An abstract elementary class is a class 
of structures in a fixed language
together with a partial order 
called strong substructure. Both are invariant under
isomorphism, and 
implies that 
is a substructure of 
. Coherence requires that 
and 
imply 
.
Increasing strong-substructure chains have unions in 
,
each chain member is strong in its union, and a common
strong upper bound contains the union
as a strong substructure. There must also be an infinite cardinal 
such that every subset 
of a structure 
is contained in a strong substructure of cardinality
at most 
.
The least such cardinal, required to be at least
the size of the language plus 
, is the Löwenheim-Skolem number 
.
The class is categorical in a cardinal 
when it has exactly one structure
of that cardinality up to isomorphism,
including existence. The eventual
categoricity conjecture concerns uniform transfer of this property in sufficiently
large cardinals.
