Derivator

A derivator is a formal framework for abstract homotopy theory that aims to organize how a homotopy theory behaves with respect to diagrams. It was introduced by Alexander Grothendieck and later developed by others (notably Heller) to address limitations of triangulated categories in handling diagrams and derived functors.

In brief, a derivator D is a 2-functor D: Cat^op → CAT from the opposite of the category of small categories to the category of (large) categories. For each small category I, D(I) is interpreted as the category of diagrams of shape I in a fixed homotopy theory. For a functor u: I → J, D assigns a functor D(u): D(J) → D(I), representing pullback along u. The framework provides, for each u, left and right Kan extension functors u_! and u_* between the corresponding diagram categories, playing the roles of homotopy colimits and homotopy limits, and these operations satisfy coherence conditions with respect to composition and base change.

Derivators come in several flavors. Grothendieck’s original derivators, Heller’s variant, and the theory of stable derivators (where the homotopy categories carry a triangulated structure) are common examples. A prototypical source of a derivator is a model category M, where D(I) is the homotopy category ho(M^I). Stable model categories give rise to stable derivators, reflecting triangulated diagrammatic behavior.

Purpose and use include providing a robust setting for derived functors, base change, and coherent diagrammatic calculus, while avoiding some shortcomings of triangulated categories. They are employed in areas of homotopical algebra, algebraic geometry, and higher category theory.