The co-stability manifold of a triangulated category

TL;DR

This paper introduces co-stability conditions as a new framework generalizing co-t-structures, demonstrating that their set forms a manifold, exemplified by the compact derived category of dual numbers having a co-stability manifold equal to the complex plane.

Contribution

It defines co-stability conditions for triangulated categories and proves that their set forms a manifold, extending the theory of stability conditions.

Findings

The co-stability manifold of the compact derived category of dual numbers is the complex plane.

The set of nice co-stability conditions on a triangulated category is a manifold.

Co-stability conditions generalize stability conditions and form a continuous space.

Abstract

Stability conditions on triangulated categories were introduced by Bridgeland as a 'continuous' generalisation of t-structures. The set of locally-finite stability conditions on a triangulated category is a manifold which has been studied intensively. However, there are mainstream triangulated categories whose stability manifold is the empty set. One example is the compact derived category of the dual numbers over an algebraically closed field. This is one of the motivations in this paper for introducing co-stability conditions as a 'continuous' generalisation of co-t-structures. Our main result is that the set of nice co-stability conditions on a triangulated category is a manifold. In particular, we show that the co-stability manifold of the compact derived category of the dual numbers is the complex numbers.