  We study torsion torsionfree(=TTF) triples in abelian and triangulated
categories. (Notice that TTF triples in a triangulated category are essentially
in bijection with recollement data for this triangulated category.) In
particular, we complete Jans' characterization of split TTF triples on a
category of modules, prove a weak version of the Generalized Smashing
Conjecture, use homological epimorphisms of differential graded(=dg) categories
to give an explicite description of all the TTF triples in the derived category
of a k-flat dg category and develope an unbounded approach to Koenig's theorem
on recollements of right bounded derived categories of ordinary algebras.
