nLab abstract elementary class




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 ω 1,ωL_{\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 KK of structures for a given signature with language L(K)L(K), that is closed under isomorphisms and equipped with a strong substructure relation K\prec_K (strong substructure relation means that if M KNM\prec_K N and M 0MM_0\subset M is a substructure, then M 0 KNM_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, K\prec_K is a partial order such that

(A0) if M,NKM,N\in K, M KNM\prec_K N then MNM\subset N

(A1) (closure under isomorphisms)

  • (a) MKM\in K and NN an L(K)L(K) structure with NMN\cong M, then NKN\in K

  • (b) if N 1,N 2,M 1,M 2KN_1,N_2,M_1,M_2\in K, f i:N iM if_i : N_i\cong M_i, i=1,2i = 1,2, f 1f 2f_1\subset f_2, with M 1 KM 2M_1\prec_K M_2 then N 1 KN 2N_1\prec_K N_2

(A2) for M,N,PKM,N,P\in K, if M KPM\prec_K P, N KPN\prec_K P, and MNM\subset N, then M KNM\prec_K N

(A3) downward Loewenheim-Skolem. There exist a cardinal LS(K)=LS(K, K)|L(K)|+ 0LS(K) = LS(K,\prec_K)\geq |L(K)|+\aleph_0 such that MK\forall M\in K, A|M|\forall A\subset |M|, NK\exists N\in K with A|N|A\subset |N|, N KMN\prec_K M, N|A|+LS(K)\|N\|\leq |A|+LS(K).

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


The usual elementary classes, i.e. the classes of the form K=Mod(T)K = Mod(T) for a first-order theory TT, are abstract elementary with respect to the relation K\prec_K of being an elementary submodel, with |LS(K)|=|L(T)|+ 0{|LS(K)|} = {|L(T)|}+\aleph_0 (L(T)L(T) is the underlying language of the theory TT).


  • 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:

Last revised on March 2, 2023 at 16:52:59. See the history of this page for a list of all contributions to it.