Encontrados 10 documentos, a visualizar página 1 de 1
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...
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;
Axioms for sequential convergence
Gutierres, Gonçalo; Hofmann, DirkIt is of general knowledge that those (ultra)filter convergence relations coming from a topology can be characterized by two natural axioms. However, the situation changes considerable when moving to sequential spaces. In case of unique limit points J. Kisynski [Kis60] obtained a result for sequential convergence similar to the one for ultrafilters, but the general case seems more difficult to deal with. Finall...
One Setting for All: Metric, Topology, Uniformity, Approach Structure
Clementino, Maria; Hofmann, Dirk; Tholen, WalterAbstract For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and ...
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
Topological Features of Lax Algebras
Clementino, Maria Manuel; Hofmann, DirkHaving as starting point Barr's description of topological spaces as lax algebras for the ultrafilter monad, in this paper we present further topological examples of lax algebras – such as quasi-metric spaces, approach spaces and quasi-uniform spaces – and show that, in a suitable setting, the categories of lax algebras have indeed a topological nature. Furthermore, we generalize to this setting known propertie...
Local homeomorphisms via ultrafilter convergence
Clementino, Maria Manuel; Hofmann, Dirk; Janelidze, GeorgeUsing the ultrafilter-convergence description of topological spaces, we generalize Janelidze-Sobral characterization of local homeomorphisms between finite topological spaces, showing that local homeomorphisms are the pullback-stable discrete fibrations.
On a Generalization of the Stone–Weierstrass Theorem
Hofmann, DirkA categorical version of the famous theorem of Stone and Weierstrass is formulated and studied in detail. Several applications and examples are given. ; http://dx.doi.org/10.1023/A:1020969017551
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