In this talk I will discuss how to build models of programming languages using monads on categories of sheaves. The goal of the talk is to explain how several well-known models from different application domains, such as probability, differentiability and full abstraction, are instances of the same general framework.

I will explain what sheaves are and how in all these models partiality and recursion are treated uniformly using ideas from synthetic domain theory.

The talk is based on joint work with Sean Moss and Sam Staton.