Introduction to Turing categories

Abstract

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.

Computability structures, simulations and realizability

“…Our work overlaps significantly with recent and essentially independent work by Cockett and Hofstra [4,5]. In both their work and ours, the key idea is to generalize the definition of PCA to something with a typed, first order flavour, and to show how a general notion of 'simulation' may be defined in this setting.…”

“…An analogue of our correspondence theorem can also be obtained for the stricter equivalence: whereas lax equivalence coincides with equivalence of the associated categories of assemblies, it is shown in [4] is that strict equivalence coincides with Morita equivalence of Turing categories.…”

“…[5,Section 2]). It is known that equivalence of (strict) PCAs via strict applicative morphisms is a stronger condition than equivalence via lax ones (see [4,Section 5]). …”

“…Whereas our aim is to describe concretely a construction that directly generalizes that of the 'classical' category of assemblies, the aim in [4,5] is to give a categorical analysis of that construction in terms of several smaller steps. Indeed, most of the focus in [4,5] is not on the classical category of assemblies itself, but on an intermediate structure known as a Turing category. One of the main results of [5] is a correspondence theorem for Turing categories, closely analogous to ours for categories of assemblies.…”

“…In [4,5], Cockett and Hofstra compellingly argue that strict equivalence is more appropriate where questions of computability are concerned, while the lax notion is better adapted to questions of realizability. This is because in realizability it is never a problem if a realizer does 'more' than the job it is designed for, whereas in computability one is often interested in the precise domain of a computable operation.…”