A Categorical Approach to Convergence
نویسنده
چکیده
We define the concept of a convergence class on an object of a given category by using certain generalized nets for expressing the convergence. The resulting topological category, whose objects are the pairs consisting of objects of the original category and convergence classes on them, is then investigated. We study the full subcategories of this category which are obtained by imposing on it some natural convergence axioms. In particular, we find sufficient conditions for the subcategories to be cartesian closed. We also investigate the behavior of the closure operator associated with the convergence in a natural way.
منابع مشابه
Convergence and quantale-enriched categories
Generalising Nachbin's theory of ``topology and order'', in this paper we continue the study of quantale-enriched categories equipped with a compact Hausdorff topology. We compare these $V$-categorical compact Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that the presence of a compact Hausdorff topology guarantees Cauchy completeness and (suitably defined) ...
متن کاملThe Effect of Media Usage on Intergenerational Devaluation and Convergence in Families in Tehran
The system of social values of any society is the result of the interaction of human beings and different generations with the social environment, which factors affect the reproduction of production and its change and evolution. Media is an essential area of social life that is closely related to fundamental value. In the process of evolution of the value system of generations under the influen...
متن کاملON STRATIFIED LATTICE-VALUED CONVERGENCE SPACES
In this paper we provide a common framework for different stratified $LM$-convergence spaces introduced recently. To this end, we slightly alter the definition of a stratified $LMN$-convergence tower space. We briefly discuss the categorical properties and show that the category of these spaces is a Cartesian closed and extensional topological category. We also study the relationship of our cat...
متن کاملFurther study on $L$-fuzzy Q-convergence structures
In this paper, we discuss the equivalent conditions of pretopological and topological $L$-fuzzy Q-convergence structures and define $T_{0},~T_{1},~T_{2}$-separation axioms in $L$-fuzzy Q-convergence space. {Furthermore, $L$-ordered Q-convergence structure is introduced and its relation with $L$-fuzzy Q-convergence structure is studied in a categorical sense}.
متن کاملA COMMON FRAMEWORK FOR LATTICE-VALUED, PROBABILISTIC AND APPROACH UNIFORM (CONVERGENCE) SPACES
We develop a general framework for various lattice-valued, probabilistic and approach uniform convergence spaces. To this end, we use the concept of $s$-stratified $LM$-filter, where $L$ and $M$ are suitable frames. A stratified $LMN$-uniform convergence tower is then a family of structures indexed by a quantale $N$. For different choices of $L,M$ and $N$ we obtain the lattice-valued, probabili...
متن کاملAxioms for Sequential Convergence
It 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. Kisyński [Kis60] obtained a result for sequential convergence similar to the one for ultrafilters, but the general case seems more difficult to ...
متن کامل