We clarify the relationship between separable and covering morphisms in general categories by int... more 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.
Attended by some ninety mathematicians from all continents, a week-long Workshop on Category Theo... more Attended by some ninety mathematicians from all continents, a week-long Workshop on Category Theory was held at the University of Coimbra (Portugal) in July 2012 to celebrate the sixtieth birthday of Professor George Janelidze of the University of Cape Town. This volume constitutes the proceedings of the conference which were open for non-attendees to submit contributions. George Janelidze's career as a publishing research mathematician started in 1974 when, following the completion of his Diploma at Tbilisi State University, his first scientific article appeared on Galois theory in categories with inclusions (in Russian). This was the beginning of the mathematical field that he has influenced the most: Categorical Galois Theory (CGT). He established its pillars in his 1978 PhD thesis and in some subsequent publications, providing inter alia a categorical proof of Magid's Theorem. This work became widely known in the West only after it had been actively promoted by no one less esteemed than Saunders Mac Lane, leading to George's first major English-language publication: Pure Galois theory in categories (Journal of Algebra, 1990). George's book with Borceux on Galois Theories (Cambridge 2001) has become the standard reference for CGT and also points to the central
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.
We consider the ordinary category Span(C) of (isomorphism classes of) spans of morphisms in a cat... more We consider the ordinary category Span(C) of (isomorphism classes of) spans of morphisms in a category C with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of Span(C) to be an allegory. In particular, when C carries a pullback-stable, but not necessarily proper, (E, M)factorization system, we establish a quotient category Span E (C) that is isomorphic to the category Rel M (C) of M-relations in C, and show that it is a (unitary and tabular) allegory precisely when M is a class of monomorphisms in C. Without this restriction, one can still find a least pullback-stable and composition-closed class E • containing E such that Span E• (C) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary and tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the huge 2-category of all regular categories.
Under a general categorical procedure for the extension of dual equivalences as presented in this... more Under a general categorical procedure for the extension of dual equivalences as presented in this paper's predecessor, a new algebraically defined category is established that is dually equivalent to the category LKHaus of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category CLCA of complete local contact algebras and suitable morphisms. Thereby, a new proof is presented for the equivalence LKHaus ≃ CLCA op that was obtained by the first author more than a decade ago. Unlike the morphisms of CLCA, the morphisms of the new category and their composition law are very natural and easy to handle.
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.
In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by wh... more In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory. Motivated by some key results of the 1970s, the paper develops all needed ingredients from the theory of quantaloids in order to place essential results of categorical topology into the context of quantaloid-enriched category theory, a field that previously drew its motivation and applications from other domains, such as quantum logic and sheaf theory.
We clarify the relationship between separable and covering morphisms in general categories by int... more 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.
Attended by some ninety mathematicians from all continents, a week-long Workshop on Category Theo... more Attended by some ninety mathematicians from all continents, a week-long Workshop on Category Theory was held at the University of Coimbra (Portugal) in July 2012 to celebrate the sixtieth birthday of Professor George Janelidze of the University of Cape Town. This volume constitutes the proceedings of the conference which were open for non-attendees to submit contributions. George Janelidze's career as a publishing research mathematician started in 1974 when, following the completion of his Diploma at Tbilisi State University, his first scientific article appeared on Galois theory in categories with inclusions (in Russian). This was the beginning of the mathematical field that he has influenced the most: Categorical Galois Theory (CGT). He established its pillars in his 1978 PhD thesis and in some subsequent publications, providing inter alia a categorical proof of Magid's Theorem. This work became widely known in the West only after it had been actively promoted by no one less esteemed than Saunders Mac Lane, leading to George's first major English-language publication: Pure Galois theory in categories (Journal of Algebra, 1990). George's book with Borceux on Galois Theories (Cambridge 2001) has become the standard reference for CGT and also points to the central
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.
We consider the ordinary category Span(C) of (isomorphism classes of) spans of morphisms in a cat... more We consider the ordinary category Span(C) of (isomorphism classes of) spans of morphisms in a category C with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of Span(C) to be an allegory. In particular, when C carries a pullback-stable, but not necessarily proper, (E, M)factorization system, we establish a quotient category Span E (C) that is isomorphic to the category Rel M (C) of M-relations in C, and show that it is a (unitary and tabular) allegory precisely when M is a class of monomorphisms in C. Without this restriction, one can still find a least pullback-stable and composition-closed class E • containing E such that Span E• (C) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary and tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the huge 2-category of all regular categories.
Under a general categorical procedure for the extension of dual equivalences as presented in this... more Under a general categorical procedure for the extension of dual equivalences as presented in this paper's predecessor, a new algebraically defined category is established that is dually equivalent to the category LKHaus of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category CLCA of complete local contact algebras and suitable morphisms. Thereby, a new proof is presented for the equivalence LKHaus ≃ CLCA op that was obtained by the first author more than a decade ago. Unlike the morphisms of CLCA, the morphisms of the new category and their composition law are very natural and easy to handle.
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.
In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by wh... more In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory. Motivated by some key results of the 1970s, the paper develops all needed ingredients from the theory of quantaloids in order to place essential results of categorical topology into the context of quantaloid-enriched category theory, a field that previously drew its motivation and applications from other domains, such as quantum logic and sheaf theory.
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Papers by W. Tholen