Observationally-induced Effects in Cartesian Closed Categories

Alex Simpson has suggested an observationally-induced approach towards obtaining monads for computational effects in denotational semantics. The underlying idea of this approach is to use a single observation algebra as computational prototype and to obtain a computational monad as a free algebra construction derived from this prototype. Recently, it has been shown that free observationally-induced algebras exist in the category of continuous maps between topological spaces for arbitrary pre-chosen computational prototypes.In this work we transfer these results to cartesian closed categories. In particular, we show that, provided the category under consideration satisfies suitable completeness conditions, it supports a free observationally-induced algebra construction for arbitrary computational prototypes. We also show that the free algebras are obtained as certain subobjects of double exponentials involving the computational prototype as result type. Finally, we apply these results to show that in topological domain theory an observationally-induced lower powerspace construction over a QCB-space X is given by the space of nonempty closed subsets of X topologised suitably.