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Lawvere–Tierney sheaves in Algebraic Set Theory

Published online by Cambridge University Press:  12 March 2014

S. Awodey ,
N. Gambino ,
P. L. Lumsdaine  and
M. A. Warren
S. Awodey
Affiliation:
Department of Philosophy, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pa 15213, USA, E-mail: awodey@cmu.edu
N. Gambino
Affiliation:
Department of Computer Science, University of Leicester, University Road, Leicester Lei 7Rh, UK, E-mail: nicola.gambino@gmail.com
P. L. Lumsdaine
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pa 15213, USA, E-mail: plumsdai@andrew.cmu.edu
M. A. Warren
Affiliation:
Department of Philosophy, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pa 15213, USA, E-mail: mwarren@andrew.cmu.edu
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Abstract

We present a solution to the problem of denning 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 861 - 890
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Aczel, P. and Rathjen, M., Notes on constructive set theory, Technical Report 40, Mittag–Leffler Institut, The Swedish Royal Academy of Sciences, 2001.Google Scholar
[2]Awodey, S., Butz, C., Simpson, A., and Streicher, T., Relating topos theory and set theory via categories of classes, Technical Report CMU-PHIL-146, Department of Philosophy, Carnegie Mellon University, 2003.Google Scholar
[3]Awodey, S. and Warren, M. A., Predicative algebraic set theory, Theory and applications of categories, vol. 15 (2005), no. 1, pp. 1–39.Google Scholar
[4]van den Berg, B., Sheaves for predicative toposes, Archive for Mathematical Logic, to appear. ArXiv:math.L0/0507480vl, 2005.Google Scholar
[5]van den Berg, B. and Moerdijk, I., A unified approach to algebraic set theory, arXiv: 0710.3066vl, 2007, to appear in the Proceedings of the Logic Colloquium 2006.Google Scholar
[6]van den Berg, B., Aspects of predicative algebraic set theory I: Exact completions, Annals of Pure and Applied Logic, vol. 156 (2008), no. 1, pp. 123–159.CrossRefGoogle Scholar
[7]Carboni, A., Some free constructions in readability and proof theory, Journal of Pure and Applied Algebra, vol. 103 (1995), pp. 117–148.CrossRefGoogle Scholar
[8]Carboni, A. and Vitale, E. M., Regular and exact completions, Journal of Pure and Applied Algebra, vol. 125 (1998), pp. 79–116.CrossRefGoogle Scholar
[9]Fourman, M. P., Sheaf models for set theory, Journal of Pure and Applied Algebra, vol. 19 (1980), pp. 91–101.CrossRefGoogle Scholar
[10]Freyd, P., The axiom of choice, Journal of Pure and Applied Algebra, vol. 19 (1980), pp. 103–125.CrossRefGoogle Scholar
[11]Gambino, N., Presheaf models for constructive set theories, From sets and types to topology and analysis (Crosilla, L. and Schuster, P., editors), Oxford University Press, 2005, pp. 62–77.CrossRefGoogle Scholar
[12]Gambino, N., Heyting-valued interpretations for constructive set theory, Annals of Pure and Applied Logic, vol. 137 (2006), no. 1-3, pp. 164–188.CrossRefGoogle Scholar
[13]Gambino, N., The associated sheaf functor theorem in algebraic set theory, Annals of Pure and Applied Logic, vol. 156 (2008), no. 1, pp. 68–77.CrossRefGoogle Scholar
[14]Gambino, N. and Aczel, P., The generalized type-theoretic interpretation of constructive set theory, this Journal, vol. 71 (2006), no. 1, pp. 67–103.Google Scholar
[15]Grayson, R. J., Forcing for intuitionistic systems without power-set, this Journal, vol. 48 (1983), no. 3, pp. 670–682.Google Scholar
[16]Johnstone, P. T., Sketches of an elephant: A topos theory compendium, Oxford University Press, 2002.Google Scholar
[17]Joyal, A. and Moerdijk, I., Algebraic set theory, Cambridge University Press, 1995.CrossRefGoogle Scholar
[18]Lane, S. Mac and Moerdijk, I., Sheaves in geometry and logic: A first introduction to topos theory, Springer, 1992.Google Scholar
[19]Lubarsky, R. S., Independence results around constructive ZF, Annals of Pure and Applied Logic, vol. 132 (2005), no. 2-3, pp. 209–225.CrossRefGoogle Scholar
[20]Makkai, M. and Reyes, G., First-order categorical logic, Lecture Notes in Mathematics, vol. 611, Springer, 1977.CrossRefGoogle Scholar
[21]McCarty, D. C., Readability and recursive mathematics, Ph.D. thesis, University of Oxford, 1984.Google Scholar
[22]Moerduk, I. and Palmgren, E., Type theories, toposes, and constructive set theories: predicative aspects of AST, Annals of Pure and Applied Logic, vol. 114 (2002), pp. 155–201.CrossRefGoogle Scholar
[23]Nordström, B., Petersson, K., and Smith, J. M., Martin-Löf type theory, Handbook of logic in computer science (Abramsky, S., Gabbay, D. M., and Maibaum, T. S. E., editors), vol. 5, Oxford University Press, 2000.Google Scholar
[24]Rathjen, M., Realizability for constructive Zermelo–Fraenkel set theory, Logic Colloquium '03 (Väänänen, J. and Stoltenberg-Hansen, V., editors), Lecture Notes in Logic, vol. 24, Association for Symbolic Logic and AK Peters, 2006, pp. 282–314.CrossRefGoogle Scholar
[25]Simpson, A. K., Elementary axioms for categories of classes, 14th Symposium on Logic in Computer Science, IEEE Computer Society Press, 1999, pp. 77–85.Google Scholar
[26]Warren, M. A., Coalgebras in a category of classes, Annals of Pure and Applied Logic, vol. 146 (2007), no. 1, pp. 60–71.CrossRefGoogle Scholar