Deriving DG categories

Abstract

We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a ''triangulated analogue'' (4.3) of a theorem of Freyd's [5], Ex. 5.3 H, and Gabriel's [6], Ch. V, characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a ''Morita theorem'' (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories.