  One aim of this paper is to develop some aspects of the theory of monoidal
derivators. The passages from categories and model categories to derivators
both respect monoidal objects and hence give rise to natural examples. We also
introduce additive derivators and show that the values of strong, additive
derivators are canonically pretriangulated categories. Moreover, the center of
additive derivators allows for a convenient formalization of linear structures
and graded variants thereof in the stable situation. As an illustration of
these concepts, we discuss some derivators related to chain complexes and
symmetric spectra.
