Given a triangulated category \({\mathcal {D}}\) with an action of a fusion category \({\mathcal {C}}\) , we study the moduli space \({{\,\textrm{Stab}\,}}_{{\mathcal {C}}}({\mathcal {D}})\) of fusion-equivariant Bridgeland stability conditions on \({\mathcal {D}}\) . The main theorem is that the fusion-equivariant stability conditions form a closed, complex submanifold of the moduli space of stability conditions on \({\mathcal {D}}\) . As an application of this framework to finite group actions on categories, we generalise a result of Macrì–Mehrotra–Stellari by establishing a biholomorphism between the space of G-invariant stability conditions on \({\mathcal {D}}\) and the space of \({\textsf{rep}}(G)\) -equivariant stability conditions on the equivariant category \({\mathcal {D}}^G\) . We also describe applications to the study of stability conditions associated to McKay quivers and to geometric stability conditions on free quotients of smooth projective varieties.
