In
mathematics, derivators are a proposed framework
pg 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
derived categories
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
(such as the non-functoriality of the cone construction) and provide at the same time a language for
homotopical algebra
In mathematics, homotopical algebra is a collection of concepts comprising the ''nonabelian'' aspects of homological algebra as well as possibly the abelian aspects as special cases. The ''homotopical'' nomenclature stems from the fact that a ...
.
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Alexander Grothendieck in his long unpublished 1983 manuscript ''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 They were then further developed by him in the huge unpublished 1991 manuscript ''Les Dérivateurs'' of almost 2000 pages.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.
Motivations
One of the motivating reasons for considering derivators is the lack of functoriality with the cone construction with
triangulated categories In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy cate ...
. Derivators are able to solve this problem, and solve the inclusion of general
homotopy colimit
In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfc ...
s, by keeping track of all possible diagrams in a category with
weak equivalences and their relations between each other. Heuristically, given the diagram
which is a category with two objects and one non-identity arrow, and a functor
to a category
with a class of weak-equivalences
(and satisfying the right hypotheses), we should have an associated functor
where the target object is unique up to weak equivalence in