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The fusion orbit category $\overline{\mathcal F _{\mathcal C}} (G)$ of adiscrete group $G$ over a collection $\mathcal C$ is the category whose objectsare the subgroups $H$ in $\mathcal C$, and whose morphisms $H \to K$ are givenby the $G$-maps $G/H \to G/K$ modulo the action of the centralizer group$C_G(H)$. We show that the higher limits over $\overline{\mathcal F _{\mathcalC}} (G)$ can be computed using the hypercohomology spectral sequences comingfrom the Dwyer $G$-spaces for centralizer and normalizer decompositions for$G$.  If $G$ is the discrete group realizing a saturated fusion system $\mathcalF$, then these hypercohomology spectral sequences give two spectral sequencesthat converge to the cohomology of the centric orbit category ${\mathcal O}^c(\mathcal F)$. This allows us to apply our results to the sharpness problem forthe subgroup decomposition of a $p$-local finite group. We prove that thesubgroup decomposition for every $p$-local finite group is sharp (over$\mathcal F$-centric subgroups) if it is sharp for every $p$-local finite groupwith nontrivial center. We also show that for every $p$-local finite group $(S,\mathcal F, \mathcal L)$, the subgroup decomposition is sharp if and only ifthe normalizer decomposition is sharp.

Higher limits over the fusion orbit category