Exercices de style: A homotopy theory for set theory
We construct a model category (in the sense of Quillen) for set theory, starting from two arbitrary, but natural, conventions. It is the simplest category satisfying our conventions and modelling the notions of finiteness, countability and infinite equi-cardinality. We argue that from the homotopy theoretic point of view our construction is essentially automatic following basic existing methods, and so is (almost all) the verification that the construction works.
We use the posetal model category to introduce homotopy-theoretic intuitions to set theory. Our main observation is that the homotopy invariant version of cardinality is the covering number of Shelah’s PCF theory, and that other combinatorial objects, such as Shelah’s revised power function—the cardinal function featuring in Shelah’s revised GCH theorem—can be obtained using similar tools. We include a small “dictionary” for set theory in QtNaamen, hoping it will help in finding more meaningful homotopy-theoretic intuitions in set theory.
Unable to display preview. Download preview PDF.
- M. Bays, Categoricity Results for Exponential Maps of 1-Dimensional Algebraic Groups & Schanuel Conjectures for Powers and the CIT, Ph.D. thesis, Oxford University, 2009, Available at http://people.maths.ox.ac.uk/bays/dist/thesis/.
- M. Gavrilovich, Model Theory of the Universal Covering Spaces of Complex Algebraic Varieties Misha Gavrilovich, Ph.D. thesis, Oxford University, 2006, Available at http://people.maths.ox.ac.uk/bays.
- M. Gavrilovich and A. Hasson, Exercises de style: a homotopy theory for set theory I, available on arXiv, 2010.Google Scholar
- M. Gavrilovich and A. Hasson, Exercises de style: a homotopy theory for set theory II, available on http://corrigenda.ru/by:gavrilovich-and-hasson/what:a-homotopy-theory-for-set-theory/a-homotopy-theory-for-set-theory.pdf, 2010.
- M. Gromov, Structures, Learning and Ergosystems, available at http://www.ihes.fr/gromov/PDF/ergobrain.pdf, 2009.
- M. Kojman, The A,B,C of PCF, available on http://de.arxiv.org/pdf/math/9512201.pdf, 1995.
- M. Kojman, Singular cardinals: from Hausdorff’s gaps to Shelah’s PCF theory, in The Handbook of the History of Logic (Vol. 6), Sets and Extensions in the Twentieth Century, Elsevier, Amsterdam, 2012, to appear.Google Scholar
- M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited—2001 and Beyond, Springer, Berlin, 2001, pp. 771–808.Google Scholar
- J.-P. Serre, Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques, in Motives (Seattle, WA, 1991), Proceedings of Symposia in Pure Mathematics, Vol. 55, American Mathematical Society, Providence, RI, 1994, pp. 377–400.Google Scholar