Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to

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This paper is about purely categorical approaches to realizability, and contrasts with recent work particularly by Longley [14] and Lietz and Streicher [13], in which the basis is taken as a typed generalisation of a partial combinatory algebra. We, like they, will be interested in when the construction yields a topos, and hence gives a full interpretation of higher-order logic. This is also a theme of Birkedal's work, see [1, 2], and his joint work in [3]. Birkedal makes considerable use of the construction we study. We present realizability toposes as the product of two constructions. First one takes a category (which corresponds to the typed partial combinatory algebra), and then one glues Set to it in a variant of the comma construction. This, as we shall see, has the eect of improving the categorical properties of the algebra category. Then one takes an exact completion of the result. This also has the eect of improving the categorical properties. Formally the main result of the paper is that the result is a topos just (modulo some technical conditions) when the original category has a universal object. Early work on realizability (e.g.[12, 22], or see [23]) is characterised by its largely syntactic nature. The core denition is when a sentence of some formal logic is realised, and the main interest is in when certain deductive principles (such as Markov's rule) are validated. Martin Hyland's invention y The authors wish to acknowledge the support of the EPSRC, EU Working Group 26142 APPSEM, and MURST 1 2 of realizability toposes [10] advances on this, not only in the simplicity of the construction, but by providing a semantic framework in which the formal logics can naturally be interpreted. Hyland was strongly motivated in his work by a then recent approach...

We define a constructive topos to be a locally cartesian closed pretopos. The terminology is supported by the fact that constructive toposes enjoy a relationship with constructive set theory similar to the relationship between elementary toposes and (impredicative) intuitionistic set theory. This paper elaborates upon one aspect of the relationship between constructive toposes and constructive set theory. We show that any constructive topos with countable coproducts provides a model of a standard constructive set theory, CZFExp (that is, the variant of Aczel’s Constructive Zermelo-Fraenkel set theory obtained by weakening Subset Collection to the Exponentiation axiom). The model is constructed as a category of classes, using ideas derived from Joyal and Moerdijk’s programme of algebraic set theory. A curiosity is that our model always validates the axiom V = Vω1 (in an appropriate formulation). Hence the full Separation schema is always refuted. 1.