodology, a duality is established between the subtoposes of the classifying topos of a geometric theory and the geometric theory extensions (also called ‘quotients’) of the theory. Many concepts of elementary topos theory which apply to the lattice of subtoposes of a given topos are then transferred via this duality into the context of geometric theories. A second very general implementation of the ‘bridge’ technique is the investigation of the class of theories of Presheaf type (i.e. classified by a Presheaf topos). After establishing a number of preliminary results on flat functors in relation to classifying toposes, the book carries out a systematic investigation of this class resulting in a number of general results and a characterization theorem allowing one to test whether a given theory is of Presheaf type by considering its models in arbitrary Grothendieck toposes. Expansions of geometric theories and faithful interpretations of theories of Presheaf type are also investigated. As geometric theories can always be written (in many ways) as quotients of Presheaf type theories, the study of quotients of a given theory of Presheaf type is undertaken. Lastly, the book presents a number of applications in different fields of mathematics of the theory it develops.
-
Oxford Scholarship Online - Expansions and faithful interpretations
Oxford Scholarship Online, 2018Co-Authors: Olivia CaramelloAbstract:This chapter introduces the concept of expansion of a geometric theory and develops some basic theory about it; it proves in particular that expansions of geometric theories induce geometric morphisms between the respective classifying toposes and that conversely every geometric morphism to the classifying topos of a geometric theory can be seen as arising from an expansion of that theory. The notion of hyperconnected-localic factorization of a geometric morphism is then investigated and shown to admit a natural description in the context of geometric theories. Further, the preservation, by ‘faithful interpretations’ of theories, of each of the conditions in the characterization theorem for theories of Presheaf type established in Chapter 6 is discussed, leading to results of the form ‘under appropriate conditions, a geometric theory in which a theory of Presheaf type faithfully interprets is again of Presheaf type’.
-
Oxford Scholarship Online - Examples of theories of Presheaf type
Oxford Scholarship Online, 2018Co-Authors: Olivia CaramelloAbstract:This chapter discusses several classical as well as new examples of theories of Presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of Presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.
Andreas Döring - One of the best experts on this subject based on the ideXlab platform.
-
Spectral presheaves as quantum state spaces
Philosophical transactions. Series A Mathematical physical and engineering sciences, 2015Co-Authors: Andreas DöringAbstract:For each quantum system described by an operator algebra [Formula: see text] of physical quantities, we provide a (generalized) state space, notwithstanding the Kochen-Specker theorem. This quantum state space is the spectral Presheaf [Formula: see text]. We formulate the time evolution of quantum systems in terms of Hamiltonian flows on this generalized space and explain how the structure of the spectral Presheaf [Formula: see text] geometrically mirrors the double role played by self-adjoint operators in quantum theory, as quantum random variables and as generators of time evolution.
-
Two New Complete Invariants of von Neumann Algebras
arXiv: Operator Algebras, 2014Co-Authors: Andreas DöringAbstract:We show that the oriented context category and the oriented spectral Presheaf are complete invariants of a von Neumann algebra not isomorphic to C ⊕ C and with no direct summand of type I2.
Parsa Bakhtary - One of the best experts on this subject based on the ideXlab platform.
-
On the cohomology of a simple normal crossings divisor
Journal of Algebra, 2010Co-Authors: Parsa BakhtaryAbstract:AbstractWe establish a formula which decomposes the cohomologies of various sheaves on a simple normal crossings divisor (SNC) D in terms of the simplicial cohomologies of the dual complex Δ(D) with coefficients in a Presheaf of vector spaces. This Presheaf consists precisely of the corresponding cohomology data on the components of D and on their intersections. We use this formula to give a Hodge decomposition for SNC divisors and investigate the toric setting. We also conjecture the existence of such a formula for effective non-reduced divisors with SNC support, and show that this would imply the vanishing of the higher simplicial cohomologies of the dual complex associated to a resolution of an isolated rational singularity
-
On the cohomology of a simple normal crossings divisor
arXiv: Algebraic Geometry, 2008Co-Authors: Parsa BakhtaryAbstract:We establish a formula which decomposes the cohomologies of various sheaves on a simple normal crossings divisor (SNC) $D$ in terms of the simplicial cohomologies of the dual complex $\Delta(D)$ with coefficients in a Presheaf of vector spaces. This Presheaf consists precisely of the corresponding cohomology data on the components of $D$ and on their intersections. We use this formula to give a Hodge decomposition for SNC divisors and investigate the toric setting. We also conjecture the existence of such a formula for effective non-reduced divisors with SNC support, and show that this would imply the vanishing of the higher simplicial cohomologies of the dual complex associated to a resolution of an isolated rational singularity.
