Generalized stability for abstract homotopy theories

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.

This article incorporates most of arXiv:1602.07632

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Cited by in corpus (3)

• Abstract cubical homotopy theory
• Higher symmetries in abstract stable homotopy theories
• The bivariant parasimplicial ${\mathsf{S}}_{\bullet }$-construction