15 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
Geometric constructions preserve fibrations
Let be a representable 2-category, and a
2-endofunctor of the arrow 2-category such that (i)
and (ii)
preserves proneness of morphisms in
. Then preserves fibrations and
opfibrations in .
The proof takes Street's characterization of (e.g.) opfibrations as
pseudoalgebras for 2-monads on slice categories
and develops it by defining a 2-monad on
that takes change of base into account, and uses known
results on the lifting of 2-functors to pseudoalgebras.Comment: 29 page
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