The purpose of this work is to prove the existence of an algebraic moduli classifying objects in a given triangulated category.
To any dg-category T (over some base ring k), we define a VKH-4PSBWST-2&_mathId=mml1&_user=10&_cdi=6123&_rdoc=2&_acct=C000050221&_version=1&_userid=10&md5=81fba0f3ee757059858236d60d74f82f" title="Click to view the MathML source">D−-stack 
in the sense of [Toën B., Vezzosi G., Homotopical algebraic geometry II: Geometric stacks and applications, Mem. Amer. Math. Soc., in press], classifying certain Top-dg-modules. When T is saturated, 
classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness conditions on T (e.g. if it is saturated) the D−-stack 
is locally geometric (i.e. union of open and geometric sub-stacks). As a consequence we prove the algebraicity of the group of auto-equivalences of saturated dg-categories. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as well as complexes of representations of a finite quiver.
