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 Tameness, powerful images, and large cardinals

2020 ◽  
Vol 21 (01) ◽  
pp. 2050024
Author(s):  
Will Boney ◽  
Michael Lieberman
Keyword(s):  
Large Cardinals ◽  
Classification Theory ◽  
Closure Properties ◽  
Weakly Compact ◽  
Accessible Categories ◽  
Elementary Classes ◽  
Comprehensive Level ◽  
Theoretic Characterization ◽  
Measurable Cardinals

We provide comprehensive, level-by-level characterizations of large cardinals, in the range from weakly compact to strongly compact, by closure properties of powerful images of accessible functors. In the process, we show that these properties are also equivalent to various forms of tameness for abstract elementary classes. This systematizes and extends results of [W. Boney and S. Unger, Large cardinal axioms from tameness in AECs, Proc. Amer. Math. Soc. 145(10) (2017) 4517–4532; A. Brooke-Taylor and J. Rosický, Accessible images revisited, Proc. AMS 145(3) (2016) 1317–1327; M. Lieberman, A category-theoretic characterization of almost measurable cardinals (Submitted, 2018), http://arxiv.org/abs/1809.06963; M. Lieberman and J. Rosický, Classification theory for accessible categories. J. Symbolic Logic 81(1) (2016) 1647–1648].

scholarly journals METRIC ABSTRACT ELEMENTARY CLASSES AS ACCESSIBLE CATEGORIES

2017 ◽  
Vol 82 (3) ◽  
pp. 1022-1040 ◽  
Author(s):  
M. LIEBERMAN ◽  
J. ROSICKÝ
Keyword(s):  
Abstract Elementary Classes ◽  
Accessible Categories ◽  
Elementary Classes

AbstractWe 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.

scholarly journals Ramsey-like cardinals II

2011 ◽  
Vol 76 (2) ◽  
pp. 541-560 ◽  
Author(s):  
Victoria Gitman ◽  
P. D. Welch
Keyword(s):  
Upper Bound ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Core Model ◽  
Consistency Strength ◽  
The Core ◽  
Countable Ordinal ◽  
Elementary Embeddings ◽  
Remarkable Cardinals

AbstractThis paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α ≤ ω1, that they are downward absolute to L for , and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.

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
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