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On toposes generated by cardinal finite objects

Published online by Cambridge University Press:  23 May 2017

SIMON HENRY
SIMON HENRY*
Affiliation:
College de France, 3 Rue d'Ulm 75005 Paris, France. e-mail: [email protected]
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Abstract

We give a characterisations of toposes which admit a generating set of objects which are internally cardinal finite (i.e. Kuratowski finite and decidable) in terms of “topological” conditions. The central result is that, constructively, a hyperconnected separated locally decidable topos admit a generating set of cardinal finite objects. The main theorem is then a generalisation obtained as an application of this result internally in the localic reflection of an arbitrary topos: a topos is generated by cardinal finite objects if and only if it is separated, locally decidable, and its localic reflection is zero dimensional.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 2 , September 2018 , pp. 209 - 223
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Borceux, F. Handbook of Categorical Algebra 3: Sheaf Theory, volume 3 (Cambridge University Press, 1994).Google Scholar
[2] Coquand, T. Compact spaces and distributive lattices. J. Pure Appl. Algebra. 184 (1) 2003, 1–6.Google Scholar
[3] Henry, S. Measure theory over boolean toposes. arXiv preprint arXiv:1411.1605 (2014).Google Scholar
[4] Henry, S. Complete C*-categories and a topos theoretic Green-Julg theorem. arXiv preprint arXiv:1512.03290 (2015).Google Scholar
[5] Henry, S. Toward a non-commutative Gelfand duality: Boolean locally separated toposes and monoidal monotone complete C*-categories. arXiv preprint arXiv:1501.07045 (2015).Google Scholar
[6] Johnstone, P. T. Stone Spaces (Cambridge University Press, 1986).Google Scholar
[7] Johnstone, P. T. Sketches of an Elephant: A Topos Theory Compendium (Clarendon Press, 2002).Google Scholar
[8] MacLane, S. and Moerdijk, I. Sheaves in Geometry and Logic: a First Introduction to Topos Theory (Springer, 1992).Google Scholar
[9] Moerdijk, I. and Vermeulen, J. J. C. Proper maps of toposes, vol. 705 (AMS Bookstore, 2000).Google Scholar
[10] Shulman, M. A. Stack semantics and the comparison of material and structural set theories. arXiv preprint arXiv:1004.3802 (2010).Google Scholar