295,505 research outputs found
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
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
Limits of Ordered Graphs and their Applications
The emerging theory of graph limits exhibits an analytic perspective on
graphs, showing that many important concepts and tools in graph theory and its
applications can be described more naturally (and sometimes proved more easily)
in analytic language. We extend the theory of graph limits to the ordered
setting, presenting a limit object for dense vertex-ordered graphs, which we
call an \emph{orderon}. As a special case, this yields limit objects for
matrices whose rows and columns are ordered, and for dynamic graphs that expand
(via vertex insertions) over time. Along the way, we devise an ordered
locality-preserving variant of the cut distance between ordered graphs, showing
that two graphs are close with respect to this distance if and only if they are
similar in terms of their ordered subgraph frequencies. We show that the space
of orderons is compact with respect to this distance notion, which is key to a
successful analysis of combinatorial objects through their limits.
We derive several applications of the ordered limit theory in extremal
combinatorics, sampling, and property testing in ordered graphs. In particular,
we prove a new ordered analogue of the well-known result by Alon and Stav
[RS\&A'08] on the furthest graph from a hereditary property; this is the first
known result of this type in the ordered setting. Unlike the unordered regime,
here the random graph model with an ordering over the vertices is
\emph{not} always asymptotically the furthest from the property for some .
However, using our ordered limit theory, we show that random graphs generated
by a stochastic block model, where the blocks are consecutive in the vertex
ordering, are (approximately) the furthest. Additionally, we describe an
alternative analytic proof of the ordered graph removal lemma [Alon et al.,
FOCS'17].Comment: Added a new application: An Alon-Stav type result on the furthest
ordered graph from a hereditary property; Fixed and extended proof sketch of
the removal lemma applicatio
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