Generalized stability for abstract homotopy theories

TL;DR

This paper characterizes stability in derivators through the commutation and adjoint properties of homotopy finite limits and colimits, extending to a broader notion of stability relative to classes of functors.

Contribution

It introduces new characterizations of derivator stability and develops the theory of derivators enriched over monoidal left derivators with weighted limits and colimits.

Findings

Homotopy finite limits and colimits commute in stable derivators.

Homotopy finite limit functors have right adjoints in stable derivators.

Homotopy finite colimit functors have left adjoints in stable derivators.

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.