TITLE: Is there a relationship between model theory and category theory?
QUESTION [23 upvotes]: According to Chang and Keisler's "Model Theory", Model Theory = Universal Algebra + Logic. Model theory generalized Universal Algebra in the sense that we allow relations while in Universal Algebra we only allow functions.
Also, we know that Category Theory generalized Universal Algebra. From wikipedia:

Blockquote Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category. Category theory applies to many situations where universal algebra does not, extending the reach of the theorems. Conversely, many theorems that hold in universal algebra do not generalise all the way to category theory.

So this suggest there might be some overlapping between Model Theory and Category Theory. I hope some one can elaborate about the relation (if there is)?

REPLY [2 votes]: The theory of Abstract Elementary Classes, which were introduced by Shelah as an abstract axiomatization of elementary classes was recently connected with Accessible categories.

Beke, Rosicky. Abstract elementary classes and accessible categories.
M. Lieberman's PhD Thesis.

This connection was better understood since the recent join of Sebastian Vasey.

Rosicky Lieberman Vasey. Universal abstract elementary classes and locally multipresentable categories
Rosicky Lieberman Vasey. Internal sizes in μ-abstract elementary classes
Rosicky Lieberman Vasey. Forking independence from the categorical point of view