Localic separation and the duality between closedness and fittedness

[EN] There are a number of localic separation axioms which are roughly analogous to the T1-axiom from classical topology. For instance, besides the well-known subfitness and fitness, there are also Rosický-Šmarda's T1-locales, totally unordered locales and, more categorically, the recently introduced F-separated locales (i.e., those with a fitted diagonal) - a property strictly weaker than fitness. It has recently been shown that the strong Hausdorff property and F-separatedness are in a certain sense dual to each other. In this paper, we provide further instances of this duality - e.g., we introduce a new first-order separation property which is to F-separatedness as the Johnstone–Sun-shu-Hao–Paseka–Šmarda conservative Hausdorff axiom is to the strong Hausdorff property, and which can be of independent interest. Using this, we tie up the loose ends of the theory by establishing all the possible implications between these properties and other T1-type axioms occurring in the literature. In particular, we show that the strong Hausdorff property does not imply F-separatedness, a question which remained open and shows a remarkable difference with its counterpart in the category of topological spaces.The author acknowledges support from the Basque Government (grant IT1483-22 and a postdoctoral fellowship of the Basque Government, grant POS-2022-1-0015)

A study of localic subspaces, separation, and variants of normality and their duals.

198 p.As in classical topology, in localic topology one often needs to restrict to locales satisfyinga certain degree of separation. In fact, the study of separation in the category of localesconstitutes a non-trivial and important piece of the theory. For instance, it is sometimesimpossible to give an exact counterpart of a classical axiom, while other times a singleproperty for spaces yields multiple non-equivalent localic versions.The main goal of this thesis is to investigate several classes of separated locales and theirconnections with different classes of sublocales, that is, the regular subobjects in the categoryof locales.In particular, we introduce a new diagonal separation and show that it is, in a certainsense, dual to Isbell¿s (strong) Hausdorff property. The duality between suplattices andpreframes, and that between normality and extremal disconnectedness, turn out to be ofspecial interest in this context.Regarding higher separation, we introduce cardinal generalizations of normality andtheir duals (e.g., properties concerning extensions of disjoint families of cozero elements),and give characterizations via suitable insertion or extension results.The lower separation property known as the TD-axiom, also plays an important role inthe thesis. Namely, we investigate the TD-duality between the category of TD-spaces and acertain (non-full) subcategory of the category of locales, identifying the regular subobjects inthe localic side, and provide several applications in point-free topology.Tal como na topologia clássica, também na topologia dos locales (reticulados locais) éfrequente termos que nos restringir a locales que satisfaçam um certo grau de separação.De facto, o estudo de axiomas de separação na categoria dos locales constitui um aspectonão trivial e relevante da teoria. Por exemplo, em alguns casos é impossível termos acontrapartida exacta de um axioma clássico, enquanto noutros casos uma única propriedadepara espaços topológicos produz, na categoria dos locales, diversas versões não equivalentesentre si.O objectivo principal desta tese é investigar várias classes de locales separados e suasconexões com diferentes classes de sublocales (os subobjetos regulares na categoria doslocales).Em particular, introduzimos uma nova propriedade de separação diagonal e mostramosque se trata, em certo sentido, de uma propriedade dual do axioma (forte) de Hausdorffintroduzido por Isbell. As dualidades entre semi-reticulados e reticulados pré-locais, e entrenormalidade e desconexão extrema, acabam por ter um papel relevante neste contexto.Relativamente a axiomas de separação fortes, introduzimos generalizações de normalidade,em função de um cardinal arbitrário, e suas duais (por exemplo, propriedadesenvolvendo extensões de famílias disjuntas de elementos co-zero), e apresentamos caracterizaçõesem termos de propriedades de inserção ou extensão de funções.O axioma TD, uma propriedade de separação muito fraca, também desempenha umpapel importante nesta tese. Especificamente, investigamos a dualidade TD entre a categoriados espaços topológicos TD e uma determinada subcategoria (não plena) da categoria doslocales, identificando os subobjetos regulares na subcategoria de locales, e apresentamosvárias aplicações à topologia sem pontos.Tal y como ocurre en topología clásica, en topología locálica frecuentemente uno tiene querestringir su atención a locales que cumplen cierto grado de separación. De hecho, el estudiode la separación en la categoría de locales es un aspecto no trivial y relevante de la teoría. Enalgunos casos, es imposible dar una contrapartida exacta a un axioma clásico, mientras queen otros casos, una sola propiedad produce multitud de versiones locálicas no equivalentesentre sí.El principal objetivo de esta tesis es investigar varias clases de locales separados y susrelaciones con diferentes clases de sublocales, esto es, los subobjetos regulares en la categoríade locales.En particular, introducimos una nueva separación diagonal, y probamos que es, en ciertosentido, dual al axioma Hausdorff (fuerte) de Isbell. En este contexto, la dualidad entreretículos completos y premarcos, y aquella entre la normalidad y la desconexión extremaresultan ser de especial interés.En cuanto a la separación más fuerte, introducimos generalizaciones cardinales de lanormalidad y sus duales (por ejemplo, propiedades que consisten en la extensión de familiasdisjuntas de elementos cozero), y damos caracterizaciones de las mismas en términos deteoremas de extensión o inserción.Ciertas propiedades de separación más débiles, especialmente el axioma TD, tambiéndesempeñan un papel importante en esta tesis. Específicamente, investigamos la dualidad TDentre la categoría de espacios topológicos TD y cierta subcategoría (no plena) de la categoríade locales, identificando los subobjetos regulares en la categoría de locales, y proporcionamosalgunas aplicaciones en la topología sin puntos

Limits in categories of Vietoris coalgebras

Motivated by the need to reason about hybrid systems, we study limits in categories of coalgebras whose underlying functor is a Vietoris polynomial one - intuitively, the topological analogue of a Kripke polynomial functor. Among other results, we prove that every Vietoris polynomial functor admits a final coalgebra if it respects certain conditions concerning separation axioms and compactness. When the functor is restricted to some of the categories induced by these conditions the resulting categories of coalgebras are even complete. As a practical application, we use these developments in the specification and analysis of non-deterministic hybrid systems, in particular to obtain suitable notions of stability, and behaviour.publishe

Localic separation and the duality between closedness and fittedness

There are a number of localic separation axioms which are roughly analogous to the $T_1$-axiom from classical topology. For instance, besides the well-known subfitness and fitness, there are also Rosicky-Smarda's $T_1$-locales, totally unordered locales and, more categorically, the recently introduced $\mathcal{F}$-separated locales (i.e., those with a fitted diagonal) - a property strictly weaker than fitness. It has recently been shown that the strong Hausdorff property and $\mathcal{F}$-separatedness are in a certain sense dual to each other. In this paper, we provide further instances of this duality - e.g., we introduce a new first-order separation property which is to $\mathcal{F}$-separatedness as the Johnstone-Sun-shu-Hao-Paseka-Smarda conservative Hausdorff axiom is to the strong Hausdorff property, and which can be of independent interest. Using this, we tie up the loose ends of the theory by establishing all the possible implications between these properties and other $T_1$-type axioms occurring in the literature. In particular, we show that the strong Hausdorff property does not imply $\mathcal{F}$-separatedness, a question which remained open and shows a remarkable difference with its counterpart in the category of topological spaces

Quantales of open groupoids

It is well known that inverse semigroups are closely related to \'etale groupoids. In particular, it has recently been shown that there is a (non-functorial) equivalence between localic \'etale groupoids, on one hand, and complete and infinitely distributive inverse semigroups (abstract complete pseudogroups), on the other. This correspondence is mediated by a class of quantales, known as inverse quantal frames, that are obtained from the inverse semigroups by a simple join completion that yields an equivalence of categories. Hence, we can regard abstract complete pseudogroups as being essentially ``the same'' as inverse quantal frames, and in this paper we exploit this fact in order to find a suitable replacement for inverse semigroups in the context of open groupoids that are not necessarily \'etale. The interest of such a generalization lies in the importance and ubiquity of open groupoids in areas such as operator algebras, differential geometry and topos theory, and we achieve it by means of a class of quantales, called open quantal frames, which generalize inverse quantal frames and whose properties we study in detail. The resulting correspondence between quantales and open groupoids is not a straightforward generalization of the previous results concerning \'etale groupoids, and it depends heavily on the existence of inverse semigroups of local bisections of the quantales involved.Comment: 55 page

On binary coproducts of frames

summary:The structure of binary coproducts in the category of frames is analyzed, and the results are then applied widely in the study of compactness, local compactness (continuous frames), separatedness, pushouts and closed frame homomorphisms