turkmath.org

We first discuss actions of monoidal categories on homotopical categories by a novel approach; namely, via bi-actions that behave like one sided actions. This approach allows us to view equivariant functors between such homotopical categories as fixed points. Then we give a stabilization of a homotopy theory with respect to a suspension action (or a looping action) as a form of category of fixed points of monoidal category actions. The (co)homology theories with exotic gradings appear as particular cases of this stabilization; more precisely, stabilizations of (co)homology functors with actions induced on functor categories. This new approach unifies many of the known genuine stabilizations and exotic gradings of (co)homology theories. Moreover, our point of view presents a new categorical machinery for obtaining axioms for cohomology theories with exotic gradings. This is a joint work with Özgün Ünlü.