• Categorical simulations (joint work with Robin Cockett, submitted for publication). Abstract: a notion of simulations between partial map categories over a fixed base category is introduced; this notion comprises simulations between PCAs as well as other examples from logic and computer science. Then it is shown how this notion may be analysed as a Kleisli construction in terms of preorder fibrations. This in turn provides a new perspective on the universal property of categories of (partitioned) assemblies. Finally, we apply the machinery to prove that the category of Turing categories and simulations is equivalent (in a suitable 2-categorical sense) to the category of relative PCAs. Click here to download.
• Introduction to abstract computability (brief notes for an invited tutorial at MFPS, Philadelphia, May 2008). These notes contain some of the key definitions and results about Turing categories and PCAs as well as a small glimpse into how to do recursion theory in these settings. Click here to download.
• Introduction to Turing categories (joint work with Robin Cockett, to appear in APAL). Abstract: we give a detailed treatment of the fundamentals of Turing categories, which are categories embodying an abstract notion of computation. In such categories, elementary recursion theory may be developed. Aside from an exposition of the basic theory, the paper explains in detail how the study of Turing categories is related to the study of partial combinatory algebras. Click here to download.
• An introduction to partial lambda algebras (joint work with Robin Cockett). Abstract: the aim of this paper is to give a unified treatment of partial combinatory logic, partial combinatory algebras, Turing categories, partial lambda calculus and partial lambda algebras. The main vehicle is the concept of a Cartesian closed restriction category. Currently, we are rewriting this material in order to add various new results. (Note: this preprint is partially superseded by the above paper.)
• Iterated realizability as a comma construction. (Math. Proc. Cam. Phil. Soc., January 2008) Abstract: we introduce a version of the comma construction for partial combinatory algebras and various generalizations. It is shown how, on the level of realizability toposes, the comma construction corresponds to iteration of the Effective topos-construction. This gives a generalization as well as a conceptual explanation of a result by Andy Pitts. Click here to download.
• All realizability is relative. (Math. Proc. Cam. Phil. Soc, July 2006.) A class of basic combinatorial objects (BCOs) is introduced, generalizing both (ordered) PCAs and locales. To every BCO one associates a Set-indexed preorder. This is 2-functorial. The resulting indexed preorder is a tripos (and thus to a topos) precisely when the BCO arises in a canonical way from an ordered PCA with a filter. A density condition on the morphisms completely characterizes when a morphism of BCOs gives rise to a geometric morphism of triposes, and hence we have obtained a reduction of the category of realizability triposes (toposes) to a category of BCOs with dense morphisms. Click here to download.
• Descent for monads. (Joint work with Federico de Marchi., TAC 2006) Abstract: it as been observed that certain monads which play an important role in higher category theory, such as the free omega-category monad on globular sets, and the free operad monad on collections, admit a dimension-wise decomposition. We analyse this by studying descent properties of monads on cocomplete categories. Click here to download.
• Relative Completions. (JPAA, March 2004). It is a well-known result that the Effective topos can be obtained as the exact completion of the category of partitioned assemblies. However, this result relies on the Axiom of Choice in Set, and hence fails when we work over an arbitrary base topos. In order to fix this, we introduce a relative version of the exact completion, which clarifies the relation between the Effective topos and partitioned assemblies over an arbitrary base. Click here to download.