Vol. 6, No. 1, 2021

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Generalized stability for abstract homotopy theories

Moritz Rahn and Michael Shulman

Vol. 6 (2021), No. 1, 1–28
Abstract

We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left adjoints. These characterizations generalize to an abstract notion of “stability relative to a class of functors”, which includes in particular pointedness, semiadditivity, and ordinary stability. To prove them, we develop the theory of derivators enriched over monoidal left derivators and weighted homotopy limits and colimits therein.

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Keywords
derivator, stable derivator, stable $\infty$-category, absolute colimit
Mathematical Subject Classification 2010
Primary: 55U35
Milestones
Received: 29 April 2019
Revised: 29 July 2020
Accepted: 17 October 2020
Published: 8 July 2021
Authors
Moritz Rahn
Mathematisches Institut
Johannes Gutenberg-Universität
Mainz
Germany
Michael Shulman
Department of Mathematics
University of San Diego
San Diego, CA
United States