On the theory of derivators

Abstract

In Part 1, we develop some aspects of the theory of derivators, pointed derivators, and stable derivators. As a main result, we show that the values of a stable derivator can be canonically endowed with the structure of a triangulated category. Moreover, the functors belonging to the stable derivator can be turned into exact functors with respect to these triangulated structures. Along the way, we give a simplification of the axioms of a pointed derivator and a reformulation of the base change axiom in terms of Grothendieck (op)fibration. Furthermore, we have a new proof that a combinatorial model category has an underlying derivator.<br /> In Part 2, we develop the theory of monoidal derivators and the related notions of derivators being tensored, cotensored, or enriched over a monoidal derivator. The passage from model categories to derivators respects these notions and, hence, gives rise to natural examples. Moreover, we introduce the notion of the center of additive derivators which allows for a convenient formalization of linear structures on additive derivators and graded variants thereof in the stable situation. As an illustration we discuss some derivators related to chain complexes and symmetric spectra.<br /> In the last part, we take a closer look of the derivator associated to a differential-graded algebra over a field. A theorem of Kadeishvili ensures that the homotopy type of such a dga A can be encoded by a minimal A_∞-algebra structure on the homology algebra. Moreover, a result of Renaudin guarantees that the derivator DA of differential-graded A-modules essentially captures the homotopy theory associated to A. This motivates that these two structures –the A_∞-algebra and the derivator DA – should determine each other, and we give a first step towards such a comparison result.