Title: Catalan families in categories

Abstract: The Catalan numbers are known to count many different families of objects, with two important examples being the binary rooted trees, and the finitely branching rooted trees. These families of objects arise as the initial models of two different algebraic theories, and so the question is: why should the initial models coincide? We attempt a category-theoretic answer to this question which, while not wholly satisfactory, does at least allow for some interesting generalizations to Catalan families in other categories and to more general kinds of Catalan number.

This is joint work with Geoff Edington-Cheater.