425 research outputs found
The enriched Vietoris monad on representable spaces
Employing a formal analogy between ordered sets and topological spaces, over
the past years we have investigated a notion of cocompleteness for topological,
approach and other kind of spaces. In this new context, the down-set monad
becomes the filter monad, cocomplete ordered set translates to continuous
lattice, distributivity means disconnectedness, and so on. Curiously, the
dual(?) notion of completeness does not behave as the mirror image of the one
of cocompleteness; and in this paper we have a closer look at complete spaces.
In particular, we construct the "up-set monad" on representable spaces (in the
sense of L. Nachbin for topological spaces, respectively C. Hermida for
multicategories); we show that this monad is of Kock-Z\"oberlein type; we
introduce and study a notion of weighted limit similar to the classical notion
for enriched categories; and we describe the Kleisli category of our "up-set
monad". We emphasize that these generic categorical notions and results can be
indeed connected to more "classical" topology: for topological spaces, the
"up-set monad" becomes the upper Vietoris monad, and the statement " is
totally cocomplete if and only if is totally complete"
specialises to O. Wyler's characterisation of the algebras of the Vietoris
monad on compact Hausdorff spaces.Comment: One error in Example 1.9 is corrected; Section 4 works now without
the assuming core-compactnes
Convergence and quantale-enriched categories
Generalising Nachbin's theory of "topology and order", in this paper we
continue the study of quantale-enriched categories equipped with a compact
Hausdorff topology. We compare these -categorical compact
Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that
the presence of a compact Hausdorff topology guarantees Cauchy completeness and
(suitably defined) codirected completeness of the underlying quantale enriched
category
Topology from enrichment: the curious case of partial metrics
For any small quantaloid \Q, there is a new quantaloid \D(\Q) of
diagonals in \Q. If \Q is divisible then so is \D(\Q) (and vice versa),
and then it is particularly interesting to compare categories enriched in \Q
with categories enriched in \D(\Q). Taking Lawvere's quantale of extended
positive real numbers as base quantale, \Q-categories are generalised metric
spaces, and \D(\Q)-categories are generalised partial metric spaces, i.e.\
metric spaces in which self-distance need not be zero and with a suitably
modified triangular inequality. We show how every small quantaloid-enriched
category has a canonical closure operator on its set of objects: this makes for
a functor from quantaloid-enriched categories to closure spaces. Under mild
necessary-and-sufficient conditions on the base quantaloid, this functor lands
in the category of topological spaces; and an involutive quantaloid is
Cauchy-bilateral (a property discovered earlier in the context of distributive
laws) if and only if the closure on any enriched category is identical to the
closure on its symmetrisation. As this now applies to metric spaces and partial
metric spaces alike, we demonstrate how these general categorical constructions
produce the "correct" definitions of convergence and Cauchyness of sequences in
generalised partial metric spaces. Finally we describe the Cauchy-completion,
the Hausdorff contruction and exponentiability of a partial metric space, again
by application of general quantaloid-enriched category theory.Comment: Apart from some minor corrections, this second version contains a
revised section on Cauchy sequences in a partial metric spac
Enriched Stone-type dualities
A common feature of many duality results is that the involved equivalence
functors are liftings of hom-functors into the two-element space resp. lattice.
Due to this fact, we can only expect dualities for categories cogenerated by
the two-element set with an appropriate structure. A prime example of such a
situation is Stone's duality theorem for Boolean algebras and Boolean
spaces,the latter being precisely those compact Hausdorff spaces which are
cogenerated by the two-element discrete space. In this paper we aim for a
systematic way of extending this duality theorem to categories including all
compact Hausdorff spaces. To achieve this goal, we combine duality theory and
quantale-enriched category theory. Our main idea is that, when passing from the
two-element discrete space to a cogenerator of the category of compact
Hausdorff spaces, all other involved structures should be substituted by
corresponding enriched versions. Accordingly, we work with the unit interval
and present duality theory for ordered and metric compact Hausdorff
spaces and (suitably defined) finitely cocomplete categories enriched in
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
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