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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 acomplete bi-interpretation invariant for such theories, taking the form of anopen Polish topological groupoid. More explicitly, for every such theory $T$ weconstruct a groupoid $\mathbf{G}^*(T)$ that only depends on thebi-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) isalways homeomorphic to the Lelek fan. We break the construction of theinvariant 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 previouslyestablished ones, which only differ by their different choice of areconstruction sort.

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