On weak Fraisse limits

TL;DR

This paper characterizes when a countable class of structures has a weak Fra"iss"e limit using the action of $S__$, linking properties like JEP and WAP to the existence of a comeager structure with a specific automorphism property.

Contribution

It establishes a new characterization of weak Fra"iss"e limits via topological and group action methods, extending previous results to a broader class of structures.

Findings

Characterizes weak Fra"iss"e limits using $S__$ action.

Links JEP and WAP to the existence of a comeager structure.

Recovers and generalizes a result by Kechris and Rosendal.

Abstract

Using the natural action of $S_\infty$ we show that a countable hereditary class $\mathcal C$ of finitely generated structures has the joint embedding property (JEP) and the weak amalgamation property (WAP) if and only if there is a structure $M$ whose isomorphism type is comeager in the space of all countable, infinitely generated structures with age in $\mathcal C$. In this case, $M$ is the weak Fra\"iss\'e limit of $\mathcal C$. This applies in particular to countable structures with generic automorphisms and recovers a result by Kechris and Rosendal [Proc. Lond. Math. Soc., 2007].