Structural Algebraic Compactness

Alas, the motivating examples of algebraically compact categories are not algebraically compact, and the enriched setting does not directly support the theory of algebraic compactness. We show that the structural setting, the same setting developed independently for models of linear logic, directly supports the theory of algebraic cocompleteness. We extend the structural setting to a bistructural setting that supports the full theory of algebraic compactness. These results provide an elementary theory that includes Freyd’s original theory together with the domain-theoretic models that motivated it, without reference to enriched, indexed or internal categories. We then describe how the structural setting fits together with the more sophisticated indexed and enriched settings in which domain-theoretic models are commonly studied.