  We define a topological Hochschild (THH) and cyclic (TC) homology theory for
differential graded (dg) categories and construct several non-trivial natural
transformations from algebraic K-theory to THH(-).
  In an intermediate step, we prove that the homotopy theory of dg categories
is Quillen equivalent, through a four step zig-zag of Quillen equivalences, to
the homotopy theory of Eilenberg-Mac Lane spectral categories.
  Finally, we show that over the rationals two dg categories are topological
equivalent if and only if they are quasi-equivalent.
