bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element. Other examples are partially ordered sets, where (INT) is satisfied because of reflexivity. We will shortly identify posets as being exactly the bases of compact elements of algebraic domains. In what follows we will use the terminology developed at the beginning of this chapter, even though the relation ≺ on an abstract basis need neither be reflexive nor antisymmetric. This is convenient but in some instances looks more innocent than it is. An idealA in a basis, for example, has the property (following from directedness) that for everyx ∈ A there is another element y ∈ A with x ≺ y. In posets this doesn’t mean anything but here it becomes an important feature. Sometimes this is stressed by using the expression ‘ A is a round ideal’. Note that a set of the form↓x is always an ideal because of (INT) but that it need not contain x itself. We will refrain from calling ↓x ‘principal’ in these circumstances. Definition 2.2.21. For a basis〈B,≺〉 let Idl(B) be the set of all ideals ordered by inclusion. It is called theideal completionof B. Furthermore, leti : B → Idl(B) denote the function which maps x ∈ B to ↓x. If we want to stress the relation with whichB is equipped then we write Idl(B,≺) for the ideal completion. Proposition 2.2.22.Let 〈B,≺〉 be an abstract basis.

Domain theory

Preliminaries. Categories. Functors. Diagrams. Naturality and Sketches. Products and Sums. Catesian Closed Categories. Finite Discrete Sketches. Limits and Colimits. More About Sketches. Fibrations. Adjoints. Algebras for Endofunctors. Toposes.

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Category Theory For Computing Science

We place simple axioms on an elementary topos which suffice for it to provide a denotational model of call-by-value PCF with sum and product types. The model is synthetic in the sense that types are interpreted by their set-theoretic counterparts within the topos. The main result characterises when the model is computationally adequate with respect to the operational semantics of the programming language. We prove that computational adequacy holds if and only if the topos is 1-consistent (i.e. its internal logic validates only true Σ\(^{\rm 0}_{\rm 1}\)-sentences).

/pdf/computational-adequacy-in-an-elementary-topos-59ke3tx03y.pdf

Computational Adequacy in an Elementary Topos

/pdf/the-lazy-lambda-calculus-4pm08ds80s.pdf

The lazy lambda calculus

Classification theory and the number of non-isomorphic models

This paper attempts to reconcile the various abstract notions of “category of partial maps” which appear in the literature. First a particular algebraic theory ( p -categories) is introduced and a representation theorem proved. This gives the authors a coherent framework in which to place the various other definitions. Both algebraic theories and theories which make essential use of the poset-enriched structure of partial maps are discussed. Proofs of equivalence are given where possible and counterexamples where known. The paper concludes with brief sections on the representation of partial maps and on partial algebras.

Categories of partial maps

The original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two non-trivial small (i.e. internal) full subcategories which are closed under all small limits in the topos (and hence in particular are internally complete). The interest in such subcategories arises from the fact that they provide a very natural notion of model for many of the strongly polymorphic type theories. The nature of these models is that types are interpreted as objects of the topos, and hence as 'sets' in an intuitionistic sense. They in fact provide 'set-theoretic' models in the sense of Reynolds [16], except that the set-theory involved is non-classical. At the end of the paper we indicate briefly how one (only) of the two full subcategories gives rise to a model of the theory of constructions. This sheds some light on how far we can travel with a simple set-theoretic picture of data types. However, our main concern is to present a clean mathematical characterization of the complete subcategories. The basic idea, due to Peter Freyd, remains very simple—it is that the collection of objects orthogonal to any given object is automatically closed under all existing limits, and to look at the objects orthogonal to the subobject classifier in the effective topos. However, one of our original motivations for writing this paper was that we did not quite believe Freyd's identification of this category. During the course of the paper we shall show that the category of (families of) objects in ^"orthogonal to Q is the category of (families of) subquotients of N. The chain of reasoning now proceeds by saying that this second category is equivalent to the externalisation of an internal category, and that the internal category is therefore small complete. Somewhat to our surprise, it was this second stage, trivial for an internal category in Sfe&, that we found difficulty in making precise, and it is in the discussion of notions of equivalence and completeness for internal categories and fibrations that much of the content of the paper now lies. We should warn the reader that even so our treatment is not exhaustive. We discuss only two notions (weak and strong) of equivalence for internal or fibred categories, and only two notions of completeness. Our definitions are welladapted for our present purposes, corresponding more or less to the distinction between being told for any instance of a given set of parameters that there is an object with certain properties, and being given a function that produces one. Let C be a small full subcategory of a topos (see § 3 for details) which is closed under arbitrary products. Then C cannot contain Q, the object of truth values (in fact C cannot contain any object into which Q embeds)—for if A is the disjoint union of the sets in C, then Q^eC, and hence Q^y^A, which contradicts one of the constructive versions of Cantor's theorem. This argument runs parallel

The Discrete Objects in the Effective Topos

Locally cartesian closed exact completions (

Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a category-theoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework.

Type theory via exact categories

A search is conducted for a class of PERs (partial equivalence relations on the natural numbers) such that the resulting full subcategory has the expected properties of any good category of CPOs: it should be a CCC (Cartesian closed category) and every endomorphism should have a canonical fixed point. Moreover the reflection functor (usually called the lifting operation) should yield a good notion of partial map. The following topics are discussed: conventions, partial-map classifiers, ExPERS, ExPERS as domains, reflectivity of strict maps, multicorreflectivity of strict maps, the extensional natural numbers, domain equations, and intrinsic descriptions. >

http://ieeexplore.ieee.org/iel2/305/3367/00113760.pdf

Giuseppe Rosolini

Papers

Categories of partial maps

The Discrete Objects in the Effective Topos

Locally cartesian closed exact completions (

Type theory via exact categories

Extensional PERs