13 research outputs found
Generalized Vietoris Bisimulations
We introduce and study bisimulations for coalgebras on Stone spaces [14]. Our
notion of bisimulation is sound and complete for behavioural equivalence, and
generalizes Vietoris bisimulations [4]. The main result of our paper is that
bisimulation for a coalgebra is the topological closure of
bisimulation for the underlying coalgebra
Coalgebraic Geometric Logic: Basic Theory
Using the theory of coalgebra, we introduce a uniform framework for adding
modalities to the language of propositional geometric logic. Models for this
logic are based on coalgebras for an endofunctor on some full subcategory of
the category of topological spaces and continuous functions. We investigate
derivation systems, soundness and completeness for such geometric modal logics,
and we we specify a method of lifting an endofunctor on Set, accompanied by a
collection of predicate liftings, to an endofunctor on the category of
topological spaces, again accompanied by a collection of (open) predicate
liftings. Furthermore, we compare the notions of modal equivalence, behavioural
equivalence and bisimulation on the resulting class of models, and we provide a
final object for the corresponding category
Coalgebraic Fuzzy geometric logic
The paper aims to develop a framework for coalgebraic fuzzy geometric logic
by adding modalities to the language of fuzzy geometric logic. Using the
methods of coalgebra, the modal operators are introduced in the language of
fuzzy geometric logic. To define the modal operators, we introduce a notion of
fuzzy-open predicate lifting. Based on coalgebras for an endofunctor on the
category of fuzzy topological spaces and fuzzy continuous
maps, we build models for the coalgebraic fuzzy geometric logic. Bisimulations
for the defined models are discussed in this work
Coalgebraic Geometric Logic
Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor T on some full subcategory of the category Top of topological spaces and continuous functions. We compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category. Furthermore, we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces
Localic completion of uniform spaces
We extend the notion of localic completion of generalised metric spaces by
Steven Vickers to the setting of generalised uniform spaces. A generalised
uniform space (gus) is a set X equipped with a family of generalised metrics on
X, where a generalised metric on X is a map from the product of X to the upper
reals satisfying zero self-distance law and triangle inequality.
For a symmetric generalised uniform space, the localic completion lifts its
generalised uniform structure to a point-free generalised uniform structure.
This point-free structure induces a complete generalised uniform structure on
the set of formal points of the localic completion that gives the standard
completion of the original gus with Cauchy filters.
We extend the localic completion to a full and faithful functor from the
category of locally compact uniform spaces into that of overt locally compact
completely regular formal topologies. Moreover, we give an elementary
characterisation of the cover of the localic completion of a locally compact
uniform space that simplifies the existing characterisation for metric spaces.
These results generalise the corresponding results for metric spaces by Erik
Palmgren.
Furthermore, we show that the localic completion of a symmetric gus is
equivalent to the point-free completion of the uniform formal topology
associated with the gus.
We work in Aczel's constructive set theory CZF with the Regular Extension
Axiom. Some of our results also require Countable Choice.Comment: 39 page
Logical Berkovich Geometry: A Point-free Perspective
Extending our insights from \cite{NVOstrowski}, we apply point-free
techniques to sharpen a foundational result in Berkovich geometry. In our
language, given the ring of convergent power series
over a suitable non-Archimedean field , the points of its Berkovich Spectrum
correspond to -good filters. The surprise is
that, unlike the original result by Berkovich, we do not require the field
to be non-trivially valued. Our investigations into non-Archimedean geometry
can be understood as being framed by the question: what is the relationship
between topology and logic
Adelic Geometry via Topos Theory
Our starting point has to do with a key tension running through number theory: although all completions of the rationals Q should be treated symmetrically, this is complicated by fundamental disanalogies between the p-adics vs. the reals. Whereas prior work has typically been guided by classical point-set reasoning, this thesis explores various ways of pulling this problem away from the underlying set theory, revealing various surprises that are obscured by the classical perspective. Framing these investigations is the following test problem: construct and describe the topos of completions of Q (up to equivalence).
Chapter 2 begins with the preliminaries: we set up the topos-theoretic framework of point-free topology, with a view towards highlighting the distinction between classical vs. geometric mathematics, before introducing the number-theoretic context. A key theme is that geometric mathematics possesses an intrinsic continuity, which forces us to think more carefully about the topological character of classical algebraic constructions.
Chapter 3 represents the first step towards constructing the topos of completions. Here, we provide a pointfree account of real exponentiation and logarithms, which will allow us to define the equivalence of completions geometrically. Chapter 4 provides a geometric proof of Ostrowski's Theorem for both upper-valued abosolute values on Z as well as Dedekind-valued absolute values on Q, along with some key insights about the relationship between the multiplicative seminorms and upper reals.
In a slightly more classical interlude, Chapter 5 extends these insights to obtain a surprising generalisation of a foundational result in Berkovich geometry. Namely, by replacing the use of classical rigid discs with formal balls, we obtain a classification of the points of Berkovich Spectra M(K{R^{-1}T}) via the language of filters [more precisely, what we call: R-good filters] even when the base field K is trivially-valued.
Returning to geometricity, Chapter 6 builds upon Chapters 3 and 4 to investigate the space of places of Q via descent arguments. Here, we uncover an even deeper surprise. Although the non-Archimedean places correspond to singletons (as is classically expected), the Archimedean place corresponds to the subspace of upper reals in [0, 1], a sort of blurred unit interval. The chapter then analyses the topological differences between the non-Archimedean vs. Archimedean places. In particular, we discover that while the topos corresponding to Archimedean place witnesses non-trivial forking in the connected components of its sheaves, the topos corresponding to the non-Archimedean place eliminates all kinds of forking phenomena. We then conclude with some insights and observations, framed by the question: "How should the connected and the disconnected interact?