Categories of directed spaces

Krzysztof Ziemiański¹

¹ Faculty of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2, 02-097 Warszawa, Poland

Fundamenta Mathematicae, Tome 217 (2012) no. 1, pp. 55-71

Résumé

The main goal of the present paper is to unify two commonly used models of directed spaces: d-spaces and streams. To achieve this, we provide certain “goodness” conditions for d-spaces and streams. Then we prove that the categories of good d-spaces and good streams are isomorphic. Next, we prove that the category of good d-spaces is complete, cocomplete, and cartesian closed (assuming we restrict to compactly generated weak Hausdorff spaces). The category of good d-spaces is large enough to contain many interesting examples of directed spaces, including probably all which are interesting from the point of view of concurrency theory. However it fails to contain some spaces having applications to non-commutative geometry. Next, we define the class of locally d-path-connected spaces (ldpc-spaces); the additional condition allows us to eliminate some exotic examples of directed spaces. Again, we prove that ldpc-spaces and good ldpc-spaces form a category which is complete, cocomplete and cartesian closed.

Détail

DOI : 10.4064/fm217-1-5

Keywords: main present paper unify commonly models directed spaces d spaces streams achieve provide certain goodness conditions d spaces streams prove categories d spaces streams isomorphic prove category d spaces complete cocomplete cartesian closed assuming restrict compactly generated weak hausdorff spaces category d spaces large enough contain many interesting examples directed spaces including probably which interesting point view concurrency theory however fails contain spaces having applications non commutative geometry define class locally d path connected spaces ldpc spaces additional condition allows eliminate exotic examples directed spaces again prove ldpc spaces ldpc spaces form category which complete cocomplete cartesian closed