Approximate Fraïssé theory and MU-categories

Fraïssé theory links together properties of families of structures like the amalgamation property with properties of limit objects like homogeneity and the extension property. The structures considered are not limited to be first-order structures, and the maps between the structures are not limited to be embeddings. In fact, we may consider any category consisting of abstract objects and morphisms.

We are motivated by the result of Irwin and Solecki from projective Fraïssé theory that the Fraïssé limit of the category of finite discrete linear graphs with continuous epimorphisms is a pre-space of the pseudo-arc. This yields a characterization of the pseudo-arc as a unique approximately projectively homogeneous arc-like continuum. We are interested in a “direct” counterpart of the result – that the pseudo-arc itself can be viewed as a Fraïssé limit. For this we develop an approximate framework for Fraïssé theory. This was already done by Kubiś in the metric-enriched setting. Now we generalize the setting to so-called MU-categories, and besides the pseudo-arc we also realize the pseudo-solenoids as Fraïssé limits.

In this talk I will give an overview of the Fraïssé theory and its various flavors, introduce the notion of MU-category as a base for the approximate Fraïssé theoretical framework, and illustrate the framework on the example of the pseudo-arc and pseudo-solenoids.

This is a joint work in progress with Wiesław Kubiś.