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In this paper we develop a theory of stability for $G$-categories (presheafof categories on the orbit category of $G$), where $G$ is a finite group. Wegive a description of Mackey functors as $G$-commutative monoids exploit it tocharacterize $G$-spectra as the $G$-stabilization of $G$-spaces. As anapplication of this we provide an alternative proof of a theorem by Guillou andMay. The theory here is developed in the more general setting of orbitalcategories.

Parametrized higher category theory and higher algebra: Expos\'e IV -- Stability with respect to an orbital $\infty$-category