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1 results

  • METRIC ABSTRACT ELEMENTARY CLASSES AS ACCESSIBLE CATEGORIES

  • M. LIEBERMAN, J. ROSICKÝ
  • Journal:
    The Journal of Symbolic Logic / Volume 82 / Issue 3 / September 2017
    Published online by Cambridge University Press:
    08 May 2017, pp. 1022-1040
    Print publication:
    September 2017

  • 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.