Cohomological finiteness conditions for Mackey and cohomological Mackey functors

We study cohomological finiteness conditions for groups associated to Mackey and cohomological Mackey functors, proving that the cohomological dimension associated to cohomological Mackey functors is always equal to the $\mathcal{F}$-cohomological dimension, and characterising the conditions Mackey-$\mathrm{FP}_n$ and cohomological Mackey-$\mathrm{FP}_n$. We show that all finiteness conditions for cohomological Mackey functors are unchanged when considering only the family of $p$-subgroups, and we characterise cohomological Mackey-$\mathrm{FP}_n$ conditions over the finite field $\mathbb{F}_p$.