Construction - periodic-fractal categories

Recall from defn - fractal infinity category, a fractal $\infty$-category is a category with finite products, such that the map

$\Omega :C\to Gp(C^{Cart})$ is an equivalence.

We introduce a relaxation of this notion as follows.

Definition An n-periodic fractal $\infty$-category is $C$ such that the map ${\Omega }^n:C\to G{p}_{E_n}(C^{Cart})$ is an equivalence.