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We study concrete sheaf models for a call-by-value higher-order language withrecursion. Our family of sheaf models is a generalization of many examples fromthe literature, such as models for probabilistic and differentiableprogramming, and fully abstract logical relations models. We treat recursion inthe spirit of synthetic domain theory. We provide a general construction of alifting monad starting from a class of admissible monomorphisms in the site ofthe sheaf category. In this way, we obtain a family of models parametrized by aconcrete site and a class of monomorphisms, for which we prove a generalcomputational adequacy theorem.

Concrete categories and higher-order recursion (With applications including probability, differentiability, and full abstraction)