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Properly injective spaces and function spaces
 TO APPEAR IN TOPOLOGY AND ITS APPLICATIONS
, 1997
"... Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j: X → Y, Dana Scott asked whether the higherorder function [X → D] → [Y → D] which takes a continuous map f: X → D to its greatest continuous extension ¯ f: Y → D along j is Scott continuous ..."
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Cited by 28 (12 self)
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Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j: X → Y, Dana Scott asked whether the higherorder function [X → D] → [Y → D] which takes a continuous map f: X → D to its greatest continuous extension ¯ f: Y → D along j is Scott continuous. In this case the extension map is a subspace embedding. We show that the extension map is Scott continuous iff D is the trivial onepoint space or j is a proper map in the sense of Hofmann and Lawson. In order to avoid the ambiguous expression “proper subspace embedding”, we refer to proper maps as finitary maps. We show that the finitary sober subspaces of the injective spaces are exactly the stably locally compact spaces. Moreover, the injective spaces over finitary embeddings are the algebras of the upper power space monad on the category of sober spaces. These coincide with the retracts of upper power spaces of sober spaces. In the full subcategory of locally compact sober spaces, these are known to be the continuous meetsemilattices. In the full subcategory of stably locally compact spaces these are again the continuous lattices. The above characterization of the injective spaces over finitary embeddings is an instance of a general result on injective objects in posetenriched categories with the structure of a KZmonad established in this paper, which we also apply to various full subcategories closed under the upper power space construction and to the upper and lower power locale monads. The above results also hold for the injective spaces over dense subspace embeddings (continuous Scott domains). Moreover, we show that every sober space has a smallest finitary dense sober subspace (its support). The support always contains the subspace of maximal points, and in the stably locally compact case (which includes densely injective spaces) it is the subspace of maximal points iff that subspace is compact.
Convergence In Exponentiable Spaces
 Theory Appl. Categories
, 1999
"... . Exponentiable spaces are characterized in terms of convergence. More precisely, we prove that a relation R : UX * X between ultrafilters and elements of a set X is the convergence relation for a quasilocallycompact (that is, exponentiable) topology on X if and only if the following conditions ar ..."
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Cited by 7 (1 self)
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. Exponentiable spaces are characterized in terms of convergence. More precisely, we prove that a relation R : UX * X between ultrafilters and elements of a set X is the convergence relation for a quasilocallycompact (that is, exponentiable) topology on X if and only if the following conditions are satisfied: 1. id ` R ffi j 2. R ffi UR = R ffi ¯ where j : X ! UX and ¯ : U(UX) ! UX are the unit and the multiplication of the ultrafilter monad, and U : Rel ! Rel extends the ultrafilter functor U : Set ! Set to the category of sets and relations. (U ; j; ¯) fails to be a monad on Rel only because j is not a strict natural transformation. So, exponentiable spaces are the lax (with respect to the unit law) algebras for a lax monad on Rel . Strict algebras are exponentiable and T 1 spaces. 1. Introduction In [4] it was implicitly proved that a topological space is exponentiable if and only if its lattice of open sets is a continuous lattice [6, 8], so fixing an important topological pr...
Injective spaces via the filter monad
 Topology Proceedings
, 1997
"... An injective space is a topological space with a strong extension property for continuous maps with values on it. A certain filter space construction embeds every T0 topological space into an injective space. The construction gives rise to a monad. We show that the monad is of the KockZöberlein typ ..."
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Cited by 7 (3 self)
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An injective space is a topological space with a strong extension property for continuous maps with values on it. A certain filter space construction embeds every T0 topological space into an injective space. The construction gives rise to a monad. We show that the monad is of the KockZöberlein type and apply this to obtain a simple proof of the fact that the algebras are the continuous lattices (Alan Day, 1975, Oswald Wyler, 1976). In previous work we established an injectivity theorem for monads of this type, which characterizes the injective objects over a certain class of embeddings as the algebras. For the filter monad, the class turns out to consist precisely of the subspace embeddings. We thus obtain as a corollary that the injective spaces over subspace embeddings are the continuous lattices endowed with the Scott topology (Dana Scott, 1972). Similar results are obtained for continuous Scott domains, which are characterized as the injective spaces over dense subspace embeddings. Keywords: Extension of maps, injective space, continuous lattice, continuous Scott domain, domain theory, KockZöberlein monad. AMS classification: 54C20, 06B35, 18C20. 1
Ordered topological structures
 Topology Appl
, 2009
"... The paper discusses interactions between order and topology on a given set which do not presuppose any separation conditions for either of the two structures, but which lead to the existing notions established by Nachbin in more special situations. We pursue this discussion at the much more general ..."
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The paper discusses interactions between order and topology on a given set which do not presuppose any separation conditions for either of the two structures, but which lead to the existing notions established by Nachbin in more special situations. We pursue this discussion at the much more general level of lax algebras, so that our categories do not concern just ordered topological spaces, but also sets with two interacting orders, approach spaces with an additional metric, etc. Key words: modular topological space, closedordered topological space, openordered topological space, lax (T, V)algebra, (T, V)category
The monad of probability measures over compact ordered spaces and its EilenbergMoore algebras
, 2008
"... The probability measures on compact Hausdorff spaces K form a compact convex subset PK of the space of measures with the vague topology. Every continuous map f: K → L of compact Hausdorff spaces induces a continuous affine map Pf: PK → PL extending P. Together with the canonical embedding ε: K → PK ..."
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The probability measures on compact Hausdorff spaces K form a compact convex subset PK of the space of measures with the vague topology. Every continuous map f: K → L of compact Hausdorff spaces induces a continuous affine map Pf: PK → PL extending P. Together with the canonical embedding ε: K → PK associating to every point its Dirac measure and the barycentric map β associating to every probability measure on PK its barycenter, we obtain a monad (P, ε, β). The EilenbergMoore algebras of this monad have been characterised to be the compact convex sets embeddable in locally convex topological vector spaces by Swirszcz [31]. We generalise this result to compact ordered spaces in the sense of Nachbin [23]. The probability measures form again a compact ordered space when endowed with the stochastic order. The maps ε and β are shown to preserve the stochastic orders. Thus, we obtain a monad over the category of compact ordered spaces and order preserving continuous maps. The algebras of this monad are shown to be the compact convex ordered sets embeddable in locally convex ordered topological vector spaces. This result can be seen as a step towards the characterisation of the algebras of the monad of probability
Injective spaces via adjunction
 J. Pure Appl. Algebra
, 2011
"... Abstract. Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relationx − → x between ultrafilters and points of a top ..."
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Abstract. Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relationx − → x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from (enriched) Category Theory for the investigation of (for instance) topological spaces. In this paper we study cocompleteness, adjoint functors and Kan extensions in the context of topological theories. We show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on Set. This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.
Injective spaces and the filter monad
 of GSMCTS Radio Interface Concepts,” TDoc SMG2 WPB 69/98, ETSI
, 1998
"... An injective space is a topological space with a strong extension property for continuous maps with values on it. A certain filter space construction embeds every T0 topological space into an injective space. The construction gives rise to a monad. We show that the monad is of the KockZöberlein typ ..."
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Cited by 1 (1 self)
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An injective space is a topological space with a strong extension property for continuous maps with values on it. A certain filter space construction embeds every T0 topological space into an injective space. The construction gives rise to a monad. We show that the monad is of the KockZöberlein type and apply this to obtain a simple proof of the fact that the algebras are the continuous lattices (Alan Day, 1975). In previous work we established an injectivity theorem for monads of this type, which characterizes the injective objects over a certain class of embeddings as the algebras. For the filter monad, the class turns out to consist precisely of the subspace embeddings. We thus obtain as a corollary that the injective spaces over subspace embeddings are the continuous lattices endowed with the Scott topology (Dana Scott, 1972). Similar results are obtained for continuous Scott domains, which are characterized as the injective spaces over dense subspace embeddings. Keywords: Extension of maps, injective space, continuous lattice, continuous Scott domain, domain theory, KockZöberlein monad. AMS classification: 54C20, 54C25, 54C15, 54E99, 06B35, 18C20. 1
Injective locales over perfect embeddings and algebras of the upper powerlocale monad
, 2002
"... We show that the locales which are injective over perfect sublocale embeddings coincide with the underlying objects of the algebras of the upper powerlocale monad, and we characterize them as those whose frames of opens enjoy a property analogous to stable supercontinuity. ..."
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We show that the locales which are injective over perfect sublocale embeddings coincide with the underlying objects of the algebras of the upper powerlocale monad, and we characterize them as those whose frames of opens enjoy a property analogous to stable supercontinuity.
Theory and Applications of Categories, Vol. 5, No. 6, pp. 148–162. CONVERGENCE IN EXPONENTIABLE SPACES
"... ABSTRACT. Exponentiable spaces are characterized in terms of convergence. More precisely, we prove that a relation R: UX ⇀Xbetween ultrafilters and elements of asetX is the convergence relation for a quasilocallycompact (that is, exponentiable) topology on X if and only if the following conditions ..."
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ABSTRACT. Exponentiable spaces are characterized in terms of convergence. More precisely, we prove that a relation R: UX ⇀Xbetween ultrafilters and elements of asetX is the convergence relation for a quasilocallycompact (that is, exponentiable) topology on X if and only if the following conditions are satisfied: 1. id ⊆ R ◦ η 2. R ◦UR = R ◦ µ where η: X →UX and µ: U(UX) →UX are the unit and the multiplication of the ultrafilter monad, and U: Rel → Rel extends the ultrafilter functor U: Set → Set to the category of sets and relations. (U,η,µ) fails to be a monad on Rel only because η is not a strict natural transformation. So, exponentiable spaces are the lax (with respect to the unit law) algebras for a lax monad on Rel. Strict algebras are exponentiable and T1 spaces. 1.