Decompositions of the stable module ∞$\infty$‐category

We show that the stable module $\infty$-category of a finite group $G$ decomposes in three different ways as limit $\infty$-categories certain subgroups $G$. Analogously to Dwyer's terminology for homology decompositions, we call these centraliser, normaliser, and subgroup decompositions. construct centraliser normaliser decompositions extend decomposition (constructed by Mathew) more collections subgroups. The key step proof is extending be defined any $G$-space, then showing this extension only depends on $S$-equivariant homotopy type $G$-space. methods used are not specific $\infty$-category, so may also applicable other settings where an functorially