Given a(n $\infty$-)category, there is a process called "stabilitazion" which spits out a stable $\infty$-category (as one can read about in either Higher Algebra or the nlab).

The famous example is Spectra, which is the stabilization of Top (and I guess the various enhancements of derived categories, out of categories of modules).

If I start instead with the plain old category of pointed sets, what is its stabilization? And if I start with finite pointed sets?