  We introduce a new device in the study of abstract elementary classes (AECs):
Galois Morleyization, which consists in expanding the models of the class with
a relation for every Galois type of length less than a fixed cardinal $\kappa$.
We show:
  $\mathbf{Theorem}$ (The semantic-syntactic correspondence)
  An AEC $K$ is fully $(<\kappa)$-tame and type short if and only if Galois
types are syntactic in the Galois Morleyization.
  This exhibits a correspondence between AECs and the syntactic framework of
stability theory inside a model. We use the correspondence to make progress on
the stability theory of tame and type short AECs. The main theorems are:
  $\mathbf{Theorem}$
  Let $K$ be a $\text{LS}(K)$-tame AEC with amalgamation. The following are
equivalent:
  * $K$ is Galois stable in some $\lambda \ge \text{LS}(K)$.
  * $K$ does not have the order property (defined in terms of Galois types).
  * There exist cardinals $\mu$ and $\lambda_0$ with $\mu \le \lambda_0 <
\beth_{(2^{\text{LS}(K)})^+}$ such that $K$ is Galois stable in any $\lambda
\ge \lambda_0$ with $\lambda = \lambda^{<\mu}$.
  $\mathbf{Theorem}$
  Let $K$ be a fully $(<\kappa)$-tame and type short AEC with amalgamation,
$\kappa = \beth_{\kappa} > \text{LS} (K)$. If $K$ is Galois stable, then the
class of $\kappa$-Galois saturated models of $K$ admits an independence notion
($(<\kappa)$-coheir) which, except perhaps for extension, has the properties of
forking in a first-order stable theory.
