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A CLASSIFICATION OF 2-CHAINS HAVING 1-SHELL BOUNDARIES IN ROSY THEORIES

  • BYUNGHAN KIM, SUNYOUNG KIM (a1) and JUNGUK LEE

Abstract

We classify, in a nontrivial amenable collection of functors, all 2-chains up to the relation of having the same 1-shell boundary. In particular, we prove that in a rosy theory, every 1-shell of a Lascar strong type is the boundary of some 2-chain, hence making the 1st homology group trivial. We also show that, unlike in simple theories, in rosy theories there is no upper bound on the minimal lengths of 2-chains whose boundary is a 1-shell.

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[1]Adler, Hans, Explanations of Independence, Ph. D. Thesis, University of Freiburg, 2005.
[2]Casanovas, Enrique, Lascar, Daniel, Pillay, Anand, and Ziegler, Martin, Galois groups of first order theories. Journal of Mathematical Logic, vol. 1 (2001), pp. 305319.
[3]Diestel, Reinhard, Graph Theory, 2nd ed., Springer, New York, 2000.
[4]Ealy, Clifton and Onshuus, Alf, Characterizing rosy theories, this Journal, vol. 72 (2007), pp. 919–940.
[5]Goodrick, John, Kim, Byunghan, and Kolesnikov, Alexei, Homology groups of types in model theory and the computation of H 2 (p), this Journal, vol. 78 (2013), pp. 1086–1114.
[6]Goodrick, John, Kim, Byunghan, and Kolesnikov, Alexei, Amalgamation functors and homology groups in model theory. Proceedings of ICM 2014, to appear.
[7]Kim, SunYoung and Lee, Junguk, More on classification of 2-chains having 1-shell boundaries in rosy theories, preprint.

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