Notre Dame Journal of Formal Logic

Ultrasheaves and Double Negation

Steve Awodey and Jonas Eliasson


Moerdijk has introduced a topos of sheaves on a category of filters. Following his suggestion, we prove that its double negation subtopos is the topos of sheaves on the subcategory of ultrafilters—the ultrasheaves. We then use this result to establish a double negation translation of results between the topos of ultrasheaves and the topos on filters.

Article information

Notre Dame J. Formal Logic, Volume 45, Number 4 (2004), 235-245.

First available in Project Euclid: 29 October 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03G30: Categorical logic, topoi [See also 18B25, 18C05, 18C10]

ultrapower Moerdijk's Topos double negation subtopos sheaf theory


Awodey, Steve; Eliasson, Jonas. Ultrasheaves and Double Negation. Notre Dame J. Formal Logic 45 (2004), no. 4, 235--245. doi:10.1305/ndjfl/1099238447.

Export citation


  • [1] Blass, A., "Two closed categories of filters", Fundamenta Mathematicae, vol. 94 (1977), pp. 129--43.
  • [2] Butz, C., "Saturated Models of Intuitionistic Theories", unpublished manuscript.
  • [3] Chang, C. C., and H. J. Keisler, Model Theory, 3d edition, vol. 73 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990.
  • [4] Eliasson, J., "Ultrapowers as sheaves on a category of ultrafilters", Archive for Mathematical Logic, vol. 43 (2004), pp. 825--843.
  • [5] Eliasson, J., "Ultrasheaves and Ultrapowers", in preparation.
  • [6] Ellerman, D. P., "Sheaves of structures and generalized ultraproducts", Annals of Mathematical Logic, vol. 7 (1974), pp. 163--95.
  • [7] Koubek, V., and J. Reiterman, "On the category of filters", Commentationes Mathematicae Universitatis Carolinae, vol. 11 (1970), pp. 19--29.
  • [8] Mac Lane, S., and I. Moerdijk, Sheaves in Geometry and Logic. A First Introduction to Topos Theory, Universitext. Springer-Verlag, New York, 1994. Corrected reprint of the 1992 edition.
  • [9] Makkai, M., "The topos of types", pp. 157--201 in Logic Year 1979--80 (Proceedings of Seminars and Conferences in Mathematical Logic, Storrs, 1979/80), vol. 859 of Lecture Notes in Mathematics, Springer, Berlin, 1981.
  • [10] Moerdijk, I., "A model for intuitionistic non-standard arithmetic. A tribute to Dirk van Dalen", Annals of Pure and Applied Logic, vol. 73 (1995), pp. 37--51.
  • [11] Moerdijk, I., and E. Palmgren, "Minimal models of Heyting arithmetic", The Journal of Symbolic Logic, vol. 62 (1997), pp. 1448--60.
  • [12] Nelson, E., "Internal set theory: A new approach to nonstandard analysis", Bulletin of the American Mathematical Society, vol. 83 (1977), pp. 1165--98.
  • [13] Palmgren, E., "A sheaf-theoretic foundation for nonstandard analysis", Annals of Pure and Applied Logic, vol. 85 (1997), pp. 69--86.
  • [14] Palmgren, E., "Developments in constructive nonstandard analysis", The Bulletin of Symbolic Logic, vol. 4 (1998), pp. 233--72.
  • [15] Palmgren, E., "Real numbers in the topos of sheaves over the category of filters", Journal of Pure and Applied Algebra, vol. 160 (2001), pp. 275--84.
  • [16] Palmgren, E., "Unifying constructive and nonstandard analysis", pp. 167--83 in Reuniting the Antipodes---Constructive and Nonstandard Views of the Continuum (Venice, 1999), vol. 306 of Synthese Library, Kluwer Academic Publishers, Dordrecht, 2001.
  • [17] Pitts, A. M., "Amalgamation and interpolation in the category of Heyting algebras", Journal of Pure and Applied Algebra, vol. 29 (1983), pp. 155--65.
  • [18] Pitts, A. M., "An application of open maps to categorical logic", Journal of Pure and Applied Algebra, vol. 29 (1983), pp. 313--26.
  • [19] Pitts, A. M., "Conceptual completeness for first-order intuitionistic logic: An application of categorical logic", Annals of Pure and Applied Logic, vol. 41 (1989), pp. 33--81.
  • [20] Troelstra, A. S., and D. van Dalen, Constructivism in Mathematics. An Introduction. Vol. I, vol. 121 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1988.