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
we construct
a groupoid
that only depends on the bi-interpretation class of
, and conversely, we
reconstruct from
a theory
that is bi-interpretable with
.
The basis of
(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 sort of codes for models, 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 sort of
codes.