Browsing by Author "Sousa, Lurdes"
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- Constructions of Solid HullsPublication . Adámek, Jirí; Sousa, LurdesFor each concrete category (K,U) an extension LIM(K,U) is constructed and under certain ‘smallness conditions’ it is proved that LIM(K,U) is a solid hull of (K,U), i.e., the least finally dense solid extension of (K,U). A full subcategory of Top_2 is presented which does not have a solid hull.
- Note on multisolid categoriesPublication . Sousa, LurdesIt is shown that a cowellpowered concrete category (A,U) over a multicocomplete category is multisolid if and only if A is multicocmplete and U is a right multi-adjoint.
- On the pullback stability of a quotient map with respect to a closure operatorPublication . Sousa, LurdesThere are well-known characterizations of hereditary quotient maps in the category of topological spaces, (that is, of quotient maps stable under pullback along embeddings), as well as of universal quotient maps (that is, of quotient maps stable under pullback). These are precisely the so-called pseudo-open maps, as shown by Arhangel'slii, and the bi-quotient maps of Michael, as shown by Day and Kelly, respectively. In this paper hereditary and stable quotient maps are characterized in the broader context given by a category eqquipped with a closure operator. To this end, we derive explicit formulae and conditions for the closure in the codomain of such a quotient map in terms of the closure in its domain.
- Orthogonality and closure operatorsPublication . Sousa, LurdesGiven a full and replete subcategory A of a category X, we present a new closure operator, which allows a characterization of the class of all morphisms to which A is orthogonal and of the orthogonal hull of A by means of density and closedness, respectively. Using this characterization, we obtain, inter alia, conditions under which the orthogonal hull coincides with the reflective hull of A. We also explore the relationship between the closure operator introduced here and the regular closure operator.
- Solid hulls of concrete categoriesPublication . Sousa, LurdesThis paper deals with the problem of the existence of solid hulls for concrete categories. Sufficient conditions are given for the existence of a solid hull of a concrete category. For concrete categories over Set with a small finally dense subcategory, we show that the existence of solid hulls is equivalent to Weak Vopenka's Principle.
- O Teorema das Quatro CoresPublication . Sousa, LurdesO Problema das Quatro Cores trata da determinação do número mínimo de cores necessárias para colorir um mapa, de países reais ou imaginários, de forma a que países com fronteira comum tenham cores diferentes. Em 1852, Francis Guthrie conjecturou que 4 era esse número mínimo. Mas, não obstante a aparente simplicidade, só ao cabo de mais de cem anos, em 1976, se conseguiu provar que realmente a conjectura estava certa, obtendo-se o chamado Teorema das Quatro Cores. O Problema das Quatro Cores tem a característica indubitavelmente fascinante de ser um problema matemático de formulação muito simples, a par duma enorme complexidade de resolução, que fez com que permanecesse por resolver durante mais de uma centena de anos. Há outros assim; por exemplo, é bem sabido que o famoso Último Teorema de Fermat só há escassos anos foi demonstrado.
- Totality of product completionsPublication . Adámek, Jirí; Sousa, Lurdes; Tholen, WalterCategories whose Yoneda embedding has a left adjoint are kmown as total categories and are characterized by a strong cocompleteness property. We introduce the notion of multitotal category A by asking the Yoneda embedding A --> [A^{op}, Set] to be right multiadjoint and prove that this property is equivalent to totality of the formal ptoduct completiom of A. We also characterize multitotal categories with various types of generators; in particular, the existence of dense generators is inherited by the formal product completion iff measurable cardinals cannot be arbitrarily large.
- α-sober spaces via the orthogonal closure operatorPublication . Sousa, LurdesEach ordinal alpha equipped with the upper topology is a T0-space. It is well known that for alpha=2 the reflective hull of alpha in Top0 is the subcategory of sober spaces. Here, we define alpha-sober space for every ordinal alpha in such a way that the reflective hull of alpha in Top0 is the subcategory of alpha-sober spaces. Moreover, we obtain an order-preserving bijective correspondence between a proper class of ordinals and the corresponding (epi)reflective hulls. Our main tool is the concept of orthogonal closure operator, introduced by the authour in a previous paper.