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  • M. LIEBERMAN (a1) and J. ROSICKÝ (a2)

We show that metric abstract elementary classes (mAECs) are, in the sense of [15], coherent accessible categories with directed colimits, with concrete ℵ1-directed colimits and concrete monomorphisms. More broadly, we define a notion of κ-concrete AEC—an AEC-like category in which only the κ-directed colimits need be concrete—and develop the theory of such categories, beginning with a category-theoretic analogue of Shelah’s Presentation Theorem and a proof of the existence of an Ehrenfeucht–Mostowski functor in case the category is large. For mAECs in particular, arguments refining those in [15] yield a proof that any categorical mAEC is μ- d -stable in many cardinals below the categoricity cardinal.

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[1] Ackerman, N., Completeness in generalized ultrametric spaces . P-Adic Numbers Ultrametric Analysis and Applications, vol. 5 (2013), no. 1, pp. 89105.
[2] Adámek, J. and Rosický, J., Locally Presentable and Accessible Categories, Cambridge University Press, Cambridge, 1994.
[3] Baldwin, J., Categoricity, American Mathematical Society, Providence, 2009.
[4] Baldwin, J., Ehrenfeucht-Mostowski models in abstract elementary classes , Logic and its Applications (Zhang, Y. and Blass, A., editors), Contemporary Mathematics, vol. 380, American Mathematical Society, Providence, 2005, pp. 117.
[5] Beke, T. and Rosický, J., Abstract elementary classes and accessible categories . Annals of Pure and Applied Logic, vol. 163 (2012), pp. 20082017.
[6] Boney, W., Tameness from large cardinal axioms , this Journal, vol. 79 (2014), no. 4, pp. 10921119.
[7] Boney, W., A presentation theorem for continuous logic and metric abstract elementary classes , arXiv:1408.3624.
[8] Boney, W., Grossberg, R., Lieberman, M., Rosický, J., and Vasey, S., μ-Abstract Elementary Classes and other generalizations . Journal of Pure and Applied Algebra, vol. 220 (2016), pp. 30483066.
[9] Garner, R. and Lack, S., Grothendieck quasitoposes . Journal of Algebra, vol. 355 (2012), pp. 111126.
[10] Hirvonen, A. and Hyttinen, T., Categoricity in homogeneous complete metric spaces . Archive of Mathematical Logic, vol. 48 (2009), pp. 269322.
[11] Henson, C. and Iovino, J.. Ultraproducts in analysis , Analysis and Logic (Mons, 1997), London Mathematical Society Lecture Note Series, vol. 262, Cambridge University Press, Cambridge, 2002, pp. 1110.
[12] Kirby, J., Abstract elementary categories, preprint, 2008.
[13] Lieberman, M., Category theoretic aspects of abstract elementary classes . Annals of Pure and Applied Logic, vol. 162 (2011), pp. 903915.
[14] Lieberman, M., A topology for types in abstract elementary classes . Mathematical Logic Quarterly, vol. 57 (2011), no. 2, pp. 204216.
[15] Lieberman, M. and Rosický, J., Classification theory for accessible categories , this Journal, vol. 81 (2016), no. 1, pp. 151165.
[16] Makkai, M. and Paré, R., Accessible Categories: The Foundations of Categorical Model Theory, American Mathematical Society, Providence, 1989.
[17] Priess-Crampe, S. and Ribenboim, P., Generalized ultrametric spaces I . Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 66 (1966), pp. 5573.
[18] Rosický, J., Accessible categories, saturation and categoricity , this Journal, vol. 62 (1997), pp. 891901.
[19] Vasey, S., Infinitary stability theory . Archive for Mathematical Logic, vol. 55 (2016), pp. 562592.
[20] Villaveces, A. and Zambrano, P., Limit models in metric abstract elementary classes: The categorical case , arXiv:1304.6797.
[21] Zambrano, P., Around superstability in metric abstract elementary classes, Ph.D. thesis, Universidad Nacional de Colombia, 2011.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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