A small complete category

Abstract

This paper is concerned with a remarkable fact. The effective topos contains a small complete subcategory, essentially the familiar category of partial equivalence realtions. This is in contrast to the category of sets (indeed to all Grothendieck toposes) where any small complete category is equivalent to a (complete) poset. Note at once that the phrase ‘a small complete subcategory of a topos’ is misleading. It is not the subcategory but the internal (small) category which matters. Indeed for any ordinary subcategory of a topos there may be a number of internal categories with global sections equivalent to the given subcategory. The appropriate notion of subcategory is an indexed (or better fibred) one, see 0.1. Another point that needs attention is the definition of completeness (see 0.2). In my talk at the Church’s Thesis meeting, and in the first draft of this paper, I claimed too strong a form of completeness for the internal category. (The elementary oversight is described in 2.7.) Fortunately during the writing of [13] my collaborators Edmund Robinson and Giuseppe Rosolini noticed the mistake. Again one needs to pay careful attention to the ideas of indexed (or fibred) categories. The idea that small (sufficiently) complete categories in toposes might exist, and would provide the right setting in which to discuss models for strong polymorphism (quantification over types), was suggested to me by Eugenio Moggi. And he first realized that the effective topos did indeed contain a small complete category. When, led by Moggi’s suggestion, I first came to consider the matter, I realized that the ‘result’ was staring me in the face. It is just a matter of putting together some well-known facts. The effective topos is the world of realizability (Kleene [15]) extended from arithmetic to general constructive mathematics. Details are in [ll], and the general context in [12] and [21]. The relevant subcategory, called the category of effective objects in [ll], is already in Kreisel [16]. Briefly the problem is to show * Paper presented at the conference “Church’s Thesis after fifty years”, Zeist, The Netherlands, June 14, 15, 1986.