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Resumen: Suppose that we are given a collection of endofunctors (e.g., loop space functors) on a pointed homotopy theory $A$. We will call such a homotopy theory stable if it is stable under these functors; that is, each functor is an auto-equivaence. Alternatively, one can consider this as an action of a monoidal category, and in this case stable means the monoidal category acts by auto-equivalences. In this talk, we discuss actions of monoidal categories on relative categories, and applications in stable homotopy theory. Given a monoidal category $I$ and an $I$-relative category $A$ (that is, a relative category with an $I$-action), the (co)stabilization of $A$ is an $I$-relative category that is universal with respect to the property that every object of $I$ acts by auto-equivalences (on homotopy category). We introduce a notion of $I$-equivariance for functors between $I$-relative categories and give constructions of stabilization and costabilization in terms of (weak) ends and coends in a $2$-category of $I$-relative categories and $I$-equivariant relative functors. Several examples existing in the literature, including various categories of spectra and cohomology theories with exotic gradings, can be seen as particular instances of this setting after fixing $A$ and the $I$-action on it. In particular, categories of sequential spectra, coordinate free spectra, genuine equivariant spectra, genuine parameterized spectra (indexed by vector bundles), and cohomology theories with various exotic gradings can be obtained in terms of weak ends. On the other hand, the costabilization of a relative category with respect to an action gives a stable relative category akin to a version of the Spanier-Whitehead category. This is a joint work with Özgün Ünlü.
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