Categorical simulations

We investigate notions of simulation between categories over a base, inspired by and directly relevant for the study of categories arising in computability and realizability, but applicable to other settings as well. Such simulations admit a conceptual description in terms of the free fibration monad; this relates them closely to fibrations of (partitioned) assemblies. Our main application is in the area of abstract computability, where we show that the category of Turing categories over a fixed base and simulations between them is 2-equivalent to the category of relative PCAs in the base.