January / February 2017.

Moritz Groth (Bonn),
Lectures on derivators.

Plan:

First lecture: From derived categories to derived Kan extensions
topics: classical triangulations encode a calculus of cones; cones are not functorial at the level of morphisms in derived categories, but on derived categories of morphism categories (as left derived cokernels); emphasize difference between derived categories of morphism categories and morphism categories of derived categories; construction of derived cokernels as composition of derived restriction functors and adjoints; categories of coherent diagrams (derived categories of diagram categories) and categories of incoherent diagrams (diagram categories of derived categories) are related by underlying diagram functors; underlying diagram functors discard relevant information; for example, even over a field the left derived cokernel functor does not factor through corresponding underlying diagram functor
conclusion: In order to enhance derived categories, let us also keep track of derived categories of diagram categories and derived restriction functors.

Second lecture: Limits, derived limits, and homotopy limits
possible topics: limits and colimits in ordinary categories, examples, construction via products and equalizers, final functors, derived limits, construction via deformations, illustration by derived pushouts, cones, and group cohomology, deformations also work in non-additive situations, as an illustration outlook on homotopy limits in topological spaces

Third lecture: Kan extensions
possible topics: limits are adjoints to diagonal functors, diagonal functors as particular restriction functors, Kan extensions are adjoints to more general restriction functors, construction of Kan extensions by pointwise formulas, examples of Kan extensions, fully faithfulness of Kan extensions along fully faithful functors, cokernels via Kan extensions

Fourth lecture: Basics on derivators
possible topics: definition of derivators, state examples from category theory and homological algebra and homotopy theory, sketch proof of construction of derivator of an exact category, limits versus derived/homotopy limits, existence of (co)products (in contrast to more general (co)limits), a few basic lemmas, derivators of representations with values in a given derivator

Fifth lecture: Pointed derivators
possible topics: definition of pointed derivators, examples, basic yoga of pushouts and pullbacks, construction of suspensions and loops, cofibers and fibers, basic properties, cofiber sequences, sketch proof that loop objects are group objects, mention action of loop objects on fibers

Sixth lecture: Stable derivators and canonical triangulations
possible topics: definition of stable derivators, examples, emphasize that stability is invisible to classical category theory, some sanity checks of the notion (detect isos by triviality of cone, derivator version of 5-lemma, suspension and cofibers are equivalences), canonical triangulation on stable derivators (fairly detailed proof)

Seventh lecture: Properties versus structures
possible topics: morphisms of derivators, interaction with colimits, exact morphisms of stable derivators and relation to exact functors of triangulated categories, characterizations of exact morphisms, natural transformations of stable derivators are always exact, characterizations of stability, recall refinements of triangulations: axioms of BBD and Maltsiniotis, sketch construction of canonical higher triangulations, relation to abstract representation theory of Dynkin quivers of type A