Concept
In mathematics, derivators are a proposed frameworkpg 190-195 for homological algebra giving a foundation for both abelian and non-abelian homological algebra and various generalizations of it. They were introduced to address the deficiencies of (such as the non-functoriality of the cone construction) and provide at the same time a language for homotopical algebra. Derivators were first introduced by Alexander Grothendieck in his long unpublished 1983 manuscript Pursuing Stacks. They were then further developed by him in the huge unpublished 1991 manuscript Les Dérivateurs of almost 2000 pages. Essentially the same concept was introduced (apparently independently) by Alex Heller. The manuscript has been edited for on-line publication by Georges Maltsiniotis. The theory has been further developed by several other people, including Heller, Franke, Keller and Groth. One of the motivating reasons for considering derivators is the lack of functoriality with the cone construction with . Derivators are able to solve this problem, and solve the inclusion of general homotopy colimits, by keeping track of all possible diagrams in a category with and their relations between each other. Heuristically, given the diagramwhich is a category with two objects and one non-identity arrow, and a functorto a category with a class of weak-equivalences (and satisfying the right hypotheses), we should have an associated functorwhere the target object is unique up to weak equivalence in . Derivators are able to encode this kind of information and provide a diagram calculus to use in and homotopy theory. Formally, a prederivator is a 2-functorfrom a suitable 2-category of indices to the category of categories. Typically such 2-functors come from considering the categories where is called the category of coefficients. For example, could be the category of small categories which are filtered, whose objects can be thought of as the indexing sets for a filtered colimit. Then, given a morphism of diagramsdenote byThis is called the inverse image functor.