In what sense is the ring of polynomials in one variable over a field k interpretable in k? In what sense is the subgroup of G generated by a definable subset D of G interpretable in G? These questions have been answered in various particular contexts by various people. We set up a general formalism to treat such piece-wise interpreted objects as stand-alone multi-sorted first-order structures. This formalism is motivated by our quest to create a model-theoretically tractable analog of sheaf theory. This is a very preliminary report on joint work with Ramin Takloo-Bighash.