Categories for collection monads

Abstract Manes (1998). Implementing Collection Classes with Monads. Mathematical Structures in Computer Science 8 (231–276) introduced the notion of a collection monad on the category of sets as a suitable semantics for collection types. The canonical example of collection monad is the finite powerset monad. In order to account for the algorithmic aspects, the category of sets should be replaced with categories whose arrows are maps computable by low-complexity algorithms. Inspired by realizability, we give a systematic way for constructing categories of small sets and low-complexity functions and define an analogue of collection monads on such categories.