Homotopy Limits and Colimits in Nature A Motivation for Derivators

An introduction to the notions of homotopy limit and colimit is given. It is explained how they can be used to neatly describe the “old” distinguished triangles and shift functors of derived categories resp. cofiber and fiber sequences in algebraic topology. One of the goals is to motivate the language of derivators from the perspective of classical homological algebra. Another one is to give elementary proofs (one brute-force in the exercises, and one a bit more abstract) that in the category of unbounded chain complexes of an (AB4, resp. AB4*) abelian category all homotopy limits (resp. colimits) exist and that this situation leads to a (stable) derivator. The heart of these proofs is an explicit formula for homotopy limits and colimits, the Bousfield-Kan formula. Later it is explained how these results fit in the framework of model categories. We sketch proofs that any model category gives rise to a derivator. We also rediscuss Bousfield-Kan’s formula and outline the proof that it is valid in any simplicial model category (even a slightly weaker structure). In the end the homotopy theory of (homotopy) limits and colimits is discussed. In particular we explain that any derivator is a module over H (the derivator associated with the homotopy theory of spaces). The reader is assumed to have seen some algebraic topology and/or homological algebra (here in particular the construction of abstract derived functors) although we will briefly recall everything. Some knowledge of category theory (limits, colimits, adjoints) is helpful but most facts are listed in an appendix. For the second part it is helpful to be familiar with model categories, but they will be briefly motivated and results presented as a black box. The notes of the fourth talk at the summer school, on fibered derivators, (co)homological descent and Grothendieck’s six functors is to be found in a subsequent document [16]. I would like to thank all participants of the summer school for the nice week, and for their very useful questions, comments and remarks.