Propounding a general categorical framework for the extension of dualities, we present a new proo... more Propounding a general categorical framework for the extension of dualities, we present a new proof of the de Vries Duality Theorem for the category KHaus of compact Hausdorff spaces and their continuous maps, as an extension of a restricted Stone duality. Then, applying a dualization of the categorical framework to the de Vries duality, we give an alternative proof of the extension of the de Vries duality to the category Tych of Tychonoff spaces that was provided by Bezhanishvili, Morandi and Olberding. In the process of doing so, we obtain new duality theorems for both categories, KHaus and Tych.
For locally ranked categories A, which include all locally presentable categories and the categor... more For locally ranked categories A, which include all locally presentable categories and the category Top, we prove that, given any set
Dedicated to Dominique Bourn on the occasion of his sixtieth birthday We clarify the relationship... more Dedicated to Dominique Bourn on the occasion of his sixtieth birthday We clarify the relationship between separable and covering morphisms in general categories by introducing and studying an intermediate class of morphisms that we call strongly separable. 1.
Une classe F d'objets d'une categorie generale C (que nous pouvons regarder comme le syst... more Une classe F d'objets d'une categorie generale C (que nous pouvons regarder comme le systeme des objets finis de C) induit naturellement des topologies Hausdorff sur les hom-ensembles (A, B). De cette facon, C devient une Haus-categorie. De plus, on a une Loc-categorie C naturellement associee dont C est le spectre; dans C, un cadre C(A, B) peut etre non-trivial egalement lorsque C(A, B) = O.
For a composition-closed and pullback-stable class S of morphisms in a category C containing all ... more For a composition-closed and pullback-stable class S of morphisms in a category C containing all isomorphisms, we form the category Span(C,S) of S-spans (s, f) in C with first “leg” s lying in S, and give an alternative construction of its quotient category C[S] of S-fractions. Instead of trying to turn S-morphisms “directly” into isomorphisms, we turn them separately into retractions and into sections, in a universal manner. Without confining S to be a class of monomorphisms of C, the second of these two quotient processes leads us to the category Par(C,S) of S-partial maps in C. Under mild additional hypotheses on S, Par(C,S) has a localization, which is a split restriction category, or even a split range category (in the sense of Cockett, Guo and Hofstra), but which is still large enough to admit C[S] as its quotient. The construction of the range category is part of a global adjunction between relatively stable factorization systems and split range categories.
For a composition-closed and pullback-stable class S of morphisms in a category C containing all ... more For a composition-closed and pullback-stable class S of morphisms in a category C containing all isomorphisms, we form the category Span ( C , S ) of S -spans ( s , f ) in C with first “leg” s lying in S , and give an alternative construction of its quotient category C [ S − 1 ] of S -fractions. Instead of trying to turn S -morphisms “directly” into isomorphisms, we turn them separately into retractions and into sections in a universal manner, thus obtaining the quotient categories Retr ( C , S ) and Sect ( C , S ) . The fraction category C [ S − 1 ] is their largest joint quotient category. Without confining S to be a class of monomorphisms of C , we show that Sect ( C , S ) admits a quotient category, Par ( C , S ) , whose name is justified by two facts. On one hand, for S a class of monomorphisms in C , it returns the category of S -spans in C , also called S -partial maps in this case; on the other hand, we prove that Par ( C , S ) is a split restriction category (in the sense o...
In order to facilitate a natural choice for morphisms created by the (left or right) lifting prop... more In order to facilitate a natural choice for morphisms created by the (left or right) lifting property as used in the definition of weak factorization systems, the notion of natural weak factorization system in the category K is introduced, as a pair (comonad, monad) over K. The link with existing notions in terms of morphism classes is given via the respective Eilenberg– Moore categories.
We prove a general categorical theorem for the extension of dualities. Applying it, we present ne... more We prove a general categorical theorem for the extension of dualities. Applying it, we present new proofs of the de Vries Duality Theorem for the category CHaus of compact Hausdorff spaces and continuous maps, and of the recent Bezhanishvili-Morandi-Olberding Duality Theorem which extends the de Vries duality to the category Tych of Tychonoff spaces and continuous maps. In the process of doing so we obtain new duality theorems for the categories CHaus and Tych.
A categorical proof of Tychonoff’s Theorem on the productivity of compact topological spaces is p... more A categorical proof of Tychonoff’s Theorem on the productivity of compact topological spaces is provided.
The paper proposes the notions of topological platform and quantalic topological theory for the p... more The paper proposes the notions of topological platform and quantalic topological theory for the presentation and investigation of categories of interest beyond the realm of algebra. These notions are nevertheless grounded in algebra, through the notions of monad and distributive law. The paper shows how they entail previously proposed concepts with similar goals.
Propounding a general categorical framework for the extension of dualities, we present a new proo... more Propounding a general categorical framework for the extension of dualities, we present a new proof of the de Vries Duality Theorem for the category KHaus of compact Hausdorff spaces and their continuous maps, as an extension of a restricted Stone duality. Then, applying a dualization of the categorical framework to the de Vries duality, we give an alternative proof of the extension of the de Vries duality to the category Tych of Tychonoff spaces that was provided by Bezhanishvili, Morandi and Olberding. In the process of doing so, we obtain new duality theorems for both categories, KHaus and Tych.
For locally ranked categories A, which include all locally presentable categories and the categor... more For locally ranked categories A, which include all locally presentable categories and the category Top, we prove that, given any set
Dedicated to Dominique Bourn on the occasion of his sixtieth birthday We clarify the relationship... more Dedicated to Dominique Bourn on the occasion of his sixtieth birthday We clarify the relationship between separable and covering morphisms in general categories by introducing and studying an intermediate class of morphisms that we call strongly separable. 1.
Une classe F d'objets d'une categorie generale C (que nous pouvons regarder comme le syst... more Une classe F d'objets d'une categorie generale C (que nous pouvons regarder comme le systeme des objets finis de C) induit naturellement des topologies Hausdorff sur les hom-ensembles (A, B). De cette facon, C devient une Haus-categorie. De plus, on a une Loc-categorie C naturellement associee dont C est le spectre; dans C, un cadre C(A, B) peut etre non-trivial egalement lorsque C(A, B) = O.
For a composition-closed and pullback-stable class S of morphisms in a category C containing all ... more For a composition-closed and pullback-stable class S of morphisms in a category C containing all isomorphisms, we form the category Span(C,S) of S-spans (s, f) in C with first “leg” s lying in S, and give an alternative construction of its quotient category C[S] of S-fractions. Instead of trying to turn S-morphisms “directly” into isomorphisms, we turn them separately into retractions and into sections, in a universal manner. Without confining S to be a class of monomorphisms of C, the second of these two quotient processes leads us to the category Par(C,S) of S-partial maps in C. Under mild additional hypotheses on S, Par(C,S) has a localization, which is a split restriction category, or even a split range category (in the sense of Cockett, Guo and Hofstra), but which is still large enough to admit C[S] as its quotient. The construction of the range category is part of a global adjunction between relatively stable factorization systems and split range categories.
For a composition-closed and pullback-stable class S of morphisms in a category C containing all ... more For a composition-closed and pullback-stable class S of morphisms in a category C containing all isomorphisms, we form the category Span ( C , S ) of S -spans ( s , f ) in C with first “leg” s lying in S , and give an alternative construction of its quotient category C [ S − 1 ] of S -fractions. Instead of trying to turn S -morphisms “directly” into isomorphisms, we turn them separately into retractions and into sections in a universal manner, thus obtaining the quotient categories Retr ( C , S ) and Sect ( C , S ) . The fraction category C [ S − 1 ] is their largest joint quotient category. Without confining S to be a class of monomorphisms of C , we show that Sect ( C , S ) admits a quotient category, Par ( C , S ) , whose name is justified by two facts. On one hand, for S a class of monomorphisms in C , it returns the category of S -spans in C , also called S -partial maps in this case; on the other hand, we prove that Par ( C , S ) is a split restriction category (in the sense o...
In order to facilitate a natural choice for morphisms created by the (left or right) lifting prop... more In order to facilitate a natural choice for morphisms created by the (left or right) lifting property as used in the definition of weak factorization systems, the notion of natural weak factorization system in the category K is introduced, as a pair (comonad, monad) over K. The link with existing notions in terms of morphism classes is given via the respective Eilenberg– Moore categories.
We prove a general categorical theorem for the extension of dualities. Applying it, we present ne... more We prove a general categorical theorem for the extension of dualities. Applying it, we present new proofs of the de Vries Duality Theorem for the category CHaus of compact Hausdorff spaces and continuous maps, and of the recent Bezhanishvili-Morandi-Olberding Duality Theorem which extends the de Vries duality to the category Tych of Tychonoff spaces and continuous maps. In the process of doing so we obtain new duality theorems for the categories CHaus and Tych.
A categorical proof of Tychonoff’s Theorem on the productivity of compact topological spaces is p... more A categorical proof of Tychonoff’s Theorem on the productivity of compact topological spaces is provided.
The paper proposes the notions of topological platform and quantalic topological theory for the p... more The paper proposes the notions of topological platform and quantalic topological theory for the presentation and investigation of categories of interest beyond the realm of algebra. These notions are nevertheless grounded in algebra, through the notions of monad and distributive law. The paper shows how they entail previously proposed concepts with similar goals.
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Papers by W. Tholen