  In this paper we investigate homologically finite-dimensional objects in the
derived category of a given small dg-enhanced triangulated category. Using
these we define reflexivity, hfd-closedness, and the Gorenstein property for
triangulated categories, and discuss crepant categorical contractions. We
illustrate the introduced notions on examples of categories of geometric and
algebraic origin and provide geometric applications. In particular, we apply
our results to prove a bijection between semiorthogonal decompositions of the
derived category of a singular variety and the derived category of its
smoothing with support on the central fiber.
