Partial combinatory algebras (pcas), models for a form of Combinatory Logic with partial application, have been studied for the last thirty years because of their close connection to Intuitionistic Logic (see, for example, [11])

We give an introduction to Turing categories, which are a convenient setting for the categorical study of abstract notions of computability. The concept of a Turing category first appeared (albeit not under that name or at the level of generality we present it here) in the work of Longo and Moggi; later, Di Paolo and Heller introduced the closely related recursion categories. One of the purposes of Turing categories is that they may be used to develop categorical formulations of recursion theory, but they also include other notions of computation, such as models of (partial) combinatory logic and of the (partial) lambda calculus. In this paper our aim is to give an introduction to the basic structural theory, while at the same time illustrating how the notion is a meeting point for various other areas of logic and computation. We also give a detailed exposition of the connection between Turing categories and partial combinatory algebras and show the sense in which the study of Turing categories is equivalent to the study of PCAs.

a b s t r a c tWe 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.

We introduce a category of basic combinatorial objects, encompassing PCAs and locales. Such a basic combinatorial object is to be thought of as a pre-realizability notion. To each such object we can associate an indexed preorder, generalizing the construction of triposes for various notions of realizability. There are two main results: first, the characterization of triposes which arise in this way, in terms of ordered PCAs equipped with a filter. This will include "Effective Topos-like" triposes, but also the triposes for relative, modified and extensional realizability and the dialectica tripos. Localic triposes can be identified as those arising from ordered PCAs with a trivial filter. Second, we give a classification of geometric morphisms between such triposes in terms of maps of the underlying combinatorial objects. Altogether, this shows that the category of ordered PCAs with non-trivial filters serves as a framework for studying a wide variety of realizability notions.

We introduce a new model construction for Martin-Löf intensional type theory, which is sound and complete for the 1-truncated version of the theory. The model formally combines the syntactic model with a notion of realizability; it also encompasses the well-known Hofmann-Streicher groupoid semantics. As our main application, we use the model to analyse