The Modified Realizability Topos

Jaap van Oosten

February 1996


The modified realizability topos is the semantic (and higher order) counterpart of a variant of Kreisel's modified realizability (1957). These years, this realizability has been in the limelight again because of its possibilities for modelling type theory (Streicher, Hyland-Ong-Ritter) and strong normalization.

In this paper this topos is investigated from a general logical and topos-theoretic point of view. It is shown that Mod (as we call the topos) is the closed complement of the effective topos inside another one; this turns out to have some logical consequences. Some important subcategories of Mod are described, and a general logical principle is derived, which holds in the larger topos and implies the well-known Independence of Premiss principle.

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