Encontrados 14 documentos, a visualizar página 1 de 2
On the categorical meaning of Hausdorff and Gromov distances, I
Akhvlediani, Andrei; Clementino, Maria Manuel; Tholen, WalterHausdor and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdor 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 de...
On regular and homological closure operators
Clementino, Maria Manuel; Gutierres, GonçaloObserving that weak heredity of regular closure operators in Top and of homological closure operators in homological categories identifies torsion theories, we study these closure operators in parallel, showing that regular closure operators play the same role in topology as homological closure operators do algebraically.
Protolocalisations of homological categories
Borceux, Francis; Clementino, Maria Manuel; Gran, Marino; Sousa, LurdesA protolocalisation of a homological (resp. semi-abelian) category is a regular full reflective subcategory, whose reflection preserves short exact sequences. We study the closure operator and the torsion theory associated with such a situation. We pay special attention to the fibered, the regular epireflective and the monoreflective cases. We give examples in algebra, topos theory and functional analysis. ; ...
Relative injectivity as cocompleteness for a class of distributors
Clementino, Maria Manuel; Hofmann, DirkNotions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:math.CT/0804.0326] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadicity of the catego...
Topological protomodular algebras
Borceux, F.; Clementino, Maria ManuelTopological groups have very striking properties, which have already been generalized to weaker "group like" structures, like various kinds of loops. This paper intends to show evidence that this generalization holds for a much wider class of theories, known as the protomodular theories, and which admit both an elegant categorical characterization and an easy description in universal algebra terms. Thus we prop...
Exponentiation in V-categories
Clementino, Maria Manuel; Hofmann, DirkFor a Heyting algebra V which, as a category, is monoidal closed, we obtain characterizations of exponentiable objects and morphisms in the category of V-categories and apply them to some well-known examples. In the case these characterizations of exponentiable morphisms and objects in the categories (P)Met of (pre)metric spaces and non-expansive maps show in particular that exponentiable metric spaces are exac...
Exponentiable functors between quantaloid-enriched categories
Clementino, Maria Manuel; Hofmann, Dirk; Stubbe, IsarExponentiable functors between quantaloid-enriched categories are characterized in elementary terms. The proof goes as follows: the elementary conditions on a given functor translate into existence statements for certain adjoints that obey some lax commutativity; this, in turn, is precisely what is needed to prove the existence of partial products with that functor; so that the functor’s exponentiability follow...
On some special classes of continuous maps
Clementino, Maria Manuel; Hofmann, DirkWe present a survey on recent study of special continuous maps, like biquotient, triquotient, proper, perfect, open and étale maps and a selection of open problems in this area. ; Centro de Matemática da Universidade de Coimbra/FCT; Unidade de Investigação e Desenvolvimento Matemática e Aplicações da Universidade de Aveiro/FCT;
Topological semi-abelian algebras
Borceux, F.; Clementino, Maria ManuelGiven an algebraic theory whose category of models is semi-abelian, we study the category of topological models of and generalize to it most classical results on topological groups. In particular, is homological, which includes Barr regularity and forces the Mal'cev property. Every open subalgebra is closed and every quotient map is open. We devote special attention to the Hausdorff, compact, locally compact, c...
Effective Descent Morphisms in Categories of Lax Algebras
Clementino, Maria Manuel; Hofmann, DirkIn this paper we investigate effective descent morphisms in categories of reflexive and transitive lax algebras. We show in particular that open and proper maps are of effective descent, result that extends the corresponding results for the category of topological spaces and continuous maps. ; http://dx.doi.org/10.1023/B:APCS.0000049310.37773.fa
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