This paper dualizes the setting of affine spaces as originally introduced by Diers for applicatio... more This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the structures of dually affine spaces. The dual of the Zariski closure operator is introduced, and the 1-sphere and its copowers together with their fundamental groups are shown to be examples of complete objects with respect to the Zariski dual closure operator.
For a quantale $\V$, first a closure-theoretic approach to completeness and separation in $\V$-ca... more For a quantale $\V$, first a closure-theoretic approach to completeness and separation in $\V$-categories is presented. This approach is then generalized to $\Tth$-categories, where $\Tth$ is a topological theory that entails a set monad $\mT$ and a compatible $\mT$-algebra structure on $\V$.
Metric approximate categories, or metagories, for short, are metrically enriched graphs. Their st... more Metric approximate categories, or metagories, for short, are metrically enriched graphs. Their structure assigns to every directed triangle in the graph a value which may be interpreted as the area of the triangle; alternatively, as the distance of a pair of consecutive arrows to any potential candidate for their composite. These values may live in an arbitrary commutative quantale. Generalizing and extending recent work by Aliouche and Simpson, we give a condition for the existence of an Yoneda-type embedding which, in particular, gives the isometric embeddability of a metagory into a metrically enriched category. The generality of the value quantale allows for applications beyond the classical metric context.
Free regular and exact completions of categories with various ranks of weak limits are presented ... more Free regular and exact completions of categories with various ranks of weak limits are presented as subcategories of presheaf categories. Their universal properties can then be derived with standard techniques as used in duality theory.
The paper discusses interactions between order and topology on a given set which do not presuppos... more 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.
We extend the Dikranjan-Uspenskij notions of c-compact and h-complete topological group to the mo... more We extend the Dikranjan-Uspenskij notions of c-compact and h-complete topological group to the morphism level, study the stability properties of the newly defined types of maps, such as closure under direct products, and compare them with their counterparts in topology. We assume Hausdorffness only when our proofs require us to do so, which leads to new results and the affirmation of some facts that were known in a Hausdorff context.
We investigate those lax extensions of a Set-monad T = (T, m, e) to the category V-Rel of sets an... more We investigate those lax extensions of a Set-monad T = (T, m, e) to the category V-Rel of sets and V-valued relations for a quantale V = (V, ⌦, k) that are fully determined by maps ⇠ : T V ! V. We pay special attention to those maps ⇠ that make V a T-algebra and, in fact, (V, ⌦, k) a monoid in the category Set T with its cartesian structure. Any such map ⇠ forms the main ingredient to Hofmann's notion of topological theory.
Hausdorff and Gromov distances are introduced and treated in the context of categories enriched o... more Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdorff functor which, for every V-category X, provides the powerset of X with a suitable V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras are order-complete. The Gromov construction may be pursued for any endofunctor K of V-Cat. In order to define the Gromov "distance" between V-categories X and Y we use V-modules between X and Y , rather than V-category structures on the disjoint union of X and Y. Hence, we first provide a general extension theorem which, for any K , yields a lax extensionK to the category V-Mod of V-categories, with V-modules as morphisms.
For a small quantaloid Q we consider four fundamental 2-monads T on Q-Cat, given by the presheaf ... more For a small quantaloid Q we consider four fundamental 2-monads T on Q-Cat, given by the presheaf 2-monad P and the copresheaf 2-monad P † , as well as by their two composite 2-monads, and establish that they all laxly distribute over P. These four 2-monads therefore admit lax extensions to the category Q-Dist of Q-categories and their distributors. We characterize the corresponding (T, Q)-categories in each of the four cases, leading us to both known and novel categorical structures.
Introduction. Totality of a category as introduced by Street and Walters [17] is known to be a st... more Introduction. Totality of a category as introduced by Street and Walters [17] is known to be a strong cocompleteness property (cf. also [21]) which goes far beyond ordinary (small) cocompleteness. It implies compactness in the sense of Isbell [11] and therefore hypercompleteness [7], that is: the existence of limits of all those (not necessarily small) diagrams which are not prevented from having a limit merely from size-considerations with respect to the homsets. In particular, arbitrary intersections of monomorphisms exist in a total category; which is part of Street's [16] characterization of totality and is used in establishing the interrelationship with topoi (cf. also [15]). This article gives solutions to two problems mentioned in Kelly's excellent survey article [12] and gives an £-generalization of Day's theorem [8] that a cocomplete category with arbitrary cointersections of epimorphisms and generator is total. Day mentions the possibility of replacing epimorphisms by £-morphisms which belong to an (£, ^)-factorization system, but gives no generalized statement; in particular, he does not specify how to generalize the notion of generator. In any case, our theorem seems to go beyond what Day had in mind since we do not even require £ to be closed under composition. This is particularly relevant in the case where £ is the class of morphisms which are composites of two regular epimorphisms. We devote special attention to this case, as it gives the theorem that a cocomplete category is total if it has a regular generator; or a small set of objects of which every other object is (somehow) a colimit (which is a strengthening of Kelly's [12, Corollary 6.5]). We give a complete solution to the problem whether the £-generalization of Day's theorem allows for converse statements, and thereby settle the questions raised in Kelly's article: a total category always allows the formation of cointersections of arbitrary families of regular epimorphisms, but not so for strong ones, even when it contains a strong generator; on the other hand, a total category need not have a generator, even when it is cowellpowered and therefore contains cointersections of arbitrary families of epimorphisms. The paper is self-contained. Although our proof of the £-version of Day's theorem relies heavily on lifting properties of solid functors (formerly called semi-topological [18]) we in fact do not require any previous knowledge of these functors, as all relevant facts about them are provided in a new concise form in this paper. In order to keep its length as limited, and the range of potential readers as unlimited, as possible, we have given all definitions and
Quillen's notion of small object and the Gabriel-Ulmer notion of finitely presentable or generate... more Quillen's notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T 1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category all topological spaces.
For a quantale V we introduce V-valued topological spaces via V-valued point-set-distance functio... more For a quantale V we introduce V-valued topological spaces via V-valued point-set-distance functions and, when V is completely distributive, characterize them in terms of both, so-called closure towers and ultrafilter convergence relations. When V is the two-element chain 2, the extended real half-line [0, ∞], or the quantale Δ of distance distribution functions, the general setting produces known and new results on topological spaces, approach spaces, and the only recently considered probabilistic approach spaces, as well as on their functorial interactions with each other.
For a quantale V, first a closure-theoretic approach to completeness and separation in V-categori... more For a quantale V, first a closure-theoretic approach to completeness and separation in V-categories is presented. This approach is then generalized to T-categories, where T is a topological theory that entails a set monad Ì and a compatible Ì-algebra structure on V.
It is shown that, for a small quantaloid Q, the category of small Q-categories and Q-functors is ... more It is shown that, for a small quantaloid Q, the category of small Q-categories and Q-functors is total and cototal, and so is the category of Q-distributors and Q-Chu transforms.
This paper dualizes the setting of affine spaces as originally introduced by Diers for applicatio... more This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the structures of dually affine spaces. The dual of the Zariski closure operator is introduced, and the 1-sphere and its copowers together with their fundamental groups are shown to be examples of complete objects with respect to the Zariski dual closure operator.
This paper dualizes the setting of affine spaces as originally introduced by Diers for applicatio... more This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the structures of dually affine spaces. The dual of the Zariski closure operator is introduced, and the 1-sphere and its copowers together with their fundamental groups are shown to be examples of complete objects with respect to the Zariski dual closure operator.
For a quantale $\V$, first a closure-theoretic approach to completeness and separation in $\V$-ca... more For a quantale $\V$, first a closure-theoretic approach to completeness and separation in $\V$-categories is presented. This approach is then generalized to $\Tth$-categories, where $\Tth$ is a topological theory that entails a set monad $\mT$ and a compatible $\mT$-algebra structure on $\V$.
Metric approximate categories, or metagories, for short, are metrically enriched graphs. Their st... more Metric approximate categories, or metagories, for short, are metrically enriched graphs. Their structure assigns to every directed triangle in the graph a value which may be interpreted as the area of the triangle; alternatively, as the distance of a pair of consecutive arrows to any potential candidate for their composite. These values may live in an arbitrary commutative quantale. Generalizing and extending recent work by Aliouche and Simpson, we give a condition for the existence of an Yoneda-type embedding which, in particular, gives the isometric embeddability of a metagory into a metrically enriched category. The generality of the value quantale allows for applications beyond the classical metric context.
Free regular and exact completions of categories with various ranks of weak limits are presented ... more Free regular and exact completions of categories with various ranks of weak limits are presented as subcategories of presheaf categories. Their universal properties can then be derived with standard techniques as used in duality theory.
The paper discusses interactions between order and topology on a given set which do not presuppos... more 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.
We extend the Dikranjan-Uspenskij notions of c-compact and h-complete topological group to the mo... more We extend the Dikranjan-Uspenskij notions of c-compact and h-complete topological group to the morphism level, study the stability properties of the newly defined types of maps, such as closure under direct products, and compare them with their counterparts in topology. We assume Hausdorffness only when our proofs require us to do so, which leads to new results and the affirmation of some facts that were known in a Hausdorff context.
We investigate those lax extensions of a Set-monad T = (T, m, e) to the category V-Rel of sets an... more We investigate those lax extensions of a Set-monad T = (T, m, e) to the category V-Rel of sets and V-valued relations for a quantale V = (V, ⌦, k) that are fully determined by maps ⇠ : T V ! V. We pay special attention to those maps ⇠ that make V a T-algebra and, in fact, (V, ⌦, k) a monoid in the category Set T with its cartesian structure. Any such map ⇠ forms the main ingredient to Hofmann's notion of topological theory.
Hausdorff and Gromov distances are introduced and treated in the context of categories enriched o... more Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdorff functor which, for every V-category X, provides the powerset of X with a suitable V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras are order-complete. The Gromov construction may be pursued for any endofunctor K of V-Cat. In order to define the Gromov "distance" between V-categories X and Y we use V-modules between X and Y , rather than V-category structures on the disjoint union of X and Y. Hence, we first provide a general extension theorem which, for any K , yields a lax extensionK to the category V-Mod of V-categories, with V-modules as morphisms.
For a small quantaloid Q we consider four fundamental 2-monads T on Q-Cat, given by the presheaf ... more For a small quantaloid Q we consider four fundamental 2-monads T on Q-Cat, given by the presheaf 2-monad P and the copresheaf 2-monad P † , as well as by their two composite 2-monads, and establish that they all laxly distribute over P. These four 2-monads therefore admit lax extensions to the category Q-Dist of Q-categories and their distributors. We characterize the corresponding (T, Q)-categories in each of the four cases, leading us to both known and novel categorical structures.
Introduction. Totality of a category as introduced by Street and Walters [17] is known to be a st... more Introduction. Totality of a category as introduced by Street and Walters [17] is known to be a strong cocompleteness property (cf. also [21]) which goes far beyond ordinary (small) cocompleteness. It implies compactness in the sense of Isbell [11] and therefore hypercompleteness [7], that is: the existence of limits of all those (not necessarily small) diagrams which are not prevented from having a limit merely from size-considerations with respect to the homsets. In particular, arbitrary intersections of monomorphisms exist in a total category; which is part of Street's [16] characterization of totality and is used in establishing the interrelationship with topoi (cf. also [15]). This article gives solutions to two problems mentioned in Kelly's excellent survey article [12] and gives an £-generalization of Day's theorem [8] that a cocomplete category with arbitrary cointersections of epimorphisms and generator is total. Day mentions the possibility of replacing epimorphisms by £-morphisms which belong to an (£, ^)-factorization system, but gives no generalized statement; in particular, he does not specify how to generalize the notion of generator. In any case, our theorem seems to go beyond what Day had in mind since we do not even require £ to be closed under composition. This is particularly relevant in the case where £ is the class of morphisms which are composites of two regular epimorphisms. We devote special attention to this case, as it gives the theorem that a cocomplete category is total if it has a regular generator; or a small set of objects of which every other object is (somehow) a colimit (which is a strengthening of Kelly's [12, Corollary 6.5]). We give a complete solution to the problem whether the £-generalization of Day's theorem allows for converse statements, and thereby settle the questions raised in Kelly's article: a total category always allows the formation of cointersections of arbitrary families of regular epimorphisms, but not so for strong ones, even when it contains a strong generator; on the other hand, a total category need not have a generator, even when it is cowellpowered and therefore contains cointersections of arbitrary families of epimorphisms. The paper is self-contained. Although our proof of the £-version of Day's theorem relies heavily on lifting properties of solid functors (formerly called semi-topological [18]) we in fact do not require any previous knowledge of these functors, as all relevant facts about them are provided in a new concise form in this paper. In order to keep its length as limited, and the range of potential readers as unlimited, as possible, we have given all definitions and
Quillen's notion of small object and the Gabriel-Ulmer notion of finitely presentable or generate... more Quillen's notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T 1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category all topological spaces.
For a quantale V we introduce V-valued topological spaces via V-valued point-set-distance functio... more For a quantale V we introduce V-valued topological spaces via V-valued point-set-distance functions and, when V is completely distributive, characterize them in terms of both, so-called closure towers and ultrafilter convergence relations. When V is the two-element chain 2, the extended real half-line [0, ∞], or the quantale Δ of distance distribution functions, the general setting produces known and new results on topological spaces, approach spaces, and the only recently considered probabilistic approach spaces, as well as on their functorial interactions with each other.
For a quantale V, first a closure-theoretic approach to completeness and separation in V-categori... more For a quantale V, first a closure-theoretic approach to completeness and separation in V-categories is presented. This approach is then generalized to T-categories, where T is a topological theory that entails a set monad Ì and a compatible Ì-algebra structure on V.
It is shown that, for a small quantaloid Q, the category of small Q-categories and Q-functors is ... more It is shown that, for a small quantaloid Q, the category of small Q-categories and Q-functors is total and cototal, and so is the category of Q-distributors and Q-Chu transforms.
This paper dualizes the setting of affine spaces as originally introduced by Diers for applicatio... more This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the structures of dually affine spaces. The dual of the Zariski closure operator is introduced, and the 1-sphere and its copowers together with their fundamental groups are shown to be examples of complete objects with respect to the Zariski dual closure operator.
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Papers by Walter Tholen