arXiv subjects

Eva Colebunders

Publications and source records attributed to Eva Colebunders.

4 recordsLinked to original sources

arXiv2026

The category of probabilistic metric spaces

The paper is devoted to a categorical study of the category of probabilistic metric spaces. The study is based on an isomorphic description of the category of probabilistic metric spaces. The isomorphic description was obtained in [3] and is in terms of objects that are sets endowed with a collection of distances, where the distances involved do not satisfy the triangle inequality but fulfil a mixed triangle condition instead. The morphisms are levelwise non-expansive maps. We show that the category of probabilistic metric spaces is a monotopological category over the category of sets. We describe the regular closure on a probabilistic space and prove that it coincides with the closure in the underlying strong topology. This enables us to characterize the class of all epimorphisms as the dense maps and the class of all regular monomorphisms as the closed embeddings in terms of the closure operator. We prove that the category of extended metric spaces with non-expansive maps is both coreflectively and reflectively embedded in the category of probabilistic metric spaces.

math.GN↗

arXiv2026

A characterisation of probabilistic metrizability for approach spaces

Characterisations of metrizable topological spaces or metrizable uniform spaces are well known. A natural counterpart to being metrizable for topological spaces can be expressed in terms of probabilistic metrizability for approach spaces. The notion of a probabilistic metrizable approach space is based on a well known concrete functor $\Gamma$, as introduced in [9], from the category of probabilistic metric spaces with respect to a continuous arbitrary t-norm to the category of approach spaces. A characterization of those probabilistic metrizable approach spaces is still missing and in the first part of this paper we solve this problem. A natural counterpart to being metrizable for uniform spaces can be expressed in terms of probabilistic metrizability for uniform gauge spaces. In the second part of the paper we start from another concrete functor $\Lambda$, as described in [7], on the category of probabilistic metric spaces with respect to a continuous t-norm to the category of uniform gauge spaces. In a similar way as for the functor $\Gamma$ we obtain a characterisation of probabilistic metrizability of uniform gauge spaces. The last section of the paper is devoted to an isomorphic description of the category of probabilistic metric spaces. This problem is not new. Previous attempts in providing isomorphic descriptions of the category of probabilistic metric spaces worked with collections of (pseudo)metrics. These attempts were only formulated in restricted cases. Our isomorphic description is in terms of objects that are sets endowed with a collection of distances, where the distances involved do not satisfy the triangle inequality but fulfil a mixed triangle condition instead.

math.GN↗

arXiv2014

Function spaces and contractive extensions in Approach Theory: The role of regularity

Two classical results characterizing regularity of a convergence space in terms of continuous extensions of maps on one hand, and in terms of continuity of limits for the continuous convergence on the other, are extended to convergence-approach spaces. Characterizations are obtained for two alternative extensions of regularity to convergence-approach spaces: regularity and strong regularity. The results improve upon what is known even in the convergence case. On the way, a new notion of strictness for convergence-approach spaces is introduced.

math.GN↗

arXiv2001

State property systems and closure spaces: a study of categorical equivalence

We show that the natural mathematical structure to describe a physical entity by means of its states and its properties within the Geneva-Brussels approach is that of a state property system. We prove that the category of state property systems (and morphisms), SP, is equivalent to the category of closure spaces (and continuous maps), Cls. We show the equivalence of the 'state determination axiom' for state property systems with the 'T0 separation axiom' for closure spaces. We also prove that the category SP0 of state determined state property systems is equivalent to the category L0 of based complete lattices. In this sense the equivalence of SP and Cls generalizes the equivalence of Cls0 and L0, proven in Erne 1984.

quant-ph↗