Pieter J. W. Hofstra

Ordered partial combinatory algebras

Partial Combinatory Algebras, 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

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 provide 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.

All realizability is relative

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.

Combinatorial realizability models of type theory

We introduce a new model construction for Martin-Lof intensional type theory, which is sound and complete for the 1-truncated version of the theory. The model formally combines, by gluing along the functor from the category of contexts to the category of groupoids, the syntactic model with a notion of realizability. As our main application, we use the model to analyse the syntactic groupoid associated to the type theory generated by a graph G, showing that it has the same homotopy type as the free groupoid generated by G.

Martin-Löf complexes

Abstract In this paper we define Martin-Lof complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional Martin-Lof type theory. We then study the resulting categories of algebras for several theories. Our principal result is that there exists a cofibrantly generated Quillen model structure on the category of 1-truncated Martin-Lof complexes and that this category is Quillen equivalent to the category of groupoids. In particular, 1-truncated Martin-Lof complexes are a model of homotopy 1-types.

Iterated realizability as a comma construction

We show that the 2-category of partial combinatory algebras, as well as various related categories, admit a certain type of lax comma objects. This not only reveals some of the properties of such categories, but it also gives an interpretation of iterated realizability, in the following sense. Let f: A ? B be a morphism of PCAs, giving a comma object A f B. In the realizability topos RT(B) over B, the object (A, f) is an internal PCA, so we can construct the realizability topos over (A, f). This topos is equivalent to the realizability topos over the comma-PCA A f B. This result is both an analysis and a generalization of a special case studied by Pitts in the context of the effective monad.

Unitary Theories, Unitary Categories

This paper explores the fine structure of classifying categories of partial equational theories. The central concept is that of unitary category, and results about those are applied to the problem of completing partial algebras to total ones. We also look at the special case of partial combinatory logic and give a characterization of the global sections of the generic PCA.

Total Maps of Turing Categories

We give a complete characterization of those categories which can arise as the subcategory of total maps of a Turing category. A Turing category provides an abstract categorical setting for studying computability: its (partial) maps may be described, equivalently, as the computable maps of a partial combinatory algebra. The characterization, thus, tells one what categories can be the total functions for partial combinatory algebras. It also provides a particularly easy criterion for determining whether functions, belonging to a given complexity class, can be viewed as the class of total computable functions for some abstract notion of computability.