A Topologist’s View of Chu Spaces

Abstract

For a symmetric monoidal-closed category $\mathcal{X}$ and any object K, the category of K-Chu spaces is small-topological over $\mathcal{X}$ and small cotopological over $\mathcal{X}^{{{\text{op}}}}$. Its full subcategory of $\mathcal{M}$-extensive K-Chu spaces is topological over $\mathcal{X}$ when $\mathcal{X}$ is $\mathcal{M}$-complete, for any morphism class $\mathcal{M}$. Often this subcategory may be presented as a full coreflective subcategory of Diers’ category of affine K-spaces. Hence, in addition to their roots in the theory of pairs of topological vector spaces (Barr) and their connections with linear logic (Seely), the Dialectica categories (Hyland, de Paiva), and with the study of event structures for modeling concurrent processes (Pratt), Chu spaces seem to have a less explored link with algebraic geometry. We use the Zariski closure operator to describe the objects of the *-autonomous category of $\mathcal{M}$-extensive and $\mathcal{M}$-coextensive K-Chu spaces in terms of Zariski separation and to identify its important subcategory of complete objects.