1,040 research outputs found
A Few Notes on Formal Balls
Using the notion of formal ball, we present a few new results in the theory
of quasi-metric spaces. With no specific order: every continuous
Yoneda-complete quasi-metric space is sober and convergence Choquet-complete
hence Baire in its -Scott topology; for standard quasi-metric spaces,
algebraicity is equivalent to having enough center points; on a standard
quasi-metric space, every lower semicontinuous -valued
function is the supremum of a chain of Lipschitz Yoneda-continuous maps; the
continuous Yoneda-complete quasi-metric spaces are exactly the retracts of
algebraic Yoneda-complete quasi-metric spaces; every continuous Yoneda-complete
quasi-metric space has a so-called quasi-ideal model, generalizing a
construction due to K. Martin. The point is that all those results reduce to
domain-theoretic constructions on posets of formal balls
A few notes on formal balls
Using the notion of formal ball, we present a few easy, new results in the theory of quasi-metric spaces. With no specific order: every continuous Yoneda-complete quasi-metric space is sober and convergence Choquet-complete hence Baire in its d-Scott topology; for standard quasi-metric spaces, algebraicity is equivalent to having enough center points; on a standard quasi-metric space, every lower semicontinuous R+-valued function is the supremum of a chain of Lipschitz Yoneda-continuous maps; the continuous Yoneda-complete quasi-metric spaces are exactly the retracts of algebraic Yoneda-complete quasi-metric spaces; every continuous Yoneda-complete quasi-metric space has a so-called quasi-ideal model, generalizing a construction due to K. Martin. The point is that all those results reduce to domain-theoretic constructions on posets of formal balls
Approximation in quantale-enriched categories
Our work is a fundamental study of the notion of approximation in
V-categories and in (U,V)-categories, for a quantale V and the ultrafilter
monad U. We introduce auxiliary, approximating and Scott-continuous
distributors, the way-below distributor, and continuity of V- and
(U,V)-categories. We fully characterize continuous V-categories (resp.
(U,V)-categories) among all cocomplete V-categories (resp. (U,V)-categories) in
the same ways as continuous domains are characterized among all dcpos. By
varying the choice of the quantale V and the notion of ideals, and by further
allowing the ultrafilter monad to act on the quantale, we obtain a flexible
theory of continuity that applies to partial orders and to metric and
topological spaces. We demonstrate on examples that our theory unifies some
major approaches to quantitative domain theory.Comment: 17 page
Computational Models of Certain Hyperspaces of Quasi-metric Spaces
In this paper, for a given sequentially Yoneda-complete T_1 quasi-metric
space (X,d), the domain theoretic models of the hyperspace K_0(X) of nonempty
compact subsets of (X,d) are studied. To this end, the -Plotkin domain
of the space of formal balls BX, denoted by CBX is considered. This domain is
given as the chain completion of the set of all finite subsets of BX with
respect to the Egli-Milner relation. Further, a map is established and proved that it is an embedding whenever K_0(X) is
equipped with the Vietoris topology and respectively CBX with the Scott
topology. Moreover, if any compact subset of (X,d) is d^{-1}-precompact, \phi
is an embedding with respect to the topology of Hausdorff quasi-metric H_d on
K_0(X). Therefore, it is concluded that (CBX,\sqsubseteq,\phi) is an
-computational model for the hyperspace K_0(X) endowed with the
Vietoris and respectively the Hausdorff topology. Next, an algebraic
sequentially Yoneda-complete quasi-metric D on CBX\sqsubseteq_D\phi:({\cal
K}_0(X),H_d)\rightarrow({\bf C}{\bf B}X,D)\omega$-computational model for
(K_(X),H_d).Comment: 25 page
Weighted colimits and formal balls in generalized metric spaces
(a) Limits of Cauchy sequences in a (possibly non-symmetric) metric space are shown to be weighted colimits (a notion introduced by Borceux and Kelly, 1975). As a consequence, further insights from enriched category theory are applicable to the theory of metric spaces, thus continuing Lawvere's (1973) approach. Many of the recently proposed d
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