Moritz Rahn (former surname: Groth)

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Book project: The theory of derivators

In this on-going book project (which will be published by Cambridge University Press in the series New Mathematical Monographs) we give a systematic introduction to the basic theory of derivators. Collecting many basic results and techniques in one place, these two volumes prepare the ground for discussions of more advanced topics.

Hopefully sooner than later, close-to-final versions of the various parts constituting these volumes will appear here. These volumes are still under construction, and already at this stage I am very interested in feedback. If you found some typos, inaccuracies or actual mistakes, if you are interested in a more up-to-date version or if you have any other comments, then please do not hesitate to contact me. The first volume is based on unpolished and unfinished (!) lecture notes for this course. An older draft version of this volume is also available here.

  • Volume 1. Assuming basic acquaintance with homological algebra only, in the first volume we give a careful motivation for the notion of a derivator. We discuss the existence of canonical triangulations in stable derivators, indicating that stable derivators provide an enhancement of triangulated categories. This result is taken as a pretext to study various basic constructions in derivators, like iterated (co)fiber constructions, Barratt-Puppe sequences, refined octahedron diagrams and higher versions of these. To illustrate the added flexibility of derivators we also include the basic calculus of total (co)fibers and of cartesian and strongly cartesian n-cubes in the sense of Goodwillie. These calculi rely on a solid basic understanding of the formalism of parametrized Kan extensions as well as on a study of the interaction of limits, Kan extensions, and canonical mates with morphisms of derivators.
    • Abstract, preface, introduction and overview
    • Part 1: Motivation and background (July 2017)
    • Part 2: Basic notions
    • Part 3: The basic calculus
    • Part 4: Pointed derivators
    • Part 5: Stable derivators
    • Appendix
    • Bibliography (July 2017)
  • Volume 2. The second volume discusses the universality of the derivators of spaces, pointed spaces, and spectra, together with some implications for the calculus of homotopy Kan extensions. Here we also develop in detail the basic calculus of monoidal derivators and stable, monoidal derivators. This volume concludes by a discussion of enriched derivators and the calculus of weighted homotopy colimits, thereby providing a framework for the study of universal tilting modules in abstract representation theory. This also allows for conceptional explanations of various characterizations of pointed and stable derivators.