After reviewing the formalism of Simpson's shapes for the non-abelian Hodge theory associated to a smooth projective curve X, we will describe the construction of a certain class of functors between the derived categories of the Dolbeault, De Rahm and Betti shapes of X into the derived categories of the corresponding moduli stacks. We shall describe as well some of their main properties, in particular their interaction with the so-called Wilson operators.

This is a report on the thesis of Robert Hanson.