Abstract elementary classes and accessible categories

@article{Beke2012AbstractEC,
  title={Abstract elementary classes and accessible categories},
  author={Tibor Beke and Jir{\'i} Rosick{\'y}},
  journal={Ann. Pure Appl. Log.},
  year={2012},
  volume={163},
  pages={2008-2017}
}
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We generalize the constructions and results of Chapter 10 in Baldwin's "Categoricity" to coherent accessible categories with concrete directed colimits and concrete monomorphisms. In particular, we
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We show that the category of abstract elementary classes (AECs) and concrete functors is closed under constructions of "limit type," which generalizes the approach of Mariano, Zambrano and Villaveces
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CLASSIFICATION THEORY FOR ACCESSIBLE CATEGORIES
TLDR
A generalization of a recent result of Boney on tameness under a large cardinal assumption is proved, which shows that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits.
METRIC ABSTRACT ELEMENTARY CLASSES AS ACCESSIBLE CATEGORIES
Abstract 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
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Accessible categories admit a purely category-theoretic replacement for cardinality: the internal size. Generalizing results and methods from [LRV19b], we examine set-theoretic problems related to
Combinatorial Homotopy Categories
A model category is called combinatorial if it is cofibrantly generated and its underlying category is locally presentable. As shown in recent years, homotopy categories of combinatorial model
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Bootstrapping structural properties, via accessible images
We present several new model-theoretic applications of the fact that, under a mild large cardinal assumption, the powerful image of any accessible functor is accessible. In particular, we generalize
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