Differential graded versus simplicial categories

We establish a connection between differential graded and simplicial categories byconstructing a three-step zig-zag of Quillen adjunctions relating the homotopy theories of the two. In an intermediate step, we extend the Dold–Kan correspondence to aQuillen equivalence between categories enriched over non-negatively graded complexesand categories enriched over simplicial modules. As an application, we obtain a simplecalculation of Simpson’s homotopy fiber, which is known to be a key step in the construction of a moduli stack of perfect complexes on a smooth projective variety.