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Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
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Cited by 13 (4 self)
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Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to
The Convergence Approach to Exponentiable Maps
 352 MARIA MANUEL CLEMENTINO, DIRK HOFMANN AND WALTER
, 2000
"... Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the ..."
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Cited by 9 (7 self)
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Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the classical result for compact Hausdorff spaces. Furthermore, in generalization of the WhiteheadMichael characterization of locally compact Hausdorff spaces, we characterize exponentiable maps of Top between Hausdorff spaces as restrictions of perfect maps to open subspaces.
Exponentiable morphisms: posets, spaces, locales
 and Grothendieck toposes, Theory and Applications of Categories 8
, 2000
"... ABSTRACT. Inthis paper, we consider those morphisms p: P − → B of posets for which the induced geometric morphism of presheaf toposes is exponentiable in the category of Grothendieck toposes. In particular, we show that a necessary condition is that the induced map p ↓ : P ↓ − → B ↓ is exponentiable ..."
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Cited by 7 (5 self)
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ABSTRACT. Inthis paper, we consider those morphisms p: P − → B of posets for which the induced geometric morphism of presheaf toposes is exponentiable in the category of Grothendieck toposes. In particular, we show that a necessary condition is that the induced map p ↓ : P ↓ − → B ↓ is exponentiable in the category of topological spaces, where P ↓ is the space whose points are elements of P and open sets are downward closed subsets of P. Along the way, we show that p ↓ : P ↓ − → B ↓ is exponentiable if and only if p: P − → B is exponentiable in the category of posets and satisfies an additional compactness condition. The criteria for exponentiability of morphisms of posets is related to (but weaker than) the factorizationlifting property for exponentiability of morphisms in the
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...
Topologies on spaces of continuous functions
 Topology Proc
"... It is wellknown that a Hausdorff space is exponentiable if and only if it is locally compact, and that in this case the exponential topology is the compactopen topology. It is less wellknown that among arbitrary topological spaces, the exponentiable spaces are precisely the corecompact spaces. T ..."
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Cited by 5 (1 self)
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It is wellknown that a Hausdorff space is exponentiable if and only if it is locally compact, and that in this case the exponential topology is the compactopen topology. It is less wellknown that among arbitrary topological spaces, the exponentiable spaces are precisely the corecompact spaces. The available approaches to the general characterization are based on either category theory or continuouslattice theory, or even both. It is the main purpose of this paper to provide a selfcontained, elementary and brief development of general function spaces. The only prerequisite to this development is a basic knowledge of general topology (continuous functions, product topology and compactness). But another connection with the theory of continuous lattices lurks in this approach to function spaces, which is examined after the elementary exposition is completed. Continuity of the functionevaluation map is shown to coincide with a certain approximation property of a topology on the frame of open sets of the exponent space, and the existence of a smallest approximating topology is equivalent to exponentiability of the space. We show that the intersection of the approximating topologies of any preframe is the Scott topology. In particular, we conclude that a complete lattice is continuous if and only if it has a smallest approximating topology and finite meets distribute over directed joins. 1
Functionspace compactifications of function spaces
 Topology Appl
"... If X and Y are Hausdorff spaces with X locally compact, then the compactopen topology on the set C(X, Y) of continuous maps from X to Y is known to produce the right functionspace topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for an ..."
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Cited by 1 (1 self)
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If X and Y are Hausdorff spaces with X locally compact, then the compactopen topology on the set C(X, Y) of continuous maps from X to Y is known to produce the right functionspace topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X, Z) has a compact Hausdorff topology whose relative topology on C(X, Y) is the compactopen topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X, Y) under the compactopen topology is embedded into the Vietoris hyperspace V(X × Y). (2) The space of realvalued continuous functions on a locally compact Hausdorff space under the compactopen topology is embedded into a compact Hausdorff space whose points are pairs of extended realvalued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways.
Exponentiability Of Perfect Maps: Four Approaches
, 2002
"... Two proofs of the exponentiability of perfect maps are presented and compared to two other recent approaches. One of the proofs is an elementary approach including a direct construction of the exponentials. The other, implicit in the literature, uses internal locales in the topos of setvalued sh ..."
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Cited by 1 (1 self)
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Two proofs of the exponentiability of perfect maps are presented and compared to two other recent approaches. One of the proofs is an elementary approach including a direct construction of the exponentials. The other, implicit in the literature, uses internal locales in the topos of setvalued sheaves on a topological space.
University that enabled me to do this research. Contents
, 2008
"... I gratefully acknowledge the generous financial support received from the Killam Trusts and Dalhousie ..."
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I gratefully acknowledge the generous financial support received from the Killam Trusts and Dalhousie
Foundations for Computable Topology
, 2009
"... Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines. Stone Duality. We express the duality between al ..."
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Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines. Stone Duality. We express the duality between algebra and geometry as an abstract monadic adjunction that we turn into a new type theory. To this we add an equation that is satisfied by the Sierpiński space, which plays a key role as the classifier for both open and closed subspaces. In the resulting theory there is a duality between open and closed concepts. This captures many basic properties of compact and closed subspaces, despite the absence of any explicitly infinitary axiom. It offers dual results that link general topology to recursion theory. The extensions and applications of ASD elsewhere that this paper survey include a purely recursive theory of elementary real analysis in which, unlike in previous approaches, the real closed interval [0, 1] in ASD is compact.