An algebraic approach to stable domains
The category of domains (posets with directed joins) and Scott continuous functions between them has played an important role in the category theoretical study of denotational semantics. Recently, the notion of ``stability'' has received considerable interest, wherein pullbacks are required to exist and stable functions must preserve these. For example, stability plays a key role in J. Y. Girard's theory of coherence spaces in linear logic [\textit{J. Y. Girard}, \textit{Y. Lafont} and \textit{P. Taylor}, Proofs and types (1989; Zbl 0671.68002)]. The paper under review investigates the category of stable domains from the perspective of algebraic theories. A stable domain D is a domain with connected meets, which are required to distribute over directed joins (this is equivalent to \(\downarrow x\) being a continuous lattice for all \(x\in D)\). Stable morphisms not only preserve directed joins, but also the connected meets (equivalently, they have a left multiadjoint). The category of stable domains is shown to be equivalent to the category of algebras for a monad \({\mathcal F}\) defined on the category of locally connected spaces. \({\mathcal F}\) is constructed by looking at filters of connected open sets. Furthermore, the category of stable domains is cartesian closed (this involves consideration of the Berry order \(\sqsubset \underline {\;}\) on functions, where \(f\sqsubset \underline g\) iff whenever \(a\leq b\), we have \(f(a)=f(b)\wedge g(a))\). The article also contains a lengthy introductory section surveying stability theory, which is very informative and provides the reader a good background and perspective on the ideas introduced in the paper.
monad on the category of locally connected spaces
scientific article; zbMATH DE number 4156721
scientific article