Voir la notice de l'article provenant de la source Theory and Applications of Categories website
Recently Benno~van~den~Berg introduced a new class of realizability toposes which he christened Herbrand toposes. These toposes have strikingly different properties from ordinary realizability toposes, notably the (related) properties that the `constant object' functor from the topos of sets preserves finite coproducts, and that De Morgan's law is satisfied. In this paper we show that these properties are no accident: for any Schonfinkel algebra $\Lambda$, the Herbrand realizability topos over $\Lambda$ may be obtained as the Gleason cover (in the sense of Johnstone (1980)) of the ordinary realizability topos over $\Lambda$. As a corollary, we obtain the functoriality of the Herbrand realizability construction on the category of Schonfinkel algebras and computationally dense applicative morphisms.
@article{TAC_2013_28_a31,
author = {Peter Johnstone},
title = {The {Gleason} cover of a realizability topos},
journal = {Theory and applications of categories},
pages = {1139--1152},
publisher = {mathdoc},
volume = {28},
year = {2013},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2013_28_a31/}
}
Peter Johnstone. The Gleason cover of a realizability topos. Theory and applications of categories, Tome 28 (2013), pp. 1139-1152. http://geodesic.mathdoc.fr/item/TAC_2013_28_a31/