Representing Certain Continued Fraction AF Algebras as C*-algebras of Categories of Paths and non-AF Groupoids

C*-algebras of categories of paths were introduced by Spielberg in 2014 and generalize C*-algebras of higher rank graphs. An approximately finite dimensional (AF) C*-algebra is one which is isomorphic to an inductive limit of finite dimensional C*-algebras. In 2012, D.G. Evans and A. Sims proposed an analogue of a cycle for higher rank graphs and show that the lack of such an object is necessary for the associated C*-algebra to be AF. Here, I give a class of examples of categories of paths whose associated C*-algebras are Morita equivalent to a large number of periodic continued fraction AF algebras, first described by Effros and Shen in 1980. I then provide two examples which show that the analogue of cycles proposed by Evans and Sims is neither a necessary nor a sufficient condition for the C*-algebra of a category of paths to be AF.