Star sorts, Lelek fans, and the reconstruction of non-$\aleph\_0$-categorical theories in continuous logic

TL;DR

This paper establishes a bi-interpretation invariant for theories in continuous and classical logic using an open Polish topological groupoid, with the Lelek fan as a key component, enabling reconstruction of theories from the groupoid.

Contribution

It introduces a new reconstruction theorem for theories in continuous logic, providing a complete bi-interpretation invariant via topological groupoids, extending previous frameworks.

Findings

Constructs a groupoid invariant depending only on the bi-interpretation class of a theory.

Shows the object space of the groupoid is homeomorphic to the Lelek fan.

Provides a method to reconstruct theories from the associated groupoid.

Abstract

We prove a reconstruction theorem valid for arbitrary theories in continuous (or classical) logic in a countable language, that is to say that we provide a complete bi-interpretation invariant for such theories, taking the form of an open Polish topological groupoid. More explicitly, for every such theory $T$ we construct a groupoid $\mathbf{G}^*(T)$ that only depends on the bi-interpretation class of $T$, and conversely, we reconstruct from $\mathbf{G}^*(T)$ a theory that is bi-interpretable with $T$. The basis of $\mathbf{G}^*(T)$ (namely, the set of objects, when viewed as a category) is always homeomorphic to the Lelek fan. We break the construction of the invariant into two steps. In the second step we construct a groupoid from any \emph{reconstruction sort}, while in the first step such a sort is constructed. This allows us to place our result in a common framework with previously established ones, which only differ by their different choice of a reconstruction sort.