scholarly journals Advances in Classification Theory for Abstract Elementary Classes

2018 ◽  
Vol 24 (4) ◽  
pp. 454-455
Author(s):  
Will Boney
Keyword(s):  
Classification Theory ◽  
Abstract Elementary Classes ◽  
Elementary Classes

scholarly journals CLASSIFICATION THEORY FOR ACCESSIBLE CATEGORIES

2016 ◽  
Vol 81 (1) ◽  
pp. 151-165 ◽  
Author(s):  
M. LIEBERMAN ◽  
J. ROSICKÝ
Keyword(s):  
Recent Result ◽  
Large Cardinal ◽  
Classification Theory ◽  
Abstract Elementary Classes ◽  
Robust Version ◽  
Accessible Categories ◽  
Elementary Classes ◽  
Added Benefit ◽  
Precise Role

AbstractWe show that a number of results on abstract elementary classes (AECs) hold in accessible categories with concrete directed colimits. In particular, we prove a generalization of a recent result of Boney on tameness under a large cardinal assumption. We also show that such categories support a robust version of the Ehrenfeucht–Mostowski construction. This analysis has the added benefit of producing a purely language-free characterization of AECs, and highlights the precise role played by the coherence axiom.

scholarly journals Independence in finitary abstract elementary classes

2006 ◽  
Vol 143 (1-3) ◽  
pp. 103-138 ◽  
Author(s):  
T. Hyttinen ◽  
M. Kesälä
Keyword(s):  
Abstract Elementary Classes ◽  
Elementary Classes

scholarly journals Shelah's categoricity conjecture from a successor for tame abstract elementary classes

2006 ◽  
Vol 71 (2) ◽  
pp. 553-568 ◽  
Author(s):  
Rami Grossberg ◽  
Monica Vandieren
Keyword(s):  
Abstract Elementary Classes ◽  
Transfer Theorem ◽  
Elementary Class ◽  
Joint Embedding ◽  
Elementary Classes ◽  
Abstract Elementary Class

AbstractWe prove a categoricity transfer theorem for tame abstract elementary classes.Suppose that K is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ, LS(K+}. If K is categorical in λ and λ+, then K is categorical in λ++.Combining this theorem with some results from [37]. we derive a form of Shelah's Categoricity Conjecture for tame abstract elementary classes:Suppose K is χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ0 ≔ Hanf(K). Ifand K is categorical in somethen K is categorical in μ for all μ .

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