<p>In the list of convenience properties for topological constructs the property of being a quasitopos is one of the most interesting ones for investigations in function spaces, differential calculus, functional analysis, homotopy theory, etc. The topological construct Cls of closure spaces and continuous maps is not a quasitopos. In this article we give an explicit description of the quasitopos topological hull of Cls using a method of F. Schwarz: we first describe the extensional topological hull of Cls and of this hull we construct the cartesian closed topological hull.</p>
We introduce a fibre homotopy relation for maps in a category of cofibrant objects equipped with a choice of cylinder objects. Weak fibrations are defined to be those morphisms having the weak right lifting property with respect to weak equivalences. We prove a version of Dold's fibre homotopy equivalence theorem and give a number of examples of weak fibrations. If the category of cofibrant objects comes from a model category, we compare fibrations and weak fibrations, and we compare our fibre homotopy relation, which is defined in terms of left homotopies and cylinders, with the fibre homotopy relation defined in terms of right homotopies and path objects. We also dualize our notion of weak fibration in a category of cofibrant objects to a notion of weak cofibration in a category of fibrant objects, and give examples of these weak cofibrations. A section is devoted to the case of chain complexes in an abelian category. IntroductionThe fibre homotopy equivalence theorem of Dold [Dol63, Theorem 6.1] in Top has been generalized by various authors. Besides the original work by Dold, the book [DKP70] of tom Dieck-Kamps-Puppe gives an exposition on weak fibrations (h-Faserungen in Top). Some of the generalizations consider maps which are simultaneously over a given space and under a given space. Booth [Boo93] also obtains versions of Dold's theorem, using suitably defined generalizations of the covering homotopy property. In other cases the fibre homotopy equivalences were studied in a categorical setting, as for example in the Homology, Homotopy and Applications, vol. 5(1), 2003 346 the basic assumption is that the category has some cylinder functor. In the article [Kam72], Kamps uses cylinder functors to define a notion of weak fibration. A model category structure, a concept due to Quillen [Qui67], is another way of introducing a homotopy relation in a category. In fact in a model category there are two dual ways of defining homotopy of maps: left homotopies, defined in terms of cylinder objects, and right homotopies, defined in terms of cocylinder objects. These two methods feature in categories of cofibrant objects and, respectively, categories of fibrant objects. Of these two notions, the latter was introduced by K. S. Brown [Bro73] in 1973 and dualized into the former by Kamps and Porter (see [KP97]). We consider a notion of weak fibration in the context of a category of cofibrant objects with a cylinder object choice, i.e., a chosen cylinder object for every object of the category. Our weak fibrations, and their properties, depend on this cylinder object choice. In case this choice comes from a cylinder functor satisfying certain Kan filler conditions, our fibre homotopy relation coincides with the one used in [KP97]. This makes it possible to compare our weak fibrations with Kamps's.The aim of this article is to study fibre homotopies and weak fibrations in a category of cofibrant objects and, dually, relative homotopies and weak cofibrations in a category of fibrant objects. The presentation is a...
Abstract. Ascoli theorems characterize "precompact" subsets of the set of morphisms between two objects of a category in terms of "equicontinuity" and "pointwise precompactness," with appropriate definitions of precompactness and equicontinuity in the studied category. An Ascoli theorem is presented for sets of continuous functions from a sequential space to a uniform space. In our development we make extensive use of the natural function space structure for sequential spaces induced by continuous convergence and define appropriate concepts of equicontinuity for sequential spaces. We apply our theorem in the context of C * -algebras.2000 Mathematics Subject Classification. 54A20, 54C35, 54E15.1. Introduction. Ascoli theorems characterize "precompact" subsets of the set of morphisms between two objects of a category in terms of "equicontinuity" and "pointwise precompactness," with appropriate definitions of precompactness and equicontinuity in the studied category. Such general theorems are inspired by the classical Ascoli theorem, proved by G. Ascoli (and independently by C. Arzelà) in the 19th century (see [3,4]). It characterizes compactness of sets of continuous real-valued functions on the interval [0, 1] with respect to the topology of uniform convergence. Since then, many related theorems have been proved, for example, characterizing compactness of sets of continuous functions from a topological to a uniform space (see [6]), of uniformly continuous functions from a merotopic to a uniform space (see [5]) of continuous functions between topological spaces (see [14,17]). As was pointed out by Wyler in [24], it is clear that the setting for Ascoli theorems requires natural function space structures; the existence of nice function spaces is guaranteed by Cartesian closedness of the considered topological construct. Around 1980, Dubuc [8] and Gray [12] both proposed a general theory for Ascoli theorems in a categorical setting, but as neither of them seems to be entirely satisfactory, Wyler suggested that more examples should be constructed in order to guide the general theory. Wyler himself developed new examples of Ascoli theorems for sets of continuous functions between limit spaces and of uniformly continuous functions from a uniform convergence space to a pseudo-uniform space (see [24]).In this paper, we present another setting for an Ascoli theorem: we choose the construct L of sequential spaces. Sequential spaces were already introduced at the beginning of the century by Fréchet (see [9,10]) and Urysohn (see [2,23]), even before topological spaces were axiomatized. Since then they have extensively been used as a tool in topology and analysis. Since the 60s sequential structures have been investigated from a categorical point of view (for categorical background, we refer to [1]);
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