Stabilization and costabilization with respect to an action of a monoidal category

We study actions of monoidal categories on objects in a suitably enriched $2$-category, and applications in stable homotopy theory. Given a monoidal category $\mathcal{I}$ and an $\mathcal{I}$-object $\mathcal{A}$, the (co)stabilization of $\mathcal{A}$ is obtained by universally forcing the $\mathcal{I}$-action to be reversible so that every object of $\mathcal{I}$ acts on $\mathcal{A}$ by auto-equivalences. We introduce a notion of $\mathcal{I}$-equivariance for morphisms between $\mathcal{I}$-objects and give constructions of stabilization and costabilization in terms of weak ends and coends in an enriched $2$-category of $\mathcal{I}$-objects and $\mathcal{I}$-equivariant morphisms. We observe that the stabilization of a relative category with respect to an action coincides with the usual notion of stabilization in stable homotopy theory when the action is defined by loop space functors. We show that several examples that exist in the literature, including various categories of spectra, fit into our setting after fixing $\mathcal{A}$ and the $\mathcal{I}$-action on it. In particular, categories of sequential spectra, coordinate free spectra, genuine equivariant spectra, parameterized spectra indexed by vector bundles are obtained in terms of weak ends in the $2$-category of relative categories. On the other hand, the costabilization of a relative category with respect to an action gives a stable relative category akin to a version Spanier-Whitehead category. In particular, we establish a form of duality between constructions of stable homotopy categories by spectra and by Spanier-Whitehead like categories.