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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 27 (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.
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
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 6 (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...
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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Cited by 4 (2 self)
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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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Cited by 3 (2 self)
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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 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
Semantic domains, injective spaces and monads (Extended Abstract)
"... Many categories of semantic domains can be considered from an ordertheoretic point of view and from a topological point of view via the Scott topology. The topological point of view is particularly fruitful for considerations of computability in classical spaces such as the Euclidean real line. Whe ..."
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Many categories of semantic domains can be considered from an ordertheoretic point of view and from a topological point of view via the Scott topology. The topological point of view is particularly fruitful for considerations of computability in classical spaces such as the Euclidean real line. When one embeds topological spaces into domains, one requires that the Scott continuous maps between the host domains fully capture the continuous maps between the guest topological spaces. This property of the host domains is known as injectivity. For example, the continuous Scott domains are characterized as the injective spaces over dense subspace embeddings (Dana Scott, 1972, 1980). From a third point of view, the continuous Scott domains arise as the algebras of a monad (Wyler, 1985). The topological characterization by injectivity turns out to follow from the algebraic characterization and general category theory (Escard'o 1998). In this paper we systematically consider monads that arise ...
Continuous Categories Revisited
, 2003
"... Generalizing the fact that Scott's continuous lattices form the equational hull of the class of all algebraic lattices, we describe an equational hull of LFP, the category of locally finitely presentable categories, over CAT. Up to a settheoretical hypothesis this hull is formed by the category of ..."
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Generalizing the fact that Scott's continuous lattices form the equational hull of the class of all algebraic lattices, we describe an equational hull of LFP, the category of locally finitely presentable categories, over CAT. Up to a settheoretical hypothesis this hull is formed by the category of all precontinuous categories, i.e., categories in which limits and filtered colimits distribute. This concept is closely related to the continuous categories of P. T. Johnstone and A. Joyal. 1.
EXTENSIONS IN THE THEORY OF LAX ALGEBRAS Dedicated
"... Abstract. Recent investigations of lax algebrasin generalization of Barr's relationalalgebrasmake an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and Vrelations (where V is a unital quantale). For a given monadthere may be many such lax extensions, an ..."
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Abstract. Recent investigations of lax algebrasin generalization of Barr's relationalalgebrasmake an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and Vrelations (where V is a unital quantale). For a given monadthere may be many such lax extensions, and different constructions appear in the literature. The aim of this article is to shed a unifying light on these lax extensions, andpresent a symptomatic situation in which distinct monads yield isomorphic categories of lax algebras.