@TimCampion just adding on @PiotrPstrągowski, answer, the analogy is more like Kn modules to Fp modules. Kn local spectra is like the completion around the stratum of height n and you want things which are actually "on it" so you need to take modules I believe to get a somewhat reasonable analogy.

@BrunoStonek I think undercategory projections only create contractible colimits in general, for which see HTT.4.4.2.9.

oh, hmm. I wasn't expecting that. thanks for the reference!

@BrunoStonek in general to compute colimits in $C_{c/}$ for $p:K\to C$ you can take the pushout of $c \leftarrow \mathrm{colim}_K c \rightarrow \mathrm{colim}_K p$; when $K$ is contractible the first map is an equivalence, but not in general. From this formula it's also easy to see that you only need $n\pm \epsilon$ connectivity in an $n$-category

@PiotrPstrągowski @S.carmeli Thanks, I was being really confused. Of course the analogy should have $K(n)$-modules being compared to $H\mathbb F_p$-modules. I guess something I still find funny is that although people talk about $K(n)$-local homotopy theory all the time with $n$ finite, I rarely hear people talk about $H\mathbb F_p$-local homotopy theory. I wonder why that is?

@TimCampion Derived completion doesn't work very well in infinite codimension (i.e. when the open complement of the closed subscheme you're completing at is not q-compact). That said $H\mathbb{F}_p$-completion occasionally shows up (e.g. in the convergence of the Adams SS)