15 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
Some notes on Esakia spaces
Under Stone/Priestley duality for distributive lattices, Esakia spaces
correspond to Heyting algebras which leads to the well-known dual equivalence
between the category of Esakia spaces and morphisms on one side and the
category of Heyting algebras and Heyting morphisms on the other. Based on the
technique of idempotent split completion, we give a simple proof of a more
general result involving certain relations rather then functions as morphisms.
We also extend the notion of Esakia space to all stably locally compact spaces
and show that these spaces define the idempotent split completion of compact
Hausdorff spaces. Finally, we exhibit connections with split algebras for
related monads
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
Dualities for modal algebras from the point of view of triples
In this paper we show how the theory of monads can be used to deduce in a uniform manner several duality theorems involving categories of relations on one side and categories of algebras with homomorphisms preserving only some operations on the other. Furthermore, we investigate the monoidal structure induced by Cartesian product on the relational side and show that in some cases the corresponding operation on the algebraic side represents bimorphisms
On the axiomatisability of the dual of compact ordered spaces
We provide a direct and elementary proof of the fact that the category of
Nachbin's compact ordered spaces is dually equivalent to an Aleph_1-ary variety
of algebras. Further, we show that Aleph_1 is a sharp bound: compact ordered
spaces are not dually equivalent to any SP-class of finitary algebras.Comment: 10 pages. v3: minor changes. To appear in Applied Categorical
Structure
Representable (T,V)-categories
Working in the framework of -categories, for a symmetric monoidal closed category V and a (not necessarily cartesian) monad , we present a common account to the study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal categories and representable multicategories on the other one. In this setting we introduce the notion of dual for -categories.Working in the framework of (T, V)-categories, for a symmetric monoidal closed
category V and a (not necessarily cartesian) monad T, we present a common account to the
study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal
categories and representable multicategories on the other one. In this setting we introduce the
notion of dual for (T, V)-categories
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
Generating the algebraic theory of : the case of partially ordered compact spaces
It is known since the late 1960's that the dual of the category of compact
Hausdorff spaces and continuous maps is a variety -- not finitary, but bounded
by . In this note we show that the dual of the category of partially
ordered compact spaces and monotone continuous maps is a -ary
quasivariety, and describe partially its algebraic theory. Based on this
description, we extend these results to categories of Vietoris coalgebras and
homomorphisms. We also characterise the -copresentable partially
ordered compact spaces
A unified framework for generalized multicategories
Notions of generalized multicategory have been defined in numerous contexts
throughout the literature, and include such diverse examples as symmetric
multicategories, globular operads, Lawvere theories, and topological spaces. In
each case, generalized multicategories are defined as the "lax algebras" or
"Kleisli monoids" relative to a "monad" on a bicategory. However, the meanings
of these words differ from author to author, as do the specific bicategories
considered. We propose a unified framework: by working with monads on double
categories and related structures (rather than bicategories), one can define
generalized multicategories in a way that unifies all previous examples, while
at the same time simplifying and clarifying much of the theory.Comment: 76 pages; final version, to appear in TA