The Lelek fan and the Poulsen simplex as Fra\"iss\'e limits

We describe the Lelek fan, a smooth fan whose set of end-points is dense, and the Poulsen simplex, a Choquet simplex whose set of extreme points is dense, as Fra"{i}ss'e limits in certain natural categories of embeddings and projections. As an application we give a short proof of their uniqueness, universality, and almost homogeneity. We further show that for every two countable dense subsets of end-points of the Lelek fan there exists an auto-homeomorphism of the fan mapping one set onto the other. This improves a result of Kawamura, Oversteegen, and Tymchatyn from 1996.