Topological categories, quantaloids and Isbell adjunctions
نویسندگان
چکیده
In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory. Motivated by some key results of the 1970s, the paper develops all needed ingredients from the theory of quantaloids in order to place essential results of categorical topology into the context of quantaloid-enriched category theory, a field that previously drew its motivation and applications from other domains, such as quantum logic.
منابع مشابه
Categories Enriched over a Quantaloid: Isbell Adjunctions and Kan Adjunctions
Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory. It is proved that these two processes are functorial with infomorphisms playing as morphi...
متن کاملCategorical Structures Enriched in a Quantaloid: Tensored and Cotensored Categories
A quantaloid is a sup-lattice-enriched category; our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and ‘order-cocomplete’. In fact, tensors and cotensors in a Q-category determine, and are determin...
متن کامل$L$-enriched topological systems---a common framework of $L$-topology and $L$-frames
Employing the notions of the strong $L$-topology introduced by Zhangand the $L$-frame introduced by Yao and the concept of $L$-enrichedtopological system defined in the present paper, we constructadjunctions among the categories {bf St$L$-Top} of strong$L$-topological spaces, {bf S$L$-Loc} of strict $L$-locales and{bf $L$-EnTopSys} of $L$-enriched topological systems. All of theseconcepts are ...
متن کاملOperational Galois Adjunctions
We present a detailed synthetic overview of the utilisation of categorical techniques in the study of order structures together with their applications in operational quantum theory. First, after reviewing the notion of residuation and its implementation at the level of quantaloids we consider some standard universal constructions and the extension of adjunctions to weak morphisms. Second, we p...
متن کاملOperational Galois Adjunctions 1
We present a detailed synthetic overview of the utilisation of categorical techniques in the study of order structures together with their applications in operational quantum theory. First, after reviewing the notion of residuation and its implementation at the level of quantaloids we consider some standard universal constructions and the extension of adjunctions to weak morphisms. Second, we p...
متن کامل