Given a smooth algebraic variety X over a field of characteristic zero, one can choose its smooth compactification, and take the formal punctured neighborhood of the infinity locus. The result does not depend on the choice of a compactification. In particular, we get a triangulated DG category of perfect complexes on this formal punctured neighborhood, which depends only on X.
According to Mohammed Abouzaid, a similar construction occurs in the framework of Fukaya categories, and this motivates the question whether we can do it in purely categorical terms.
I will explain that the answer is "yes": for any small DG category one can define the corresponding "formal punctured neighborhood of infinity". When applied to the derived category of coherent sheaves on X, one gets the category described above. Our construction is closely related with (an algebraic version of) Calkin algebras.
Hopefully, this can lead to the categorical construction of the logarithmic de Rham complex, a far-reaching generalization of the Hochschild-Kostant-Rosenberg theorem.