Source: J. Symbolic Logic
Volume 74, Issue 3
We present a solution to the problem of defining a counterpart
in Algebraic Set Theory of the construction of internal
sheaves in Topos Theory. Our approach is general
in that we consider sheaves as determined by
Lawvere—Tierney coverages, rather than by Grothendieck
coverages, and assume only a weakening of the axioms for
small maps originally introduced by Joyal and Moerdijk,
thus subsuming the existing topos-theoretic results.
P. Aczel and M. Rathjen, Notes on constructive set theory, Technical Report 40, Mittag--Leffler Institut, The Swedish Royal Academy of Sciences, 2001.
S. Awodey, C. Butz, A. Simpson, and T. Streicher, Relating topos theory and set theory via categories of classes, Technical Report CMU-PHIL-146, Department of Philosophy, Carnegie Mellon University, 2003.
S. Awodey and M. A. Warren, Predicative algebraic set theory, Theory and applications of categories, vol. 15 (2005), no. 1, pp. 1--39.
B. van den Berg, Sheaves for predicative toposes, Archive for Mathematical Logic, to appear. ArXiv:math.LO /0507480v1, 2005.
B. van den Berg and I. Moerdijk, A unified approach to algebraic set theory, arXiv :0710.3066v1, 2007, to appear in the Proceedings of the Logic Colloquium 2006.
--------, Aspects of predicative algebraic set theory I: Exact completions, Annals of Pure and Applied Logic, vol. 156 (2008), no. 1, pp. 123--159.
A. Carboni, Some free constructions in realizability and proof theory, Journal of Pure and Applied Algebra, vol. 103 (1995), pp. 117--148.
A. Carboni and E. M. Vitale, Regular and exact completions, Journal of Pure and Applied Algebra, vol. 125 (1998), pp. 79--116.
M. P. Fourman, Sheaf models for set theory, Journal of Pure and Applied Algebra, vol. 19 (1980), pp. 91--101.
Mathematical Reviews (MathSciNet): MR593249
P. Freyd, The axiom of choice, Journal of Pure and Applied Algebra, vol. 19 (1980), pp. 103--125.
Mathematical Reviews (MathSciNet): MR593250
N. Gambino, Presheaf models for constructive set theories, From sets and types to topology and analysis (L. Crosilla and P. Schuster, editors), Oxford University Press, 2005, pp. 62--77.
--------, Heyting-valued interpretations for constructive set theory, Annals of Pure and Applied Logic, vol. 137 (2006), no. 1--3, pp. 164--188.
--------, The associated sheaf functor theorem in algebraic set theory, Annals of Pure and Applied Logic, vol. 156 (2008), no. 1, pp. 68--77.
N. Gambino and P. Aczel, The generalized type-theoretic interpretation of constructive set theory, Journal of Symbolic Logic, vol. 71 (2006), no. 1, pp. 67--103.
R. J. Grayson, Forcing for intuitionistic systems without power-set, Journal of Symbolic Logic, vol. 48 (1983), no. 3, pp. 670--682.
Mathematical Reviews (MathSciNet): MR716628
P. T. Johnstone, Sketches of an elephant: A topos theory compendium, Oxford University Press, 2002.
A. Joyal and I. Moerdijk, Algebraic set theory, Cambridge University Press, 1995.
S. Mac Lane and I. Moerdijk, Sheaves in geometry and logic: A first introduction to topos theory, Springer, 1992.
R. S. Lubarsky, Independence results around constructive ZF, Annals of Pure and Applied Logic, vol. 132 (2005), no. 2--3, pp. 209--225.
M. Makkai and G. Reyes, First-order categorical logic, Lecture Notes in Mathematics, vol. 611, Springer, 1977.
Mathematical Reviews (MathSciNet): MR505486
D. C. McCarty, Realizability and recursive mathematics, Ph.D. thesis, University of Oxford, 1984.
I. Moerdijk and E. Palmgren, Type theories, toposes, and constructive set theories: predicative aspects of AST, Annals of Pure and Applied Logic, vol. 114 (2002), pp. 155--201.
B. Nordström, K. Petersson, and J. M. Smith, Martin--Löf type theory, Handbook of logic in computer science (S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors), vol. 5, Oxford University Press, 2000.
M. Rathjen, Realizability for constructive Zermelo--Fraenkel set theory, Logic Colloquium '03 (J. Väänänen and V. Stoltenberg-Hansen, editors), Lecture Notes in Logic, vol. 24, Association for Symbolic Logic and AK Peters, 2006, pp. 282--314.
A. K. Simpson, Elementary axioms for categories of classes, 14th Symposium on Logic in Computer Science, IEEE Computer Society Press, 1999, pp. 77--85.
M. A. Warren, Coalgebras in a category of classes, Annals of Pure and Applied Logic, vol. 146 (2007), no. 1, pp. 60--71.