Pieter Hofstra's research while affiliated with University of Ottawa and other places

Publications (14)

Preprint
We develop a general theory of (extended) inner autoequivalences of objects of any 2-category, generalizing the theory of isotropy groups to the 2-categorical setting. We show how dense subcategories let one compute isotropy in the presence of binary coproducts, unifying various known one-dimensional results and providing tractable computational to...
Article
Full-text available
This is an expository paper in which we explain the basic ideas of topos theory in connection with semigroup theory. We focus mainly on the classifying topos of an inverse semigroup or pseudogroup, and to some extent on creating a dictionary between the language of semigroups and topos theory. We begin with the algebraic theory having to do with in...
Chapter
We present a survey of some developments in the general area of category-theoretic approaches to the theory of computation, with a focus on topics and ideas particularly close to the interests of Jim Lambek.
Preprint
We investigate the concept of definable, or inner, automorphism in the logical setting of partial Horn theories. The central technical result extends a syntactical characterization of the group of such automorphisms (called the covariant isotropy group) associated with an algebraic theory to the wider class of quasi-equational theories. We apply th...
Preprint
We present a survey of some developments in the general area of category-theoretic approaches to the theory of computation, with a focus on topics and ideas particularly close to the interests of Jim Lambek.
Article
Full-text available
To every small category or topos one may associate its isotropy group, which is an algebraic invariant capturing information about the behaviour of automorphisms. We investigate this invariant in the particular situation of algebraic theories, thus obtaining a group-theoretic invariant of algebraic theories. This invariant encodes a notion of inner...
Article
This paper is about an invariant of small categories called isotropy. Every small category C has associated with it a presheaf of groups on C, called its isotropy group, which in a sense solves the problem of making the assignment C ↦ Aut(C) functorial. Consequently, every category has a canonical congruence that annihilates the isotropy; however,...
Article
This paper continues the investigation of isotropy theory for toposes. We develop the theory of isotropy quotients of toposes, culminating in a structure theorem for a class of toposes we call locally anisotropic. The theory has a natural interpretation for inverse semigroups, which clarifies some aspects of how inverse semigroups and toposes are r...
Article
Full-text available
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,...
Article
Motivated by constructions in the theory of inverse semigroups and étale groupoids, we define and investigate the concept of isotropy from a topos-theoretic perspective. Our main conceptual tool is a monad on the category of grouped toposes. Its algebras correspond to a generalized notion of crossed module, which we call a crossed topos. As an appl...
Article
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 φ: A → B be a morphism of PCAs, giving a comma obje...
Article
These are the lecture notes for a tutorial at FMCS 2004 in Kananaskis. The aim is to give a first introduction to Partial Combinatory Algebras and the construction of Realizability Toposes. The first part, where Partial Combinatory Algebras are discussed, requires no specific background (except for some of the examples perhaps), although familiarit...
Article
Full-text available
We study the generic model for partial combinatory logic from a rewrit-ing perspective and from the point of view of classifying partial map cate-gories. We show that the classifying category has the rather strong prop-erty of being unitary; this is a consequence of the main structure theorem which states that it can be obtained by freely adjoining...

Citations

... In [9], the author and his collaborators studied the covariant isotropy group of the category Tmod of models of a single-sorted algebraic theory T, and showed that it encodes the usual notion of inner automorphism for T, if it has one. It is shown therein that the covariant isotropy group of M ∈ Tmod may be described in terms of the elements of M x (the T-model obtained from M by freely adjoining an indeterminate element x) that are substitutionally invertible and commute generically with the operations of T. In the author's recent PhD thesis [14] and the subsequent paper [10], this analysis was extended to the category of models of any finitary quasi-equational or essentially algebraic theory in the sense of [12], which is a multi-sorted equational theory in which certain operations may only be partially defined. ...
... We should also like to point out that while this work focuses mainly on explaining topos theory in the service of semigroup theory there are numerous instances of developments in category theory and topos theory having been inspired or prompted by semigroup theory. For example, the theory of isotropy groups of categories and toposes [11][12][13][14] stems from an analysis of the so-called Clifford-fundamental sequence associated with an inverse semigroup. And the theory of unitary partial map categories [7] aims to bring the theory of (E-)unitary inverse semigroups to the level of categories of partial maps. ...
... Disjoint joins in partial map categories correspond to disjoint joins of monics, which often give a coproduct (e.g. as in coherent categories). One way to model iteration is to have a traced coproduct, and this can be directly expressed using disjoint joins: this approach was used in formalizing iteration in restriction categories and to build a partial combinatory algebra by iterating a step-function in [20,13]. The formalization of iteration using disjoint joins was based on the work of Conway [22]. ...
... In this viewpoint, one thinks of an arbitrary automorphism h : X → X as an "abstract inner automorphism" if it can be extended to such a family α with α id = h; equivalently, these are the automorphisms in the image of the comparison map θ X . Modulo size issues (discussed in the next section), we therefore have a functor Z : C → Grp called the (covariant) isotropy group of C. In a topos-theoretic context this functor (or rather, the contravariant version) was studied in in [7], and in the context of (essentially) algebraic theories in [10,11]. ...
... Second, we may replace the terms and stacks of the λ c -calculus by the A-terms and A-stacks of an arbitrary classical realizability algebra A, as shown in [21,22]. Intuitively, classical realizability algebras generalize λ c -calculi (with poles) the same way as partial combinatory algebras [13] generalize the λ-calculus (or Gödel codes for partial recursive functions) in the framework of intuitionistic realizability. This broad generalization of classical realizability-in a framework where terms and stacks are not necessarily of a combinatorial nature-is essential, since it allows us to make explicit the connection between forcing and classical realizability. ...
... -The study of 'combinatory structures' generalizing pcas started with van Oosten's [60] definition of ordered pcas 4 , and ordered pcas were further developed by Hofstra and van Oosten [21,25]. Longley generalized pcas by adding types [39], and Hofstra further generalized ordered pcas into basic combinatory objects (BCOs) [23,24]. Recently, Longley [40] presented a vast generalization of his ordered pcas. ...