nLab abstract elementary class

Idea

In model theory the notion of abstract elementary classes is a vast generalizations of that of elementary classes of structures beyond first-order theories (e.g. for the infinitary logic $L_{\omega_1,\omega}$) as introduced by Saharon Shelah. Their theory is also more general than the homogeneous model theory.

An abstract elementary class is a nonempty class $K$ of structures for a given signature with language $L(K)$, that is closed under isomorphisms and equipped with a strong substructure relation $\prec_K$ (strong substructure relation means that if $M\prec_K N$ and $M_0\subset M$ is a substructure, then $M_0 \prec_K N$) that is a partial order satisfying the axioms on union of chains (Tarski-Vaught), coherence and downward Loewenheim-Skolem properties. More precisely, $\prec_K$ is a partial order such that

(A0) if $M,N\in K$, $M\prec_K N$ then $M\subset N$

(A1) (closure under isomorphisms)

• (a) $M\in K$ and $N$ an $L(K)$ structure with $N\cong M$, then $N\in K$

• (b) if $N_1,N_2,M_1,M_2\in K$, $f_i : N_i\cong M_i$, $i = 1,2$, $f_1\subset f_2$, with $M_1\prec_K M_2$ then $N_1\prec_K N_2$

(A2) for $M,N,P\in K$, if $M\prec_K P$, $N\prec_K P$, and $M\subset N$, then $M\prec_K N$

(A3) downward Loewenheim-Skolem. There exist a cardinal $LS(K) = LS(K,\prec_K)\geq |L(K)|+\aleph_0$ such that $\forall M\in K$, $\forall A\subset |M|$, $\exists N\in K$ with $A\subset |N|$, $N\prec_K M$, $\|N\|\leq |A|+LS(K)$.

(A4) (Tarski-Vaught chain condition) for every regular cardinal $\mu$

….

The usual elementary classes, i.e. the classes of the form $K = Mod(T)$ for a first-order theory $T$, are abstract elementary with respect to the relation $\prec_K$ of being an elementary submodel, with ${|LS(K)|} = {|L(T)|}+\aleph_0$ ($L(T)$ is the underlying language of the theory $T$).

Related entries

• amalgamation

References

• wikipedia abstract elementary class

• Saharon Shelah, Classification theory for elementary abstract classes I, II, Studies in Logic (London), 18, 20, College Publications, London 2009

• John Baldwin, Categoricity, Amer. Math. Soc. 2011, pdf

• D. W. Kueker, Abstract elementary classes and infinitary logic, Ann. Pure Appl. Logic 156 (2008), 274-286.

AECs can also be essentially identified with accessible categories in which all morphisms are monomorphisms. Some recent papers which study them from this viewpoint include:

• Tibor Beke, Jiří Rosický Abstract elementary classes and accessible categories, Annals of Pure and Applied Logic 163 (2012) 2008-2017, arxiv/1005.2910

• Michael Lieberman, Jiří Rosický, Sebastien Vasey, Internal sizes in μ-abstract elementary classes, arxiv/1708.06782; Set-theoretic aspects of accessible categories, arxiv/1902.06777

• M. J. Lieberman, Topological and category-theoretic aspects of abstract elementary classes, Thesis, The University of Michigan 2009, pdf; defense slides pdf; Category theoretic aspects of abstract elementary classes, Annals Pure Appl. Logic 162 (2011), 903-915; A topology for Galois types in AECs, arxiv/0906.3573

On topos theoretic methods in model theory of AECs:

• Christian Espíndola, A topos-theoretic proof of Shelah’s eventual categoricity conjecture for abstract elementary classes, arxiv/1906.09169; A short proof of Shelah’s eventual categoricity conjecture for AEC’s with amalgamation, under GCH, arxiv/1909.13713

On the homotopy types of classifying spaces of abstract elementary classes:

• Tim Campion, Jinhe Ye, Homotopy Types of Abstract Elementary Classes, Journal of Pure and Applied Algebra

  225 5 (2021) 106461 [arXiv:1909.07965, doi:10.1016/j.jpaa.2020.106461]